Момент на инерција — Википедија

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</nav> </div> </header> <main id="content" class="mw-body"> <div class="banner-container"> <div id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading">Момент на инерција</h1> <div class="tagline"></div> </div> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"><a role="button" href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#p-lang" data-mw="interface" data-event-name="menu.languages" title="Јазик" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> Јазик </a></li> <li id="page-actions-watch" class="page-actions-menu__list-item"><a 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cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> Уреди </a></li> </ul> </nav> <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="mk" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> Моментот на инерција, инаку познат како аголна маса или вртежна инерција, на <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A6%D0%B2%D1%80%D1%81%D1%82%D0%B0_%D1%81%D0%BE%D1%81%D1%82%D0%BE%D1%98%D0%B1%D0%B0_%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D1%80%D0%B8%D1%98%D0%B0%D1%82%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Цврста состојба на материјата">цврсто тело</a> е тензор кој го одредува <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D1%81%D0%B8%D0%BB%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Момент на сила">моментот на силата</a> потребен за саканото <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%90%D0%B3%D0%BE%D0%BB%D0%BD%D0%BE_%D0%B7%D0%B0%D0%B1%D1%80%D0%B7%D1%83%D0%B2%D0%B0%D1%9A%D0%B5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Аголно забрзување">аголно забрзување</a> околу вртежната оска; слично како <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%B0%D1%81%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-disambig" title="Маса">масата</a> ја одредува <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%B8%D0%BB%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Сила">силата</a> потребна за посакуваното <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%97%D0%B0%D0%B1%D1%80%D0%B7%D1%83%D0%B2%D0%B0%D1%9A%D0%B5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Забрзување">забрзување</a>. Тоа зависи од распределбата на масата на телото и одбраната оска, со поголеми моменти кои бараат поголем вртежен момент за промена на стапката на ротација на телото. Тоа е екстензивно (адитивно) својство: за точка маса маса на инерција е само маса на квадрат на нормално растојание до оската на вртење. Моментот на инерција на композитниот систем на кругот е збирот на моментите на инерција на нејзините составни потсистеми (сите земени за истата оска). Неговата наједноставна дефиниција е вториот <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Момент (математика)">момент</a> на маса во однос на растојанието од <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%92%D1%80%D1%82%D0%B5%D1%9A%D0%B5_%D0%BE%D0%BA%D0%BE%D0%BB%D1%83_%D0%BD%D0%B5%D0%BF%D0%BE%D0%B4%D0%B2%D0%B8%D0%B6%D0%BD%D0%B0_%D0%BE%D1%81%D0%BA%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Вртење околу неподвижна оска">оската</a>. За ограничените тела да се ротираат во рамнина, важно е само нивниот момент на инерција околу оската нормална на рамнината, скаларната вредност. За телата кои слободно можат да ротираат во три димензии, нивните моменти може да се опишат со симетрична 3 × 3 <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Матрица (математика)">матрица</a>, со множество на меѓусебно нормални главни оски за кои оваа матрица е <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%94%D0%B8%D1%98%D0%B0%D0%B3%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Дијагонална матрица (страницата не постои)">дијагонална</a>, а вртежите околу оските делуваат независно еден од друг. <table class="infobox" style="width:22em"> <tbody> <tr> <th colspan="2" style="text-align:center;font-size:125%;font-weight:bold;font-style:italic;">Момент на инерција</th> </tr> <tr> <td colspan="2" style="text-align:center"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:%D0%9C%D0%B0%D1%85%D0%BE%D0%B2%D0%B8%D0%BA.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/%D0%9C%D0%B0%D1%85%D0%BE%D0%B2%D0%B8%D0%BA.jpg/220px-%D0%9C%D0%B0%D1%85%D0%BE%D0%B2%D0%B8%D0%BA.jpg" decoding="async" width="220" height="293" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/d/da/%25D0%259C%25D0%25B0%25D1%2585%25D0%25BE%25D0%25B2%25D0%25B8%25D0%25BA.jpg/330px-%25D0%259C%25D0%25B0%25D1%2585%25D0%25BE%25D0%25B2%25D0%25B8%25D0%25BA.jpg 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/d/da/%25D0%259C%25D0%25B0%25D1%2585%25D0%25BE%25D0%25B2%25D0%25B8%25D0%25BA.jpg/440px-%25D0%259C%25D0%25B0%25D1%2585%25D0%25BE%25D0%25B2%25D0%25B8%25D0%25BA.jpg 2x" data-file-width="2736" data-file-height="3648"></a> <div> <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%97%D0%B0%D0%BC%D0%B0%D0%B5%D1%86&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Замаец (страницата не постои)">Замајците</a> имаат големи моменти на инерција за да го ублажат механичкото движење. Овој пример е во рускиот музеј. </div></td> </tr> <tr> <th scope="row"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"> Симболи </div></th> <td>I</td> </tr> <tr> <th scope="row"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%B5%D1%93%D1%83%D0%BD%D0%B0%D1%80%D0%BE%D0%B4%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC_%D0%BD%D0%B0_%D0%BC%D0%B5%D1%80%D0%BD%D0%B8_%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D0%B8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Меѓународен систем на мерни единици">SI-единица</a></th> <td>kg m2</td> </tr> <tr> <th scope="row"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"> Други единици </div></th> <td>lbf·ft·s2</td> </tr> <tr> <th scope="row"><a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%94%D0%B8%D0%BC%D0%B5%D0%BD%D0%B7%D0%B8%D0%BE%D0%BD%D0%B0%D0%BB%D0%BD%D0%B0_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Димензионална анализа (страницата не постои)">Димензија</a></th> <td>M L2</td> </tr> <tr> <th scope="row"><a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%95%D0%BA%D1%81%D1%82%D0%B5%D0%BD%D0%B7%D0%B8%D0%B2%D0%BD%D0%B0_%D0%B2%D0%B5%D0%BB%D0%B8%D1%87%D0%B8%D0%BD%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Екстензивна величина (страницата не постои)">Екстензивна</a>?</th> <td>да</td> </tr> <tr> <th scope="row"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"> Изведенки од други величини </div></th> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I={\frac {L}{\omega }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> I </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> L </mi> <mi> ω </mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I={\frac {L}{\omega }}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a21fa504541c8dd9a46cddd2ad2b0d657b53b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.689ex; height:5.176ex;" alt="{\displaystyle I={\frac {L}{\omega }}}"></td> </tr> </tbody> </table> <style data-mw-deduplicate="TemplateStyles:r4539682">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:#f8f9fa;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar a{white-space:nowrap}.mw-parser-output .sidebar-wraplinks a{white-space:normal}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding-bottom:0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em 0}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding-top:0.2em;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding-top:0.4em;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.4em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding-top:0}.mw-parser-output .sidebar-image{padding:0.2em 0 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em}.mw-parser-output .sidebar-content{padding:0 0.1em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.4em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%}.mw-parser-output .sidebar-collapse .sidebar-navbar{padding-top:0.6em}.mw-parser-output .sidebar-list-title{text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{text-align:center;margin:0 3.3em}@media(max-width:720px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}</style> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Samuel_Dixon_Niagara.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Samuel_Dixon_Niagara.jpg/220px-Samuel_Dixon_Niagara.jpg" decoding="async" width="220" height="197" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Samuel_Dixon_Niagara.jpg/330px-Samuel_Dixon_Niagara.jpg 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Samuel_Dixon_Niagara.jpg/440px-Samuel_Dixon_Niagara.jpg 2x" data-file-width="461" data-file-height="412"></a> <figcaption> Пешаци на јаже го користат моментот на инерција на долги прачки за рамнотежа додека одат по јажето. Семјуел Диксон ја преминува <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9D%D0%B8%D1%98%D0%B0%D0%B3%D0%B0%D1%80%D0%B8%D0%BD%D0%B8_%D0%92%D0%BE%D0%B4%D0%BE%D0%BF%D0%B0%D0%B4%D0%B8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Нијагарини Водопади">реката Нијагара</a> во 1890 година. </figcaption> </figure> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="mk" dir="ltr"> <h2 id="mw-toc-heading">Содржина</h2><label class="toctogglelabel" for="toctogglecheckbox"></label> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%92%D0%BE%D0%B2%D0%B5%D0%B4">1 Вовед</a></li> <li class="toclevel-1 tocsection-2"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%94%D0%B5%D1%84%D0%B8%D0%BD%D0%B8%D1%86%D0%B8%D1%98%D0%B0">2 Дефиниција</a></li> <li class="toclevel-1 tocsection-3"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%9F%D1%80%D0%B8%D0%BC%D0%B5%D1%80%D0%B8">3 Примери</a> <ul> <li class="toclevel-2 tocsection-4"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%95%D0%B4%D0%BD%D0%BE%D1%81%D1%82%D0%B0%D0%B2%D0%BD%D0%BE_%D0%BD%D0%B8%D1%88%D0%B0%D0%BB%D0%BE">3.1 Едноставно нишало</a></li> <li class="toclevel-2 tocsection-5"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%A4%D0%B8%D0%B7%D0%B8%D1%87%D0%BA%D0%BE_%D0%BD%D0%B8%D1%88%D0%B0%D0%BB%D0%BE">3.2 Физичко нишало</a> <ul> <li class="toclevel-3 tocsection-6"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%A6%D0%B5%D0%BD%D1%82%D0%B0%D1%80_%D0%BD%D0%B0_%D0%BE%D1%81%D1%86%D0%B8%D0%BB%D0%B0%D1%86%D0%B8%D1%98%D0%B0">3.2.1 Центар на осцилација</a></li> </ul></li> </ul></li> <li class="toclevel-1 tocsection-7"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%9C%D0%B5%D1%80%D0%B5%D1%9A%D0%B5_%D0%BD%D0%B0_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D0%BE%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0">4 Мерење на моментот на инерција</a></li> <li class="toclevel-1 tocsection-8"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%94%D0%B2%D0%B8%D0%B6%D0%B5%D1%9A%D0%B5_%D0%B2%D0%BE_%D1%84%D0%B8%D0%BA%D1%81%D0%BD%D0%B0_%D1%80%D0%B0%D0%BC%D0%BD%D0%B8%D0%BD%D0%B0">5 Движење во фиксна рамнина</a> <ul> <li class="toclevel-2 tocsection-9"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%A2%D0%BE%D1%87%D0%BA%D0%B5%D1%81%D1%82%D0%B0_%D0%BC%D0%B0%D1%81%D0%B0">5.1 Точкеста маса</a> <ul> <li class="toclevel-3 tocsection-10"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%9F%D1%80%D0%B8%D0%BC%D0%B5%D1%80%D0%B8_2">5.1.1 Примери</a></li> </ul></li> <li class="toclevel-2 tocsection-11"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%A6%D0%B2%D1%80%D1%81%D1%82%D0%BE_%D1%82%D0%B5%D0%BB%D0%BE">5.2 Цврсто тело</a> <ul> <li class="toclevel-3 tocsection-12"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%90%D0%B3%D0%BE%D0%BB%D0%B5%D0%BD_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%81">5.2.1 Аголен импулс</a></li> <li class="toclevel-3 tocsection-13"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%98%D0%B0">5.2.2 Кинетичка енергија</a></li> <li class="toclevel-3 tocsection-14"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%8A%D1%83%D1%82%D0%BD%D0%BE%D0%B2%D0%B8_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD%D0%B8">5.2.3 Њутнови закони</a></li> </ul></li> </ul></li> <li class="toclevel-1 tocsection-15"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%94%D0%B2%D0%B8%D0%B6%D0%B5%D1%9A%D0%B5_%D0%B2%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80%D0%BE%D1%82_%D0%BD%D0%B0_%D1%86%D0%B2%D1%80%D1%81%D1%82%D0%BE_%D1%82%D0%B5%D0%BB%D0%BE_%D0%B8_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0">6 Движење во просторот на цврсто тело и инерцијална матрица</a> <ul> <li class="toclevel-2 tocsection-16"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%81%D0%BE%D1%82">6.1 Момент на импулсот</a></li> <li class="toclevel-2 tocsection-17"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%98%D0%B0_2">6.2 Кинетичка енергија</a></li> <li class="toclevel-2 tocsection-18"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%A0%D0%B5%D0%B7%D1%83%D0%BB%D1%82%D0%B0%D0%BD%D1%82%D0%B5%D0%BD_%D0%B2%D1%80%D1%82%D0%B5%D0%B6%D0%B5%D0%BD_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82">6.3 Резултантен вртежен момент</a></li> <li class="toclevel-2 tocsection-19"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BD%D0%B0_%D0%BF%D0%B0%D1%80%D0%B0%D0%BB%D0%B5%D0%BB%D0%BD%D0%B0_%D0%BE%D1%81%D0%BA%D0%B0">6.4 Теорема на паралелна оска</a></li> <li class="toclevel-2 tocsection-20"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%A1%D0%BA%D0%B0%D0%BB%D0%B0%D1%80%D0%B5%D0%BD_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0_%D0%B2%D0%BE_%D1%80%D0%B0%D0%BC%D0%BD%D0%B8%D0%BD%D0%B0">6.5 Скаларен момент на инерција во рамнина</a></li> </ul></li> <li class="toclevel-1 tocsection-21"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%98%D0%BD%D0%B5%D1%80%D1%82%D0%B5%D0%BD_%D1%82%D0%B5%D0%BD%D0%B7%D0%BE%D1%80">7 Инертен тензор</a></li> <li class="toclevel-1 tocsection-22"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%98%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_%D0%B2%D0%BE_%D1%80%D0%B0%D0%B7%D0%BB%D0%B8%D1%87%D0%BD%D0%B8_%D0%BF%D0%BE%D1%98%D0%B4%D0%BE%D0%B2%D0%BD%D0%B8_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8">8 Инерцијална матрица во различни појдовни системи</a> <ul> <li class="toclevel-2 tocsection-23"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%A0%D0%B0%D0%BC%D0%BA%D0%B0_%D0%BD%D0%B0_%D1%82%D0%B5%D0%BB%D0%BE%D1%82%D0%BE">8.1 Рамка на телото</a></li> <li class="toclevel-2 tocsection-24"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%93%D0%BB%D0%B0%D0%B2%D0%BD%D0%B8_%D0%BE%D1%81%D0%BA%D0%B8">8.2 Главни оски</a></li> <li class="toclevel-2 tocsection-25"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%95%D0%BB%D0%B8%D0%BF%D1%81%D0%BE%D0%B8%D0%B4">8.3 Елипсоид</a></li> </ul></li> <li class="toclevel-1 tocsection-26"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%9F%D0%BE%D0%B2%D1%80%D0%B7%D0%B0%D0%BD%D0%BE">9 Поврзано</a></li> <li class="toclevel-1 tocsection-27"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%9D%D0%B0%D0%B2%D0%BE%D0%B4%D0%B8">10 Наводи</a></li> <li class="toclevel-1 tocsection-28"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%9D%D0%B0%D0%B4%D0%B2%D0%BE%D1%80%D0%B5%D1%88%D0%BD%D0%B8_%D0%B2%D1%80%D1%81%D0%BA%D0%B8">11 Надворешни врски</a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <h2 id="Вовед">Вовед</h2> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Вовед“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> Кога телото може слободно да ротира околу оска, мора да се примени <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D1%81%D0%B8%D0%BB%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Момент на сила">вртежен момент</a> за да се смени аголниот момент. Количината на вртежен момент што е потребна за да предизвика било кое <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%90%D0%B3%D0%BE%D0%BB%D0%BD%D0%BE_%D0%B7%D0%B0%D0%B1%D1%80%D0%B7%D1%83%D0%B2%D0%B0%D1%9A%D0%B5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Аголно забрзување">аголно забрзување</a> (стапката на промена на <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%90%D0%B3%D0%BE%D0%BB%D0%BD%D0%B0_%D0%B1%D1%80%D0%B7%D0%B8%D0%BD%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Аголна брзина">аголната брзина</a>) е пропорционална на моментот на инерција на телото. Момент на инерција може да се изрази во единици од килограм метар квадрат (kg·m2) во единици на <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%B5%D1%93%D1%83%D0%BD%D0%B0%D1%80%D0%BE%D0%B4%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC_%D0%BD%D0%B0_%D0%BC%D0%B5%D1%80%D0%BD%D0%B8_%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D0%B8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Меѓународен систем на мерни единици">SI</a> и квадратни (квадратни-секунди) квадратни (lbf · ft · s2) во <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%98%D0%BC%D0%BF%D0%B5%D1%80%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B8_%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D0%B8&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Империјални единици (страницата не постои)">империјални</a> или <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%90%D0%BC%D0%B5%D1%80%D0%B8%D0%BA%D0%B0%D0%BD%D1%81%D0%BA%D0%B8_%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D0%B8&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Американски единици (страницата не постои)">американски</a> единици. Моментот на инерција ја игра улогата во вртежната кинетика што <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%B0%D1%81%D0%B0_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Маса (физика)">масата</a> (инерција) игра во линеарна кинетика - и го карактеризираат отпорноста на телото на промени во движењето. Моментот на инерција зависи од тоа колку масата се дистрибуира околу оската на вртење и ќе варира во зависност од избраната оска. За маса-како маса, моментот на инерција околу некоја оска е даден со <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mr^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle mr^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd9d0ea2911509b014b72a7b536acb7376cb455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.143ex; height:2.676ex;" alt="{\displaystyle mr^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.143ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd9d0ea2911509b014b72a7b536acb7376cb455" data-alt="{\displaystyle mr^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , каде што r е растојанието од точката од оската, и m е масата. За проширено цврсто тело, моментот на инерција е само збир на сите мали парчиња маса помножени со квадратот на нивните растојанија од оската за која станува збор. За проширено тело со редовна форма и еднаква густина, оваа збирка понекогаш произведува едноставен израз кој зависи од димензиите, обликот и вкупната маса на објектот. Во 1673 година, <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9A%D1%80%D0%B8%D1%81%D1%82%D0%B8%D1%98%D0%B0%D0%BD_%D0%A5%D0%B0%D1%98%D0%B3%D0%B5%D0%BD%D1%81?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Кристијан Хајгенс">Кристијан Хајгенс</a> го претстави овој параметар во неговата студија за осцилацијата на едно тело висечко од столб, познато како соединето <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9D%D0%B8%D1%88%D0%B0%D0%BB%D0%BE&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Нишало (страницата не постои)">нишало</a>.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-mach-1">[1]</a> Терминот момент на инерција бил воведен од <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9B%D0%B5%D0%BE%D0%BD%D0%B0%D1%80%D0%B4_%D0%9E%D1%98%D0%BB%D0%B5%D1%80?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Леонард Ојлер">Леонард Ојлер</a> во неговата книга Theoria motus corporum solidumum seu rigidorum во 1765,<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-mach-1">[1]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Euler1730-2">[2]</a> и е вграден во <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9E%D1%98%D0%BB%D0%B5%D1%80%D0%BE%D0%B2%D0%B8_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD%D0%B8_%D0%B7%D0%B0_%D0%B4%D0%B2%D0%B8%D0%B6%D0%B5%D1%9A%D0%B5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ојлерови закони за движење">вториот закон на Ојлер</a>. Природната честота на осцилација на сложеното нишало се добива од односот на вртежниот момент наметнат од гравитацијата врз масата на нишалото до отпорноста на забрзувањето дефинирана со моментот на инерција. Споредба на оваа природна честота со онаа на едноставното нишало кое се состои од една единствена точка на маса обезбедува математичка формулација за момент на инерција на проширено тело.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Marion_1995-3">[3]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Symon_1971-4">[4]</a> Моментот на инерција, исто така, се појавува во <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%81%D0%BE%D1%82?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Момент на импулсот">моментот на импулсот</a>, <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9A%D0%B8%D0%BD%D0%B5%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Кинетичка енергија">кинетичката енергија</a> и во <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%8A%D1%83%D1%82%D0%BD%D0%BE%D0%B2%D0%B8_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD%D0%B8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Њутнови закони">Њутновите закони</a> на движење за круто тело како физички параметар кој ги комбинира својата форма и маса. Постои интересна разлика во начинот на кој моментот на инерција се појавува во рамни и просторно движење. Плодното движење има еден скалар кој го дефинира моментот на инерција, додека за просторно движење истите пресметки даваат матрица од 3 × 3 моменти на инерција, наречена инерцијална матрица или инертен тензор.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Tenenbaum_2004-5">[5]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kane-6">[6]</a> Моментот на инерција на вртечкиот замаец се користи во машина за да се спротивстави на промените во применетиот вртежен момент за да се изедначи вртежното производство. Моментот на инерција на авионот околу неговата надолжна, хоризонтална и вертикална оска определува како управувачките сили на контролните површини на нејзините крила, лифтови и опашка влијаат на рамнината во тркалање, висина и виткање. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <h2 id="Дефиниција">Дефиниција</h2> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Дефиниција“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Cup_of_Russia_2010_-_Yuko_Kawaguti_(2).jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg/170px-Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg" decoding="async" width="170" height="277" class="mw-file-element" data-file-width="307" data-file-height="500"> </noscript><span class="lazy-image-placeholder" style="width: 170px;height: 277px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg/170px-Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg" data-width="170" data-height="277" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg/255px-Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/6/68/Cup_of_Russia_2010_-_Yuko_Kawaguti_%282%29.jpg 2x" data-class="mw-file-element"> </a> <figcaption> Уметничките лизгачи може да го намалат моментот на инерција со повлекување на нивните раце, што им овозможува да се вртат побрзо поради <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%81%D0%BE%D1%82?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Момент на импулсот">зачувувањето на импулсот</a>. </figcaption> </figure> <figure class="mw-default-size mw-halign-right" typeof="mw:Error mw:File/Thumb"> <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%BE%D0%B4%D0%B8%D0%B3%D0%B0%D1%9A%D0%B5&wpDestFile=25._%D0%92%D1%80%D1%82%D0%B5%D0%B6%D0%B5%D0%BD_%D1%81%D1%82%D0%BE%D0%BB.ogv&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Податотека:25. Вртежен стол.ogv">Податотека:25. Вртежен стол.ogv</a> <figcaption> Видео со вртежен стол, на кое е прикажано зачувувањето на моментот на импулсот. Кога професорот кој се врти ги вовлекува рацете , неговиот момент на импулсот се намалува, а за да се запази аголниот момент на импулсот, неговата аголна брзина се зголемува. </figcaption> </figure> Моментот на инерција <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\displaystyle I}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> I </mi> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\displaystyle I}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66932f38fb3374fddca73712ac432cff074352e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle {\displaystyle I}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.172ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66932f38fb3374fddca73712ac432cff074352e4" data-alt="{\displaystyle {\displaystyle I}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е дефиниран како однос на <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BC%D0%BF%D1%83%D0%BB%D1%81%D0%BE%D1%82?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Момент на импулсот">моментот на импулсот</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> </noscript><span class="lazy-image-placeholder" style="width: 1.583ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" data-alt="{\displaystyle L}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на системот на неговата <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%90%D0%B3%D0%BE%D0%BB%D0%BD%D0%B0_%D0%B1%D1%80%D0%B7%D0%B8%D0%BD%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Аголна брзина">аголна брзина</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ω </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \omega } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> </noscript><span class="lazy-image-placeholder" style="width: 1.446ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" data-alt="{\displaystyle \omega }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  околу главната оска,<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Winn-7">[7]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Fullerton-8">[8]</a> што е: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\displaystyle I={\frac {L}{\omega }}.}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> I </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> L </mi> <mi> ω </mi> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\displaystyle I={\frac {L}{\omega }}.}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4062f05985241f507c1bf1c94adbb9246bd50d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.336ex; height:5.176ex;" alt="{\displaystyle {\displaystyle I={\frac {L}{\omega }}.}}"> </noscript><span class="lazy-image-placeholder" style="width: 7.336ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4062f05985241f507c1bf1c94adbb9246bd50d25" data-alt="{\displaystyle {\displaystyle I={\frac {L}{\omega }}.}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Ако аголниот моментум на системот е константен, тогаш кога моментот на инерција станува помал, аголната брзина мора да се зголеми. Ова се случува кога вртењето на лизгачите се повлекува во нивните раширени раце или <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9D%D1%83%D1%80%D0%BA%D0%B0%D1%9A%D0%B5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Нуркање">нуркачите</a> ги закопуваат нивните тела во <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9D%D1%83%D1%80%D0%BA%D0%B0%D1%9A%D0%B5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Нуркање">позиција за навивање</a> за време на нуркање, за да се вртат побрзо.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Winn-7">[7]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Fullerton-8">[8]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Wolfram-9">[9]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Hokin-10">[10]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Breithaupt-11">[11]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Crowell-12">[12]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Tipler-13">[13]</a> Ако обликот на телото не се промени, тогаш неговиот момент на инерција се појавува во <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%8A%D1%83%D1%82%D0%BD%D0%BE%D0%B2%D0%B8_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD%D0%B8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Њутнови закони">Њутновиот закон на движење</a> како однос на применетиот <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D1%81%D0%B8%D0%BB%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Момент на сила">вртежен момент</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> τ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \tau } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"> </noscript><span class="lazy-image-placeholder" style="width: 1.202ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" data-alt="{\displaystyle \tau }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на телото до аголното забрзување <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> α </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \alpha } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> </noscript><span class="lazy-image-placeholder" style="width: 1.488ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" data-alt="{\displaystyle \alpha }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  околу главната оска, што е <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =I\alpha }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> τ </mi> <mo> = </mo> <mi> I </mi> <mi> α </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \tau =I\alpha } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01a662a7ad75feca822305e985c0d19c277b0d55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.96ex; height:2.176ex;" alt="{\displaystyle \tau =I\alpha }"> </noscript><span class="lazy-image-placeholder" style="width: 6.96ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01a662a7ad75feca822305e985c0d19c277b0d55" data-alt="{\displaystyle \tau =I\alpha }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . За <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9D%D0%B8%D1%88%D0%B0%D0%BB%D0%BE&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Нишало (страницата не постои)">едноставно нишало</a>, оваа дефиниција дава формула за моментот на инерција <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> I </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> </noscript><span class="lazy-image-placeholder" style="width: 1.172ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" data-alt="{\displaystyle I}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  во однос на масата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> </noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" data-alt="{\displaystyle m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на матичниот агол и нејзиното растојание <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  од клучната точка како, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=mr^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> I </mi> <mo> = </mo> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I=mr^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc416bcccf5f63d22af13befd74e5c653ab82f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.413ex; height:2.676ex;" alt="{\displaystyle I=mr^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.413ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc416bcccf5f63d22af13befd74e5c653ab82f1b" data-alt="{\displaystyle I=mr^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Така, моментот на инерција зависи од масата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> </noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" data-alt="{\displaystyle m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на телото и неговата геометрија или форма, како што е дефинирано со растојанието <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на оската на вртење. Оваа едноставна формула е генерализирана за да го дефинира моментот на инерција на произволно обликуваното тело како збир на сите елементарни точки маси <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dm}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle dm} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85b992fcc988fb290f5734f9a67def09cff8550c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.256ex; height:2.176ex;" alt="{\displaystyle dm}"> </noscript><span class="lazy-image-placeholder" style="width: 3.256ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85b992fcc988fb290f5734f9a67def09cff8550c" data-alt="{\displaystyle dm}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  секој помножен со квадратот на нејзиното нормално растојание <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на оската <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> </noscript><span class="lazy-image-placeholder" style="width: 1.211ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" data-alt="{\displaystyle k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Општо земено, со оглед на објект на маса <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> </noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" data-alt="{\displaystyle m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , делотворен полупречник <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> </noscript><span class="lazy-image-placeholder" style="width: 1.211ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" data-alt="{\displaystyle k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  може да се дефинира за оската преку нејзиниот центар на маса, со таква вредност дека нејзиниот момент на инерција е <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=mk^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> I </mi> <mo> = </mo> <mi> m </mi> <msup> <mi> k </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I=mk^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df60f37cce2762e866b4ffa67cfa527158c6b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.576ex; height:2.676ex;" alt="{\displaystyle I=mk^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.576ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df60f37cce2762e866b4ffa67cfa527158c6b2f" data-alt="{\displaystyle I=mk^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , каде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> </noscript><span class="lazy-image-placeholder" style="width: 1.211ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" data-alt="{\displaystyle k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е познат како <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%96%D0%B8%D1%80%D0%BE%D1%81%D0%BA%D0%BE%D0%BF%D1%81%D0%BA%D0%B8_%D0%BF%D0%BE%D0%BB%D1%83%D0%BF%D1%80%D0%B5%D1%87%D0%BD%D0%B8%D0%BA&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Жироскопски полупречник (страницата не постои)">жироскопски полупречник</a>. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <h2 id="Примери">Примери</h2> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Примери“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <div class="mw-heading mw-heading3"> <h3 id="Едноставно_нишало">Едноставно нишало</h3> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Едноставно нишало“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> Моментот на инерција може да се мери со едноставно нишало, бидејќи тоа е отпорност на ротација предизвикана од гравитацијата. Математички, моментот на инерција на нишалото е односот на вртежниот момент кој се должи на гравитацијата околу вртењето на нишалото до неговото аголно забрзување околу таа точка на вртење. За едноставно нишало, ова е резултат на производ на масата на честичката <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> </noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" data-alt="{\displaystyle m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  со квадратот на растојанието <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на клучот, што е <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=mr^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> I </mi> <mo> = </mo> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I=mr^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc416bcccf5f63d22af13befd74e5c653ab82f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.413ex; height:2.676ex;" alt="{\displaystyle I=mr^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.413ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc416bcccf5f63d22af13befd74e5c653ab82f1b" data-alt="{\displaystyle I=mr^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Ова може да се прикаже на следниов начин: Силата на гравитацијата на масата на едноставното нишало генерира вртежен момент <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =r\times F}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> τ </mi> <mo> = </mo> <mi> r </mi> <mo> × </mo> <mi> F </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \tau =r\times F} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddddfe5ab1252efa4d4faab3b64ae8a0b9be5659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.93ex; height:2.176ex;" alt="{\displaystyle \tau =r\times F}"> </noscript><span class="lazy-image-placeholder" style="width: 9.93ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddddfe5ab1252efa4d4faab3b64ae8a0b9be5659" data-alt="{\displaystyle \tau =r\times F}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  околу оската нормална на рамнината на движењето на нишалото. Тука <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е векторот на растојанието, нормалниот кон и од силата на оската на вртежниот момент, и <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> </noscript><span class="lazy-image-placeholder" style="width: 1.741ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" data-alt="{\displaystyle F}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  сила на масата. Поврзан со овој вртежен момент е <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%90%D0%B3%D0%BE%D0%BB%D0%BD%D0%BE_%D0%B7%D0%B0%D0%B1%D1%80%D0%B7%D1%83%D0%B2%D0%B0%D1%9A%D0%B5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Аголно забрзување">аголно забрзување</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> α </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \alpha } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> </noscript><span class="lazy-image-placeholder" style="width: 1.488ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" data-alt="{\displaystyle \alpha }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , на низата и масата околу оваа оска. Бидејќи масата е ограничена на круг, тангентното забрзување на масата е <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=\alpha \times r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> <mo> = </mo> <mi> α </mi> <mo> × </mo> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a=\alpha \times r} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/750fc97deb372d956fbff2a1a5a300fc98af64ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.705ex; height:1.676ex;" alt="{\displaystyle a=\alpha \times r}"> </noscript><span class="lazy-image-placeholder" style="width: 9.705ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/750fc97deb372d956fbff2a1a5a300fc98af64ef" data-alt="{\displaystyle a=\alpha \times r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Бидејќи <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=ma}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo> = </mo> <mi> m </mi> <mi> a </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F=ma} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca4e42b7d6d66f52294364928cb5f7c590f514c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.109ex; height:2.176ex;" alt="{\displaystyle F=ma}"> </noscript><span class="lazy-image-placeholder" style="width: 8.109ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca4e42b7d6d66f52294364928cb5f7c590f514c" data-alt="{\displaystyle F=ma}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =r\times F=r\times (m\alpha \times r)=m((r\cdot r)\alpha -(r\cdot \alpha )r)=mr^{2}\alpha =I\alpha k,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> τ </mi> <mo> = </mo> <mi> r </mi> <mo> × </mo> <mi> F </mi> <mo> = </mo> <mi> r </mi> <mo> × </mo> <mo stretchy="false"> ( </mo> <mi> m </mi> <mi> α </mi> <mo> × </mo> <mi> r </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> m </mi> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mi> r </mi> <mo> ⋅ </mo> <mi> r </mi> <mo stretchy="false"> ) </mo> <mi> α </mi> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi> r </mi> <mo> ⋅ </mo> <mi> α </mi> <mo stretchy="false"> ) </mo> <mi> r </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> α </mi> <mo> = </mo> <mi> I </mi> <mi> α </mi> <mi> k </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \tau =r\times F=r\times (m\alpha \times r)=m((r\cdot r)\alpha -(r\cdot \alpha )r)=mr^{2}\alpha =I\alpha k,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdbd93fa710d281f15806d544874a4293ed00373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:66.425ex; height:3.176ex;" alt="{\displaystyle \tau =r\times F=r\times (m\alpha \times r)=m((r\cdot r)\alpha -(r\cdot \alpha )r)=mr^{2}\alpha =I\alpha k,}"> </noscript><span class="lazy-image-placeholder" style="width: 66.425ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdbd93fa710d281f15806d544874a4293ed00373" data-alt="{\displaystyle \tau =r\times F=r\times (m\alpha \times r)=m((r\cdot r)\alpha -(r\cdot \alpha )r)=mr^{2}\alpha =I\alpha k,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> </noscript><span class="lazy-image-placeholder" style="width: 1.211ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" data-alt="{\displaystyle k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е единичен вектор нормален на рамнината на нишалото. (Вториот до последен чекор ја користи <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A2%D1%80%D0%BE%D0%B5%D0%BD_%D0%BF%D1%80%D0%BE%D0%B4%D1%83%D0%BA%D1%82&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Троен продукт (страницата не постои)">векторското тројно проширување</a> на производот со нормалност на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> α </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \alpha } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> </noscript><span class="lazy-image-placeholder" style="width: 1.488ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" data-alt="{\displaystyle \alpha }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) Количината <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=mr^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> I </mi> <mo> = </mo> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I=mr^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc416bcccf5f63d22af13befd74e5c653ab82f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.413ex; height:2.676ex;" alt="{\displaystyle I=mr^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.413ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc416bcccf5f63d22af13befd74e5c653ab82f1b" data-alt="{\displaystyle I=mr^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е момент на инерција на оваа единствена маса околу точка на вртење. Количината <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=mr^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> I </mi> <mo> = </mo> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I=mr^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc416bcccf5f63d22af13befd74e5c653ab82f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.413ex; height:2.676ex;" alt="{\displaystyle I=mr^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.413ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc416bcccf5f63d22af13befd74e5c653ab82f1b" data-alt="{\displaystyle I=mr^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  се појавува и во <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%90%D0%B3%D0%BE%D0%BB%D0%B5%D0%BD_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Аголен момент">аголниот момент</a> на едноставно нишало, кое се пресметува од брзината <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=\omega \times r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> <mo> = </mo> <mi> ω </mi> <mo> × </mo> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v=\omega \times r} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/427de270f2b38c63ec43a3961df1a27c2df9d02c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.561ex; height:1.676ex;" alt="{\displaystyle v=\omega \times r}"> </noscript><span class="lazy-image-placeholder" style="width: 9.561ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/427de270f2b38c63ec43a3961df1a27c2df9d02c" data-alt="{\displaystyle v=\omega \times r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на масата на нишалото околу оската, каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ω </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \omega } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> </noscript><span class="lazy-image-placeholder" style="width: 1.446ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" data-alt="{\displaystyle \omega }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е аголната брзина на масата околу точка на вртење. Овој аголен импулс е даден од <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =r\times p=r\times (m\omega \times r)=m((r\cdot r)\omega -(r\cdot \omega )r)=mr^{2}\omega =I\omega k,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> τ </mi> <mo> = </mo> <mi> r </mi> <mo> × </mo> <mi> p </mi> <mo> = </mo> <mi> r </mi> <mo> × </mo> <mo stretchy="false"> ( </mo> <mi> m </mi> <mi> ω </mi> <mo> × </mo> <mi> r </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> m </mi> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mi> r </mi> <mo> ⋅ </mo> <mi> r </mi> <mo stretchy="false"> ) </mo> <mi> ω </mi> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi> r </mi> <mo> ⋅ </mo> <mi> ω </mi> <mo stretchy="false"> ) </mo> <mi> r </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> ω </mi> <mo> = </mo> <mi> I </mi> <mi> ω </mi> <mi> k </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \tau =r\times p=r\times (m\omega \times r)=m((r\cdot r)\omega -(r\cdot \omega )r)=mr^{2}\omega =I\omega k,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7db061e8d3f6187f216d54d2f74143eb9256f325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:65.644ex; height:3.176ex;" alt="{\displaystyle \tau =r\times p=r\times (m\omega \times r)=m((r\cdot r)\omega -(r\cdot \omega )r)=mr^{2}\omega =I\omega k,}"> </noscript><span class="lazy-image-placeholder" style="width: 65.644ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7db061e8d3f6187f216d54d2f74143eb9256f325" data-alt="{\displaystyle \tau =r\times p=r\times (m\omega \times r)=m((r\cdot r)\omega -(r\cdot \omega )r)=mr^{2}\omega =I\omega k,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  користејќи слични деривации на претходната равенка. Слично на тоа, кинетичката енергија на масовната низа е дефинирана со брзината на нишалото околу вртењето за да се добие <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ek=1/2(mv)\cdot v=1/2(mr^{2})\omega =1/2(I\omega ^{2})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> E </mi> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mi> m </mi> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mi> v </mi> <mo> = </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mi> ω </mi> <mo> = </mo> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mi> I </mi> <msup> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle Ek=1/2(mv)\cdot v=1/2(mr^{2})\omega =1/2(I\omega ^{2})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/727e0497668dcfae1ef51b5a7ab11eb40262cb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.408ex; height:3.176ex;" alt="{\displaystyle Ek=1/2(mv)\cdot v=1/2(mr^{2})\omega =1/2(I\omega ^{2})}"> </noscript><span class="lazy-image-placeholder" style="width: 43.408ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/727e0497668dcfae1ef51b5a7ab11eb40262cb36" data-alt="{\displaystyle Ek=1/2(mv)\cdot v=1/2(mr^{2})\omega =1/2(I\omega ^{2})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Ова покажува дека количината <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=mr^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> I </mi> <mo> = </mo> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I=mr^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc416bcccf5f63d22af13befd74e5c653ab82f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.413ex; height:2.676ex;" alt="{\displaystyle I=mr^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.413ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc416bcccf5f63d22af13befd74e5c653ab82f1b" data-alt="{\displaystyle I=mr^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е како масата се комбинира со обликот на телото за да се дефинира вртежната инерција. Моментот на инерција на произволно обликуваното тело е збирот на вредностите <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mr^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle mr^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd9d0ea2911509b014b72a7b536acb7376cb455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.143ex; height:2.676ex;" alt="{\displaystyle mr^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.143ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd9d0ea2911509b014b72a7b536acb7376cb455" data-alt="{\displaystyle mr^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  за сите елементи на масата во телото. <div class="mw-heading mw-heading3"> <h3 id="Физичко_нишало">Физичко нишало</h3> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Физичко нишало“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Mendenhall_gravimeter_pendulums.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Mendenhall_gravimeter_pendulums.jpg/220px-Mendenhall_gravimeter_pendulums.jpg" decoding="async" width="220" height="210" class="mw-file-element" data-file-width="712" data-file-height="679"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 210px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Mendenhall_gravimeter_pendulums.jpg/220px-Mendenhall_gravimeter_pendulums.jpg" data-width="220" data-height="210" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Mendenhall_gravimeter_pendulums.jpg/330px-Mendenhall_gravimeter_pendulums.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Mendenhall_gravimeter_pendulums.jpg/440px-Mendenhall_gravimeter_pendulums.jpg 2x" data-class="mw-file-element"> </a> <figcaption> Нишала употребени во Менденхаловиот апарат за <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%93%D1%80%D0%B0%D0%B2%D0%B8%D0%BC%D0%B5%D1%82%D0%B0%D1%80?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Гравиметар">гравиметрија</a>, објавен во научно списание од 1897 година. Преносливиот гравиметер развиен во 1890 година од страна на Томас К. Менденхал обезбеди најточни релативни мерења на локалното гравитациско поле на Земјата. </figcaption> </figure> <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9D%D0%B8%D1%88%D0%B0%D0%BB%D0%BE&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Нишало (страницата не постои)">Физичко нишало</a> е тело формирано од збир на честички со континуирана форма кои ротираат круто околу стожерот. Нејзиниот момент на инерција е збирот на моментите на инерција на секоја од честичките од кои е составен.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-B-Paul-14">[14]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Resnick-15">[15]</a>:395–396<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-16">[16]</a>:51–53 <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A0%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%86%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Резонанца">Природната</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%90%D0%B3%D0%BE%D0%BB%D0%BD%D0%B0_%D1%87%D0%B5%D1%81%D1%82%D0%BE%D1%82%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Аголна честота">честота</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ω </mi> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \omega n} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d016c8be0690be6c6435d6e848b20bed5673aa93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.841ex; height:1.676ex;" alt="{\displaystyle \omega n}"> </noscript><span class="lazy-image-placeholder" style="width: 2.841ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d016c8be0690be6c6435d6e848b20bed5673aa93" data-alt="{\displaystyle \omega n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  од сложено нишало зависи од неговиот момент на инерција, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ip}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> I </mi> <mi> p </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle Ip} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da43a71b8fde0d7e4a20007765a46d48ce840f04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.341ex; height:2.509ex;" alt="{\displaystyle Ip}"> </noscript><span class="lazy-image-placeholder" style="width: 2.341ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da43a71b8fde0d7e4a20007765a46d48ce840f04" data-alt="{\displaystyle Ip}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\text{n}}={\sqrt {\frac {mgr}{I_{P}}}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> n </mtext> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi> m </mi> <mi> g </mi> <mi> r </mi> </mrow> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> P </mi> </mrow> </msub> </mfrac> </msqrt> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \omega _{\text{n}}={\sqrt {\frac {mgr}{I_{P}}}},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84766ee4c4f1b99c2efab4acf11b56499a0a36b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.702ex; height:6.176ex;" alt="{\displaystyle \omega _{\text{n}}={\sqrt {\frac {mgr}{I_{P}}}},}"> </noscript><span class="lazy-image-placeholder" style="width: 13.702ex;height: 6.176ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84766ee4c4f1b99c2efab4acf11b56499a0a36b0" data-alt="{\displaystyle \omega _{\text{n}}={\sqrt {\frac {mgr}{I_{P}}}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> </noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" data-alt="{\displaystyle m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е масата на објектот, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> g </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle g} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"> </noscript><span class="lazy-image-placeholder" style="width: 1.116ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" data-alt="{\displaystyle g}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">   е локално забрзување на гравитацијата, и <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е растојанието од точка на вртење до центарот на масата на објектот. Мерењето на оваа честота на осцилации преку мали аголни поместувања обезбедува ефикасен начин за мерење на моментот на инерција на телото.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Uicker-17">[17]</a>:516–517 Така, за да се одреди моментот на инерција на телото, едноставно го суспендира од удобна точка на вртење <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> P </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> </noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  така што слободно се ниша во рамнина нормална на правецот на посакуваниот момент на инерција, потоа ја мери неговата природна честота или период на осцилација ( <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> t </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle t} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> </noscript><span class="lazy-image-placeholder" style="width: 0.84ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" data-alt="{\displaystyle t}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ), за да се добие <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{P}={\frac {mgr}{\omega _{\text{n}}^{2}}}={\frac {mgrt^{2}}{4\pi ^{2}}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> P </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> m </mi> <mi> g </mi> <mi> r </mi> </mrow> <msubsup> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> n </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> m </mi> <mi> g </mi> <mi> r </mi> <msup> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow> <mn> 4 </mn> <msup> <mi> π </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{P}={\frac {mgr}{\omega _{\text{n}}^{2}}}={\frac {mgrt^{2}}{4\pi ^{2}}},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85426a7473f7d944e188ba5d27b4469cf5df1821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.31ex; height:6.343ex;" alt="{\displaystyle I_{P}={\frac {mgr}{\omega _{\text{n}}^{2}}}={\frac {mgrt^{2}}{4\pi ^{2}}},}"> </noscript><span class="lazy-image-placeholder" style="width: 21.31ex;height: 6.343ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85426a7473f7d944e188ba5d27b4469cf5df1821" data-alt="{\displaystyle I_{P}={\frac {mgr}{\omega _{\text{n}}^{2}}}={\frac {mgrt^{2}}{4\pi ^{2}}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> t </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle t} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> </noscript><span class="lazy-image-placeholder" style="width: 0.84ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" data-alt="{\displaystyle t}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е периодот (времетраењето) на осцилацијата (обично во просек во текот на повеќе периоди). Моментот на инерција на телото околу неговиот <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A6%D0%B5%D0%BD%D1%82%D0%B0%D1%80_%D0%BD%D0%B0_%D0%BC%D0%B0%D1%81%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Центар на маса (страницата не постои)">центар на маса</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{C}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{C}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcad0811b1e635343b4ac72fe5b75ac4a160b318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.504ex; height:2.509ex;" alt="{\displaystyle I_{C}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.504ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcad0811b1e635343b4ac72fe5b75ac4a160b318" data-alt="{\displaystyle I_{C}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , потоа се пресметува со употреба на <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%B7%D0%B0_%D0%BF%D0%B0%D1%80%D0%B0%D0%BB%D0%B5%D0%BB%D0%BD%D0%B0_%D0%BE%D1%81%D0%BA%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Теорема за паралелна оска (страницата не постои)">теорема за паралелна оска</a> за да биде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{C}=I_{P}-mr^{2},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> P </mi> </mrow> </msub> <mo> − </mo> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{C}=I_{P}-mr^{2},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b620abd41e5b8f61bbba1687aa11c8a14dcef1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.723ex; height:3.009ex;" alt="{\displaystyle I_{C}=I_{P}-mr^{2},}"> </noscript><span class="lazy-image-placeholder" style="width: 15.723ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b620abd41e5b8f61bbba1687aa11c8a14dcef1c6" data-alt="{\displaystyle I_{C}=I_{P}-mr^{2},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> </noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" data-alt="{\displaystyle m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е масата на телото и р е растојанието од вртежната точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> P </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> </noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  до центарот на масата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> C </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle C} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> </noscript><span class="lazy-image-placeholder" style="width: 1.766ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" data-alt="{\displaystyle C}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Моментот на инерција на телото често се дефинира во однос на неговиот <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%96%D0%B8%D1%80%D0%BE%D1%81%D0%BA%D0%BE%D0%BF%D1%81%D0%BA%D0%B8_%D0%BF%D0%BE%D0%BB%D1%83%D0%BF%D1%80%D0%B5%D1%87%D0%BD%D0%B8%D0%BA&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Жироскопски полупречник (страницата не постои)">жироскопски полупречник</a>, кој е полупречник на прстен со еднаква маса околу центарот на масата на телото што го има истиот момент на инерција. Полупречник на гурација к се пресметува од моментот на инерција на телото <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{C}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{C}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcad0811b1e635343b4ac72fe5b75ac4a160b318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.504ex; height:2.509ex;" alt="{\displaystyle I_{C}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.504ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcad0811b1e635343b4ac72fe5b75ac4a160b318" data-alt="{\displaystyle I_{C}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и маса <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> </noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" data-alt="{\displaystyle m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  како должина,<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Beer-18">[18]</a>:1296–1297 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\sqrt {\frac {I_{C}}{m}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> </mrow> </msub> <mi> m </mi> </mfrac> </msqrt> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k={\sqrt {\frac {I_{C}}{m}}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d33e3c713151dd31bcddd2f3d2f1bf310da47cb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.621ex; height:6.176ex;" alt="{\displaystyle k={\sqrt {\frac {I_{C}}{m}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 10.621ex;height: 6.176ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d33e3c713151dd31bcddd2f3d2f1bf310da47cb4" data-alt="{\displaystyle k={\sqrt {\frac {I_{C}}{m}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <div class="mw-heading mw-heading4"> <h4 id="Центар_на_осцилација">Центар на осцилација</h4> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Центар на осцилација“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> Едноставно нишало кое ја има истата природна честота како сложено нишало ја дефинира должината <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> </noscript><span class="lazy-image-placeholder" style="width: 1.583ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" data-alt="{\displaystyle L}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  од вртење до точка наречена <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A6%D0%B5%D0%BD%D1%82%D0%B0%D1%80_%D0%BD%D0%B0_%D0%BE%D1%81%D1%86%D0%B8%D0%BB%D0%B0%D1%86%D0%B8%D1%98%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Центар на осцилација (страницата не постои)">центар на осцилација</a> на сложеното нишало. Оваа точка, исто така, одговара на <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A6%D0%B5%D0%BD%D1%82%D0%B0%D1%80_%D0%BD%D0%B0_%D0%BF%D0%B5%D1%80%D0%BA%D1%83%D1%81%D0%B8%D0%B8&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Центар на перкусии (страницата не постои)">центарот на перкусии</a>. Должина <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> </noscript><span class="lazy-image-placeholder" style="width: 1.583ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" data-alt="{\displaystyle L}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  се определува од формулата, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\text{n}}={\sqrt {\frac {g}{L}}}={\sqrt {\frac {mgr}{I_{P}}}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> n </mtext> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi> g </mi> <mi> L </mi> </mfrac> </msqrt> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi> m </mi> <mi> g </mi> <mi> r </mi> </mrow> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> P </mi> </mrow> </msub> </mfrac> </msqrt> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \omega _{\text{n}}={\sqrt {\frac {g}{L}}}={\sqrt {\frac {mgr}{I_{P}}}},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af13f8f7fc5c36205648dfcc8e1ae74d68d1ee13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.543ex; height:6.343ex;" alt="{\displaystyle \omega _{\text{n}}={\sqrt {\frac {g}{L}}}={\sqrt {\frac {mgr}{I_{P}}}},}"> </noscript><span class="lazy-image-placeholder" style="width: 21.543ex;height: 6.343ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af13f8f7fc5c36205648dfcc8e1ae74d68d1ee13" data-alt="{\displaystyle \omega _{\text{n}}={\sqrt {\frac {g}{L}}}={\sqrt {\frac {mgr}{I_{P}}}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  или <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L={\frac {g}{\omega _{\text{n}}^{2}}}={\frac {I_{P}}{mr}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> g </mi> <msubsup> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> n </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> P </mi> </mrow> </msub> <mrow> <mi> m </mi> <mi> r </mi> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L={\frac {g}{\omega _{\text{n}}^{2}}}={\frac {I_{P}}{mr}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae9270c69ece90445dd11b9d6a65d045a3dd092" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.78ex; height:5.843ex;" alt="{\displaystyle L={\frac {g}{\omega _{\text{n}}^{2}}}={\frac {I_{P}}{mr}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 15.78ex;height: 5.843ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae9270c69ece90445dd11b9d6a65d045a3dd092" data-alt="{\displaystyle L={\frac {g}{\omega _{\text{n}}^{2}}}={\frac {I_{P}}{mr}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  На <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9D%D0%B8%D1%88%D0%B0%D0%BB%D0%BE_%D0%B7%D0%B0_%D1%81%D0%B5%D0%BA%D1%83%D0%BD%D0%B4%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Нишало за секунда (страницата не постои)">нишалото за секунда</a>, кое овозможува "крлеж" и "тока" на дедо часовник, трае една секунда за да се сврти од рамо до рамо. Ова е период од две секунди, или природна честота на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \ \mathrm {rad/s} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> π </mi> <mtext>   </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> r </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> d </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mi mathvariant="normal"> s </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \pi \ \mathrm {rad/s} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9165b16312df8cb79a9f7b2fa7146744de9390b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.358ex; height:2.843ex;" alt="{\displaystyle \pi \ \mathrm {rad/s} }"> </noscript><span class="lazy-image-placeholder" style="width: 7.358ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9165b16312df8cb79a9f7b2fa7146744de9390b6" data-alt="{\displaystyle \pi \ \mathrm {rad/s} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">   за нишалото. Во овој случај, растојанието до центарот на осцилацијата, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> </noscript><span class="lazy-image-placeholder" style="width: 1.583ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" data-alt="{\displaystyle L}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , може да се пресмета да биде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L={\frac {g}{\omega _{\text{n}}^{2}}}\approx {\frac {9.81\ \mathrm {m/s^{2}} }{(3.14\ \mathrm {rad/s} )^{2}}}\approx 0.99\ \mathrm {m} .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> g </mi> <msubsup> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> n </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mfrac> </mrow> <mo> ≈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 9.81 </mn> <mtext>   </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <msup> <mi mathvariant="normal"> s </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mrow> <mrow> <mo stretchy="false"> ( </mo> <mn> 3.14 </mn> <mtext>   </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> r </mi> <mi mathvariant="normal"> a </mi> <mi mathvariant="normal"> d </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mi mathvariant="normal"> s </mi> </mrow> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> ≈ </mo> <mn> 0.99 </mn> <mtext>   </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> m </mi> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L={\frac {g}{\omega _{\text{n}}^{2}}}\approx {\frac {9.81\ \mathrm {m/s^{2}} }{(3.14\ \mathrm {rad/s} )^{2}}}\approx 0.99\ \mathrm {m} .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6daff307d9615e49828965aef674f12679ebc65f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.464ex; height:6.676ex;" alt="{\displaystyle L={\frac {g}{\omega _{\text{n}}^{2}}}\approx {\frac {9.81\ \mathrm {m/s^{2}} }{(3.14\ \mathrm {rad/s} )^{2}}}\approx 0.99\ \mathrm {m} .}"> </noscript><span class="lazy-image-placeholder" style="width: 35.464ex;height: 6.676ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6daff307d9615e49828965aef674f12679ebc65f" data-alt="{\displaystyle L={\frac {g}{\omega _{\text{n}}^{2}}}\approx {\frac {9.81\ \mathrm {m/s^{2}} }{(3.14\ \mathrm {rad/s} )^{2}}}\approx 0.99\ \mathrm {m} .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Забележете дека растојанието до центарот на осцилацијата на секундарното нишало мора да се прилагоди за да се приспособат на различни вредности за локалното забрзување на гравитацијата. <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9A%D0%B0%D1%82%D0%B5%D1%80%D0%BE%D0%BB%D0%BE%D0%B2%D0%BE_%D0%BD%D0%B8%D1%88%D0%B0%D0%BB%D0%BE&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Катеролово нишало (страницата не постои)">Катеролово нишало</a> е сложено нишало кое го користи овој својство за мерење на локалното забрзување на гравитацијата и се нарекува <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%93%D1%80%D0%B0%D0%B2%D0%B8%D0%BC%D0%B5%D1%82%D0%B0%D1%80?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Гравиметар">гравиметар</a>. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <h2 id="Мерење_на_моментот_на_инерција">Мерење на моментот на инерција</h2> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Мерење на моментот на инерција“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> Моментот на инерција на комплексен систем, како што е возило или авион околу неговата вертикална оска, може да се мери со суспендирање на системот од три точки за да се формира трифиларно нишало. Трифиларното нишало е платформа поддржана од три жици дизајнирани да осцилираат во торзија околу вертикалната централна оска.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-19">[19]</a> Периодот на осцилација на трифиларното нишало го дава моментот на инерција на системот.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-20">[20]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <h2 id="Движење_во_фиксна_рамнина">Движење во фиксна рамнина</h2> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Движење во фиксна рамнина“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <div class="mw-heading mw-heading3"> <h3 id="Точкеста_маса">Точкеста маса</h3> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Точкеста маса“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <figure typeof="mw:File/Thumb"> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Rolling_Racers_-_Moment_of_inertia.gif?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Rolling_Racers_-_Moment_of_inertia.gif/216px-Rolling_Racers_-_Moment_of_inertia.gif" decoding="async" width="216" height="122" class="mw-file-element" data-file-width="480" data-file-height="270"> </noscript><span class="lazy-image-placeholder" style="width: 216px;height: 122px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Rolling_Racers_-_Moment_of_inertia.gif/216px-Rolling_Racers_-_Moment_of_inertia.gif" data-alt="" data-width="216" data-height="122" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Rolling_Racers_-_Moment_of_inertia.gif/324px-Rolling_Racers_-_Moment_of_inertia.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Rolling_Racers_-_Moment_of_inertia.gif/432px-Rolling_Racers_-_Moment_of_inertia.gif 2x" data-class="mw-file-element"> </a> <figcaption> Четири објекти со идентични маси и полупречници, се тркалаат по рамнина додека се тркалаат без лизгање. <style data-mw-deduplicate="TemplateStyles:r4948454">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="plainlist"> <ul> <li>    шуплива сфера,</li> <li>     цврста сфера,</li> <li>     цилиндричен прстен, и</li> <li>     цврст цилиндар.</li> </ul> </div>Времето за секој предмет да стигне до завршната линија зависи од нивниот момент на инерција. </figcaption> </figure> Моментот на инерција околу оската на телото се пресметува со собирање <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mr^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle mr^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd9d0ea2911509b014b72a7b536acb7376cb455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.143ex; height:2.676ex;" alt="{\displaystyle mr^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.143ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd9d0ea2911509b014b72a7b536acb7376cb455" data-alt="{\displaystyle mr^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> за секоја честичка во телото, каде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е нормално растојание до одредената оска. За да видиме како се јавува моментот на инерција во проучувањето на движењето на проширено тело, удобно е да се разгледа круто собрание на точка маси. (Оваа равенка може да се користи за оските кои не се главни оски под услов да се сфати дека ова не го опишува целосно моментот на инерција.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-21">[21]</a>) Размислете за кинетичката енергија на едно собрание на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> N </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle N} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> </noscript><span class="lazy-image-placeholder" style="width: 2.064ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" data-alt="{\displaystyle N}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  маси <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m_{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec8e804f69706d3f5ad235f4f983220c8df7c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;" alt="{\displaystyle m_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.84ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec8e804f69706d3f5ad235f4f983220c8df7c2" data-alt="{\displaystyle m_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> кои лежат на растојанија <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r_{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.848ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" data-alt="{\displaystyle r_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> од точка на вртење <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> P </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> </noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  , што е најблиската точка на оската на вртење. Тоа е збир на кинетичката енергија на поединечните маси,<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Uicker-17">[17]</a>:516–517<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Beer-18">[18]</a>:1084–1085<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Beer-18">[18]</a>:1296–1300 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{K}}=\sum _{i=1}^{N}{\frac {1}{2}}\,m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}=\sum _{i=1}^{N}{\frac {1}{2}}\,m_{i}\left(\omega r_{i}\right)^{2}={\frac {1}{2}}\,\omega ^{2}\sum _{i=1}^{N}m_{i}r_{i}^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> K </mtext> </mrow> </msub> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mrow> <mo> ( </mo> <mrow> <mi> ω </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle E_{\text{K}}=\sum _{i=1}^{N}{\frac {1}{2}}\,m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}=\sum _{i=1}^{N}{\frac {1}{2}}\,m_{i}\left(\omega r_{i}\right)^{2}={\frac {1}{2}}\,\omega ^{2}\sum _{i=1}^{N}m_{i}r_{i}^{2}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0b6efec911bdb60f83cfb6eac131cf2020523f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.32ex; height:7.343ex;" alt="{\displaystyle E_{\text{K}}=\sum _{i=1}^{N}{\frac {1}{2}}\,m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}=\sum _{i=1}^{N}{\frac {1}{2}}\,m_{i}\left(\omega r_{i}\right)^{2}={\frac {1}{2}}\,\omega ^{2}\sum _{i=1}^{N}m_{i}r_{i}^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 57.32ex;height: 7.343ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0b6efec911bdb60f83cfb6eac131cf2020523f" data-alt="{\displaystyle E_{\text{K}}=\sum _{i=1}^{N}{\frac {1}{2}}\,m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}=\sum _{i=1}^{N}{\frac {1}{2}}\,m_{i}\left(\omega r_{i}\right)^{2}={\frac {1}{2}}\,\omega ^{2}\sum _{i=1}^{N}m_{i}r_{i}^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Ова покажува дека моментот на инерција на телото е збирот на секоја од нив <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mr^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle mr^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd9d0ea2911509b014b72a7b536acb7376cb455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.143ex; height:2.676ex;" alt="{\displaystyle mr^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.143ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd9d0ea2911509b014b72a7b536acb7376cb455" data-alt="{\displaystyle mr^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  термини, што е <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{P}=\sum _{i=1}^{N}m_{i}r_{i}^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> P </mi> </mrow> </msub> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{P}=\sum _{i=1}^{N}m_{i}r_{i}^{2}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/442d042dcc10bf6d82bc5350bfd55e94b189ac5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.92ex; height:7.343ex;" alt="{\displaystyle I_{P}=\sum _{i=1}^{N}m_{i}r_{i}^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 14.92ex;height: 7.343ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/442d042dcc10bf6d82bc5350bfd55e94b189ac5d" data-alt="{\displaystyle I_{P}=\sum _{i=1}^{N}m_{i}r_{i}^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Друг израз ја заменува сумацијата со <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D0%BD%D0%BE_%D1%81%D0%BC%D0%B5%D1%82%D0%B0%D1%9A%D0%B5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Интегрално сметање">интеграл</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{P}=\iiint \limits _{Q}\rho \left(x,y,z\right)\left\|\mathbf {r} \right\|^{2}\mathrm {d} V}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> P </mi> </mrow> </msub> <mo> = </mo> <munder> <mo> ∭ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> Q </mi> </mrow> </munder> <mi> ρ </mi> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msup> <mrow> <mo symmetric="true"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo symmetric="true"> ‖ </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{P}=\iiint \limits _{Q}\rho \left(x,y,z\right)\left\|\mathbf {r} \right\|^{2}\mathrm {d} V} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18ed30d1d981875bfe43d0a1e678cec601738f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:27.563ex; height:7.676ex;" alt="{\displaystyle I_{P}=\iiint \limits _{Q}\rho \left(x,y,z\right)\left\|\mathbf {r} \right\|^{2}\mathrm {d} V}"> </noscript><span class="lazy-image-placeholder" style="width: 27.563ex;height: 7.676ex;vertical-align: -4.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18ed30d1d981875bfe43d0a1e678cec601738f3" data-alt="{\displaystyle I_{P}=\iiint \limits _{Q}\rho \left(x,y,z\right)\left\|\mathbf {r} \right\|^{2}\mathrm {d} V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Тука функцијата ρ ја дава густината на маса во секоја точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x,y,z)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"> </noscript><span class="lazy-image-placeholder" style="width: 7.45ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" data-alt="{\displaystyle (x,y,z)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {r} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.102ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" data-alt="{\displaystyle \mathbf {r} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е вектор нормален на оската на вртење и се протега од точка на оската на вртење до точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (x,y,z)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"> </noscript><span class="lazy-image-placeholder" style="width: 7.45ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" data-alt="{\displaystyle (x,y,z)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  во цврстата состојба, а интеграцијата се оценува преку волуменот <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> </noscript><span class="lazy-image-placeholder" style="width: 1.787ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" data-alt="{\displaystyle V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на телото <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> Q </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle Q} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"> </noscript><span class="lazy-image-placeholder" style="width: 1.838ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" data-alt="{\displaystyle Q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Моментот на инерција на рамна површина е сличен, при што густината на масата се заменува со густината на масата на површината со интегрален евалуиран над својата област. Забелешка за вториот момент на површината: Моментот на инерција на телото што се движи во рамнина и <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%92%D1%82%D0%BE%D1%80_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%BF%D0%BE%D0%B2%D1%80%D1%88%D0%B8%D0%BD%D0%B0%D1%82%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Втор момент на површината (страницата не постои)">вториот момент на површината</a> на зракот често се збунети. Моментот на инерција на тело со облик на пресек е вториот момент на оваа област околу z-оската нормална на пресекот, пондерирана според нејзината густина. Ова се нарекува и поларен момент на површината, и е збир на вторите моменти за x- и y-оски.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-22">[22]</a> Напорите во зракот се пресметуваат со користење на вториот момент на пресекот околу или околу x-оската или y-оската во зависност од товарот. <div class="mw-heading mw-heading4"> <h4 id="Примери_2">Примери</h4> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Примери“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <style data-mw-deduplicate="TemplateStyles:r4650192">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style> <div role="note" class="hatnote navigation-not-searchable"> Главна статија: <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A1%D0%BF%D0%B8%D1%81%D0%BE%D0%BA_%D0%BD%D0%B0_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D0%B8_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Список на моменти на инерција (страницата не постои)">Список на моменти на инерција</a> </div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Moment_of_inertia_rod_center.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Moment_of_inertia_rod_center.svg/220px-Moment_of_inertia_rod_center.svg.png" decoding="async" width="220" height="244" class="mw-file-element" data-file-width="512" data-file-height="569"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 244px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Moment_of_inertia_rod_center.svg/220px-Moment_of_inertia_rod_center.svg.png" data-width="220" data-height="244" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Moment_of_inertia_rod_center.svg/330px-Moment_of_inertia_rod_center.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Moment_of_inertia_rod_center.svg/440px-Moment_of_inertia_rod_center.svg.png 2x" data-class="mw-file-element"> </a> <figcaption></figcaption> </figure> Моментот на инерција на сложено нишало изградено од тенок диск монтиран на крајот од тенка прачка што осцилира околу стожерот на другиот крај на шипката започнува со пресметување на моментот на инерција на тенката прачка и тенокиот диск за нивните соодветни центри на маса.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Beer-18">[18]</a> <ul> <li>Моментот на инерција на тенка прачка со константен пресек <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> s </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle s} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> </noscript><span class="lazy-image-placeholder" style="width: 1.09ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" data-alt="{\displaystyle s}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и густината <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ρ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \rho } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"> </noscript><span class="lazy-image-placeholder" style="width: 1.202ex;height: 2.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" data-alt="{\displaystyle \rho }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и со должина <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ℓ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \ell } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"> </noscript><span class="lazy-image-placeholder" style="width: 0.97ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" data-alt="{\displaystyle \ell }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  околу нормална оска преку нејзиниот центар на маса се определува со интеграција.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Beer-18">[18]</a>:1301 Порамнувајќи ја x-оската со шипката и сместувајќи го центарот на масата во центарот на шипката, следи:</li> </ul> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{C,{\text{rod}}}=\iiint \limits _{Q}\rho \,x^{2}\,\mathrm {d} V=\int _{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}\rho \,x^{2}s\,\mathrm {d} x=\left.\rho s{\frac {x^{3}}{3}}\right|_{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}={\frac {\rho s}{3}}\left({\frac {\ell ^{3}}{8}}+{\frac {\ell ^{3}}{8}}\right)={\frac {m\ell ^{2}}{12}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> rod </mtext> </mrow> </mrow> </msub> <mo> = </mo> <munder> <mo> ∭ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> Q </mi> </mrow> </munder> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> ℓ </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> ℓ </mi> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> s </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> x </mi> <mo> = </mo> <msubsup> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mi> ρ </mi> <mi> s </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mn> 3 </mn> </mfrac> </mrow> </mrow> <mo> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> ℓ </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> ℓ </mi> <mn> 2 </mn> </mfrac> </mrow> </msubsup> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> ρ </mi> <mi> s </mi> </mrow> <mn> 3 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> ℓ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mn> 8 </mn> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> ℓ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mn> 8 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> m </mi> <msup> <mi> ℓ </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mn> 12 </mn> </mfrac> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{C,{\text{rod}}}=\iiint \limits _{Q}\rho \,x^{2}\,\mathrm {d} V=\int _{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}\rho \,x^{2}s\,\mathrm {d} x=\left.\rho s{\frac {x^{3}}{3}}\right|_{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}={\frac {\rho s}{3}}\left({\frac {\ell ^{3}}{8}}+{\frac {\ell ^{3}}{8}}\right)={\frac {m\ell ^{2}}{12}},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efcc120df3de0f96b2d79462c18bcf1444ae1731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:76.246ex; height:9.343ex;" alt="{\displaystyle I_{C,{\text{rod}}}=\iiint \limits _{Q}\rho \,x^{2}\,\mathrm {d} V=\int _{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}\rho \,x^{2}s\,\mathrm {d} x=\left.\rho s{\frac {x^{3}}{3}}\right|_{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}={\frac {\rho s}{3}}\left({\frac {\ell ^{3}}{8}}+{\frac {\ell ^{3}}{8}}\right)={\frac {m\ell ^{2}}{12}},}"> </noscript><span class="lazy-image-placeholder" style="width: 76.246ex;height: 9.343ex;vertical-align: -4.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efcc120df3de0f96b2d79462c18bcf1444ae1731" data-alt="{\displaystyle I_{C,{\text{rod}}}=\iiint \limits _{Q}\rho \,x^{2}\,\mathrm {d} V=\int _{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}\rho \,x^{2}s\,\mathrm {d} x=\left.\rho s{\frac {x^{3}}{3}}\right|_{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}={\frac {\rho s}{3}}\left({\frac {\ell ^{3}}{8}}+{\frac {\ell ^{3}}{8}}\right)={\frac {m\ell ^{2}}{12}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> каде што math>m = \rho s \ell</math> е масата на прачката. <ul> <li>Моментот на инерција на тенок диск со константна дебелина <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> s </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle s} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"> </noscript><span class="lazy-image-placeholder" style="width: 1.09ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" data-alt="{\displaystyle s}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , полупречник <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> R </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle R} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> </noscript><span class="lazy-image-placeholder" style="width: 1.764ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" data-alt="{\displaystyle R}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и густината <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ρ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \rho } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"> </noscript><span class="lazy-image-placeholder" style="width: 1.202ex;height: 2.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" data-alt="{\displaystyle \rho }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> околу оската преку неговиот центар и нормално на неговото лице (паралелно со неговата оска на <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%92%D1%80%D1%82%D0%B5%D0%B6%D0%BD%D0%B0_%D1%81%D0%B8%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%98%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Вртежна симетрија (страницата не постои)">вртежна симетрија</a>) се определува со интеграција.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Beer-18">[18]</a>:1301 Порамнувајќи ја z-оската со оската на дискот и дефинирајќи го елементот за волумен како <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} V=sr\mathrm {d} r\mathrm {d} \theta }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <mi> s </mi> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> θ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathrm {d} V=sr\mathrm {d} r\mathrm {d} \theta } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa6a262a2ab533edc9bf6be82a4250799df3fac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.041ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} V=sr\mathrm {d} r\mathrm {d} \theta }"> </noscript><span class="lazy-image-placeholder" style="width: 13.041ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa6a262a2ab533edc9bf6be82a4250799df3fac9" data-alt="{\displaystyle \mathrm {d} V=sr\mathrm {d} r\mathrm {d} \theta }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , следи:</li> </ul> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{C,{\text{disc}}}=\iiint \limits _{Q}\rho \,r^{2}\,\mathrm {d} V=\int _{0}^{2\pi }\int _{0}^{R}\rho r^{2}sr\,\mathrm {d} r\,\mathrm {d} \theta =2\pi \rho s{\frac {R^{4}}{4}}={\frac {1}{2}}mR^{2},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> disc </mtext> </mrow> </mrow> </msub> <mo> = </mo> <munder> <mo> ∭ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> Q </mi> </mrow> </munder> <mi> ρ </mi> <mspace width="thinmathspace"></mspace> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mi> π </mi> </mrow> </msubsup> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> R </mi> </mrow> </msubsup> <mi> ρ </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> s </mi> <mi> r </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> r </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> θ </mi> <mo> = </mo> <mn> 2 </mn> <mi> π </mi> <mi> ρ </mi> <mi> s </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mi> m </mi> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{C,{\text{disc}}}=\iiint \limits _{Q}\rho \,r^{2}\,\mathrm {d} V=\int _{0}^{2\pi }\int _{0}^{R}\rho r^{2}sr\,\mathrm {d} r\,\mathrm {d} \theta =2\pi \rho s{\frac {R^{4}}{4}}={\frac {1}{2}}mR^{2},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b70e507f08442801dccbc17daa7474c36c6c08a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:66.316ex; height:8.176ex;" alt="{\displaystyle I_{C,{\text{disc}}}=\iiint \limits _{Q}\rho \,r^{2}\,\mathrm {d} V=\int _{0}^{2\pi }\int _{0}^{R}\rho r^{2}sr\,\mathrm {d} r\,\mathrm {d} \theta =2\pi \rho s{\frac {R^{4}}{4}}={\frac {1}{2}}mR^{2},}"> </noscript><span class="lazy-image-placeholder" style="width: 66.316ex;height: 8.176ex;vertical-align: -4.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b70e507f08442801dccbc17daa7474c36c6c08a3" data-alt="{\displaystyle I_{C,{\text{disc}}}=\iiint \limits _{Q}\rho \,r^{2}\,\mathrm {d} V=\int _{0}^{2\pi }\int _{0}^{R}\rho r^{2}sr\,\mathrm {d} r\,\mathrm {d} \theta =2\pi \rho s{\frac {R^{4}}{4}}={\frac {1}{2}}mR^{2},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=\pi R^{2}\rho s}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <mo> = </mo> <mi> π </mi> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> ρ </mi> <mi> s </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m=\pi R^{2}\rho s} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dca17cc43732e0d5266a98ebe1fdde0a678a2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.582ex; height:3.176ex;" alt="{\displaystyle m=\pi R^{2}\rho s}"> </noscript><span class="lazy-image-placeholder" style="width: 11.582ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dca17cc43732e0d5266a98ebe1fdde0a678a2c" data-alt="{\displaystyle m=\pi R^{2}\rho s}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е неговата маса. <ul> <li>Моментот на инерција на сложеното нишало сега се добива со додавање на моментот на инерција на шипката и дискот околу точката на вртење <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> P </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> </noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  како,</li> </ul> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{P}=I_{C,{\text{rod}}}+M_{\text{rod}}\left({\frac {L}{2}}\right)^{2}+I_{C,{\text{disc}}}+M_{\text{disc}}(L+R)^{2},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> P </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> rod </mtext> </mrow> </mrow> </msub> <mo> + </mo> <msub> <mi> M </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> rod </mtext> </mrow> </msub> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> L </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> disc </mtext> </mrow> </mrow> </msub> <mo> + </mo> <msub> <mi> M </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> disc </mtext> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> L </mi> <mo> + </mo> <mi> R </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{P}=I_{C,{\text{rod}}}+M_{\text{rod}}\left({\frac {L}{2}}\right)^{2}+I_{C,{\text{disc}}}+M_{\text{disc}}(L+R)^{2},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/516751c9cd8925d27978349304a410048a9111fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.857ex; height:6.509ex;" alt="{\displaystyle I_{P}=I_{C,{\text{rod}}}+M_{\text{rod}}\left({\frac {L}{2}}\right)^{2}+I_{C,{\text{disc}}}+M_{\text{disc}}(L+R)^{2},}"> </noscript><span class="lazy-image-placeholder" style="width: 51.857ex;height: 6.509ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/516751c9cd8925d27978349304a410048a9111fc" data-alt="{\displaystyle I_{P}=I_{C,{\text{rod}}}+M_{\text{rod}}\left({\frac {L}{2}}\right)^{2}+I_{C,{\text{disc}}}+M_{\text{disc}}(L+R)^{2},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> </noscript><span class="lazy-image-placeholder" style="width: 1.583ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" data-alt="{\displaystyle L}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е должината на нишалото. Забележете дека теорема за паралелна оска се користи за поместување на моментот на инерција од центарот на масата до точка на вртење на нишалото. <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A1%D0%BF%D0%B8%D1%81%D0%BE%D0%BA_%D0%BD%D0%B0_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D0%B8_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Список на моменти на инерција (страницата не постои)">Список на моменти на инерција</a> се равенки за стандардни обици на тела и обезбедува начин да се добие моментот на инерција на сложено тело како збир на поедноставно обликувани тела. <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BD%D0%B0_%D0%BF%D0%B0%D1%80%D0%B0%D0%BB%D0%B5%D0%BB%D0%BD%D0%B0_%D0%BE%D1%81%D0%BA%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Теорема на паралелна оска (страницата не постои)">Теоремата на паралелната оска</a> се користи за поместување на референтната точка на поединечните тела на референтната точка на склопот. <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Moment_of_inertia_solid_sphere.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Moment_of_inertia_solid_sphere.svg/220px-Moment_of_inertia_solid_sphere.svg.png" decoding="async" width="220" height="203" class="mw-file-element" data-file-width="277" data-file-height="255"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 203px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Moment_of_inertia_solid_sphere.svg/220px-Moment_of_inertia_solid_sphere.svg.png" data-width="220" data-height="203" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Moment_of_inertia_solid_sphere.svg/330px-Moment_of_inertia_solid_sphere.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Moment_of_inertia_solid_sphere.svg/440px-Moment_of_inertia_solid_sphere.svg.png 2x" data-class="mw-file-element"> </a> <figcaption></figcaption> </figure> Како уште еден пример, разгледајте го моментот на инерција на цврста сфера со константна густина околу оската преку нејзиниот центар на маса. Ова се определува со сумирање на моментите на инерција на тенки дискови кои ја формираат сферата. Ако површината на топката е дефинирана со равенката<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Beer-18">[18]</a>:1301 <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}+z^{2}=R^{2},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x^{2}+y^{2}+z^{2}=R^{2},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27d27233c3e28eccb66564e293ed5d94755b7179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.988ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}+z^{2}=R^{2},}"> </noscript><span class="lazy-image-placeholder" style="width: 18.988ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27d27233c3e28eccb66564e293ed5d94755b7179" data-alt="{\displaystyle x^{2}+y^{2}+z^{2}=R^{2},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> тогаш полупречникот <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на дискот на пресекот <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> </noscript><span class="lazy-image-placeholder" style="width: 1.088ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" data-alt="{\displaystyle z}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  по должината на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> </noscript><span class="lazy-image-placeholder" style="width: 1.088ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" data-alt="{\displaystyle z}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> -оската е <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(z)^{2}=x^{2}+y^{2}=R^{2}-z^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> r </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r(z)^{2}=x^{2}+y^{2}=R^{2}-z^{2}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80e0877d0975c009e3be64c0af3de7cbd74d4468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.086ex; height:3.176ex;" alt="{\displaystyle r(z)^{2}=x^{2}+y^{2}=R^{2}-z^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 27.086ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80e0877d0975c009e3be64c0af3de7cbd74d4468" data-alt="{\displaystyle r(z)^{2}=x^{2}+y^{2}=R^{2}-z^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Затоа, моментот на инерција на топката е збирот на моментите на инерција на дисковите долж z-оската, <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}I_{C,{\text{ball}}}&=\int _{-R}^{R}{\frac {\pi \rho }{2}}r(z)^{4}\,\mathrm {d} z=\int _{-R}^{R}{\frac {\pi \rho }{2}}\left(R^{2}-z^{2}\right)^{2}\,\mathrm {d} z\\&=\left.{\frac {\pi \rho }{2}}\left(R^{4}z-{\frac {2}{3}}R^{2}z^{3}+{\frac {1}{5}}z^{5}\right)\right|_{-R}^{R}\\&=\pi \rho \left(1-{\frac {2}{3}}+{\frac {1}{5}}\right)R^{5}\\&={\frac {2}{5}}mR^{2},\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> ball </mtext> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> R </mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> π </mi> <mi> ρ </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mi> r </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> z </mi> <mo> = </mo> <msubsup> <mo> ∫ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> R </mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> π </mi> <mi> ρ </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> z </mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <msubsup> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> π </mi> <mi> ρ </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mi> z </mi> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> </mrow> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 5 </mn> </mfrac> </mrow> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 5 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> R </mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mi> π </mi> <mi> ρ </mi> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 5 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 5 </mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 2 </mn> <mn> 5 </mn> </mfrac> </mrow> <mi> m </mi> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> , </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}I_{C,{\text{ball}}}&=\int _{-R}^{R}{\frac {\pi \rho }{2}}r(z)^{4}\,\mathrm {d} z=\int _{-R}^{R}{\frac {\pi \rho }{2}}\left(R^{2}-z^{2}\right)^{2}\,\mathrm {d} z\\&=\left.{\frac {\pi \rho }{2}}\left(R^{4}z-{\frac {2}{3}}R^{2}z^{3}+{\frac {1}{5}}z^{5}\right)\right|_{-R}^{R}\\&=\pi \rho \left(1-{\frac {2}{3}}+{\frac {1}{5}}\right)R^{5}\\&={\frac {2}{5}}mR^{2},\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc34aebaa2f5f7f0cdb4a0ab2841cedcdee7312" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.717ex; margin-bottom: -0.288ex; width:49.7ex; height:25.176ex;" alt="{\displaystyle {\begin{aligned}I_{C,{\text{ball}}}&=\int _{-R}^{R}{\frac {\pi \rho }{2}}r(z)^{4}\,\mathrm {d} z=\int _{-R}^{R}{\frac {\pi \rho }{2}}\left(R^{2}-z^{2}\right)^{2}\,\mathrm {d} z\\&=\left.{\frac {\pi \rho }{2}}\left(R^{4}z-{\frac {2}{3}}R^{2}z^{3}+{\frac {1}{5}}z^{5}\right)\right|_{-R}^{R}\\&=\pi \rho \left(1-{\frac {2}{3}}+{\frac {1}{5}}\right)R^{5}\\&={\frac {2}{5}}mR^{2},\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 49.7ex;height: 25.176ex;vertical-align: -11.717ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc34aebaa2f5f7f0cdb4a0ab2841cedcdee7312" data-alt="{\displaystyle {\begin{aligned}I_{C,{\text{ball}}}&=\int _{-R}^{R}{\frac {\pi \rho }{2}}r(z)^{4}\,\mathrm {d} z=\int _{-R}^{R}{\frac {\pi \rho }{2}}\left(R^{2}-z^{2}\right)^{2}\,\mathrm {d} z\\&=\left.{\frac {\pi \rho }{2}}\left(R^{4}z-{\frac {2}{3}}R^{2}z^{3}+{\frac {1}{5}}z^{5}\right)\right|_{-R}^{R}\\&=\pi \rho \left(1-{\frac {2}{3}}+{\frac {1}{5}}\right)R^{5}\\&={\frac {2}{5}}mR^{2},\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m={\frac {4}{3}}\pi R^{3}\rho }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 4 </mn> <mn> 3 </mn> </mfrac> </mrow> <mi> π </mi> <msup> <mi> R </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mi> ρ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m={\frac {4}{3}}\pi R^{3}\rho } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92311473222cd0844b03212079c98ae36730c7b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.49ex; height:5.176ex;" alt="{\displaystyle m={\frac {4}{3}}\pi R^{3}\rho }"> </noscript><span class="lazy-image-placeholder" style="width: 12.49ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92311473222cd0844b03212079c98ae36730c7b8" data-alt="{\displaystyle m={\frac {4}{3}}\pi R^{3}\rho }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е масата на сферата. <div class="mw-heading mw-heading3"> <h3 id="Цврсто_тело">Цврсто тело</h3> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Цврсто тело“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> Ако механички систем е ограничен да се движи паралелно со фиксна рамнина, тогаш ротацијата на телото во системот се случува околу оската <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {k}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {k}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.843ex;" alt="{\displaystyle \mathbf {\hat {k}} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" data-alt="{\displaystyle \mathbf {\hat {k}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  нормално на оваа рамнина. Во овој случај, моментот на инерција на масата во овој систем е скалар кој е познат како поларниот момент на инерција. Дефиницијата на поларниот момент на инерција може да се добие со разгледување на моментумот, кинетичката енергија и законите на Њутн за плазно движење на крут систем на честички.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-B-Paul-14">[14]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Uicker-17">[17]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Goldstein-23">[23]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-24">[24]</a> Ако системот од n честички, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i},i=1,...,n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mo> . </mo> <mo> . </mo> <mo> . </mo> <mo> , </mo> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{i},i=1,...,n} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda9b5e40922a7dc3ee07ea0859c6391fcdbd4ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.954ex; height:2.509ex;" alt="{\displaystyle P_{i},i=1,...,n}"> </noscript><span class="lazy-image-placeholder" style="width: 14.954ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda9b5e40922a7dc3ee07ea0859c6391fcdbd4ee" data-alt="{\displaystyle P_{i},i=1,...,n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  се собрани во круто тело, тогаш моментумот на системот може да се напише во однос на позициите во однос на референтната точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и апсолутните брзини <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.211ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" data-alt="{\displaystyle \mathbf {v} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Delta \mathbf {r} _{i}&=\mathbf {r} _{i}-\mathbf {R} ,\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {V} ={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} ,\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mo> , </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\Delta \mathbf {r} _{i}&=\mathbf {r} _{i}-\mathbf {R} ,\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {V} ={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} ,\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/763631f93bfbb2d477b0f4bc32f6f276b38989c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.563ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\Delta \mathbf {r} _{i}&=\mathbf {r} _{i}-\mathbf {R} ,\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {V} ={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} ,\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 42.563ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/763631f93bfbb2d477b0f4bc32f6f276b38989c3" data-alt="{\displaystyle {\begin{aligned}\Delta \mathbf {r} _{i}&=\mathbf {r} _{i}-\mathbf {R} ,\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {V} ={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} ,\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде ω е аголната брзина на системот, а V е брзината на R. За рамно движење векторот на аголна брзина е насочен долж единечниот вектор k кој е нормален на рамнината на движење. Воведување на единечни вектори <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {e} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba77c5a75ef9e230d1a36183785477a2eb3c5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.025ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.025ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba77c5a75ef9e230d1a36183785477a2eb3c5c1e" data-alt="{\displaystyle \mathbf {e} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> од референтната точка R до точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {r} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.902ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" data-alt="{\displaystyle \mathbf {r} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , и единечниот вектор <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> t </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8be043bb3ba35151a191103bb68449759673fb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.622ex; height:3.176ex;" alt="{\displaystyle \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.622ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8be043bb3ba35151a191103bb68449759673fb3" data-alt="{\displaystyle \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , па така <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {\hat {e}} _{i}&={\frac {\Delta \mathbf {r} _{i}}{\Delta r_{i}}},\quad \mathbf {\hat {k}} ={\frac {\boldsymbol {\omega }}{\omega }},\quad \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i},\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} =\omega \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {V} =\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo> , </mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="bold-italic"> ω </mi> <mi> ω </mi> </mfrac> </mrow> <mo> , </mo> <mspace width="1em"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> t </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mo> = </mo> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mo> = </mo> <mi> ω </mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> t </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\mathbf {\hat {e}} _{i}&={\frac {\Delta \mathbf {r} _{i}}{\Delta r_{i}}},\quad \mathbf {\hat {k}} ={\frac {\boldsymbol {\omega }}{\omega }},\quad \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i},\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} =\omega \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {V} =\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9fa841cd3da6473a5c2721cc37981bc44ebf0f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:54.555ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {\hat {e}} _{i}&={\frac {\Delta \mathbf {r} _{i}}{\Delta r_{i}}},\quad \mathbf {\hat {k}} ={\frac {\boldsymbol {\omega }}{\omega }},\quad \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i},\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} =\omega \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {V} =\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 54.555ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9fa841cd3da6473a5c2721cc37981bc44ebf0f3" data-alt="{\displaystyle {\begin{aligned}\mathbf {\hat {e}} _{i}&={\frac {\Delta \mathbf {r} _{i}}{\Delta r_{i}}},\quad \mathbf {\hat {k}} ={\frac {\boldsymbol {\omega }}{\omega }},\quad \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i},\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} =\omega \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {V} =\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Ова го дефинира векторот на релативната позиција и векторот на брзината за крутиот систем на честичките што се движат во рамнина. Забелешка за вкрстениот производ: Кога едно тело се движи паралелно со рамнина на земјата, траекториите на сите точки во телото лежат во рамнини паралелни на оваа заземјена рамнина. Ова значи дека секоја ротација што се случува на телото мора да биде околу една оска нормална на оваа рамнина. Плодното движење често се презентира како што е проектирано на оваа основна рамнина, така што оската на вртење се појавува како точка. Во овој случај, аголната брзина и аголното забрзување на телото се скалари и фактот дека тие се вектори долж оската на вртење е игнориран. Ова обично се користи за воведување на темата. Но, во случај на момент на инерција, комбинацијата на маса и геометрија придобивки од геометриските својства на вкрстениот производ. Поради оваа причина, во овој дел за рамномерно движење аголната брзина и забрзување на телото се вектори нормални на заземјената рамнина, а операциите на вкрстените производи се исти како што се користат за проучување на движењето на просторно цврсто тело. <div class="mw-heading mw-heading4"> <h4 id="Аголен_импулс">Аголен импулс</h4> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Аголен импулс“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> Аголниот импулс на векторот за рамно движење на крут систем на честички е даден со<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-B-Paul-14">[14]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Uicker-17">[17]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i=1}^{n}m_{i}\Delta \mathbf {r} _{i}\times \mathbf {v} _{i}\\&=\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\times \left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right)\\&=\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\right)\omega \mathbf {\hat {k}} +\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\right)\times \mathbf {V} .\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> L </mi> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mi> ω </mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> t </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i=1}^{n}m_{i}\Delta \mathbf {r} _{i}\times \mathbf {v} _{i}\\&=\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\times \left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right)\\&=\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\right)\omega \mathbf {\hat {k}} +\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\right)\times \mathbf {V} .\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b74f7da5a5a9e9fcf7dd09ebbfaae0ae67076e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:48.31ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i=1}^{n}m_{i}\Delta \mathbf {r} _{i}\times \mathbf {v} _{i}\\&=\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\times \left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right)\\&=\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\right)\omega \mathbf {\hat {k}} +\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\right)\times \mathbf {V} .\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 48.31ex;height: 21.843ex;vertical-align: -10.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b74f7da5a5a9e9fcf7dd09ebbfaae0ae67076e8" data-alt="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i=1}^{n}m_{i}\Delta \mathbf {r} _{i}\times \mathbf {v} _{i}\\&=\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\times \left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right)\\&=\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\right)\omega \mathbf {\hat {k}} +\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\right)\times \mathbf {V} .\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Користете го <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A6%D0%B5%D0%BD%D1%82%D0%B0%D1%80_%D0%BD%D0%B0_%D0%BC%D0%B0%D1%81%D0%B0%D1%82%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Центар на масата (страницата не постои)">центарот на масата</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.931ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" data-alt="{\displaystyle \mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  како референтна точка така да <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Delta r_{i}\mathbf {\hat {e}} _{i}&=\mathbf {r} _{i}-\mathbf {C} ,\\\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}&=0,\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\Delta r_{i}\mathbf {\hat {e}} _{i}&=\mathbf {r} _{i}-\mathbf {C} ,\\\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}&=0,\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2142ef52e531692cc9869f02413e2ccdf95df6c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:24.06ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}\Delta r_{i}\mathbf {\hat {e}} _{i}&=\mathbf {r} _{i}-\mathbf {C} ,\\\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}&=0,\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 24.06ex;height: 9.843ex;vertical-align: -4.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2142ef52e531692cc9869f02413e2ccdf95df6c4" data-alt="{\displaystyle {\begin{aligned}\Delta r_{i}\mathbf {\hat {e}} _{i}&=\mathbf {r} _{i}-\mathbf {C} ,\\\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}&=0,\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и да го дефинира моментот на инерција во однос на центарот на масата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathbf {C} }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{\mathbf {C} }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed60543babbd8e72ba50197ad76e78856500230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.621ex; height:2.509ex;" alt="{\displaystyle I_{\mathbf {C} }}"> </noscript><span class="lazy-image-placeholder" style="width: 2.621ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed60543babbd8e72ba50197ad76e78856500230" data-alt="{\displaystyle I_{\mathbf {C} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> како <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathbf {C} }=\sum _{i}m_{i}\,\Delta r_{i}^{2},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mo> = </mo> <munder> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </munder> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{\mathbf {C} }=\sum _{i}m_{i}\,\Delta r_{i}^{2},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d74ff1b42e1668c88e66632b6cf03e2aa94cea8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.374ex; height:5.509ex;" alt="{\displaystyle I_{\mathbf {C} }=\sum _{i}m_{i}\,\Delta r_{i}^{2},}"> </noscript><span class="lazy-image-placeholder" style="width: 17.374ex;height: 5.509ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d74ff1b42e1668c88e66632b6cf03e2aa94cea8" data-alt="{\displaystyle I_{\mathbf {C} }=\sum _{i}m_{i}\,\Delta r_{i}^{2},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  тогаш се поедноставува равенката за аголниот момент<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Beer-18">[18]</a>:1028 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =I_{\mathbf {C} }\omega \mathbf {\hat {k}} .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> L </mi> </mrow> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {L} =I_{\mathbf {C} }\omega \mathbf {\hat {k}} .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/140a33b8857435797ce58aaec89d066d8e796faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.831ex; height:3.176ex;" alt="{\displaystyle \mathbf {L} =I_{\mathbf {C} }\omega \mathbf {\hat {k}} .}"> </noscript><span class="lazy-image-placeholder" style="width: 10.831ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/140a33b8857435797ce58aaec89d066d8e796faf" data-alt="{\displaystyle \mathbf {L} =I_{\mathbf {C} }\omega \mathbf {\hat {k}} .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Моментот на инерција <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathbf {C} }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{\mathbf {C} }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed60543babbd8e72ba50197ad76e78856500230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.621ex; height:2.509ex;" alt="{\displaystyle I_{\mathbf {C} }}"> </noscript><span class="lazy-image-placeholder" style="width: 2.621ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed60543babbd8e72ba50197ad76e78856500230" data-alt="{\displaystyle I_{\mathbf {C} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  околу оската нормална на движењето на цврстиот систем и низ центарот на масата е познат како поларниот момент на инерција. Поточно, тоа е втор момент на маса во однос на ортогоналното растојание од оската (или пол). За одредена количина на аголен импулс, намалувањето на моментот на инерција резултира со зголемување на аголната брзина. Сликарите можат да го променат моментот на инерција со повлекување на рацете. Така, аголната брзина постигната со лизгач со испружени раце резултира со поголема аголна брзина кога рацете се влечат, поради намалениот момент на инерција. А фигурист не е, сепак, круто тело. <div class="mw-heading mw-heading4"> <h4 id="Кинетичка_енергија">Кинетичка енергија</h4> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=13&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Кинетичка енергија“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <figure typeof="mw:File/Thumb"> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Lever_shear_flywheel.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Lever_shear_flywheel.jpg/241px-Lever_shear_flywheel.jpg" decoding="async" width="241" height="115" class="mw-file-element" data-file-width="1153" data-file-height="549"> </noscript><span class="lazy-image-placeholder" style="width: 241px;height: 115px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Lever_shear_flywheel.jpg/241px-Lever_shear_flywheel.jpg" data-width="241" data-height="115" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Lever_shear_flywheel.jpg/362px-Lever_shear_flywheel.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Lever_shear_flywheel.jpg/482px-Lever_shear_flywheel.jpg 2x" data-class="mw-file-element"> </a> <figcaption> Оваа вртежна стрижалка во 1906 го користи моментот на инерција на две замаец за складирање на кинетичката енергија, која кога се ослободува се користи за намалување на металниот фонд (Меѓународна библиотека за технологија, 1906). </figcaption> </figure> Кинетичката енергија на крут систем на честички што се движат во рамнината е дадена со<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-B-Paul-14">[14]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Uicker-17">[17]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E_{\text{K}}&={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i},\\&={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right)\cdot \left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right),\\&={\frac {1}{2}}\omega ^{2}\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\mathbf {\hat {t}} _{i}\cdot \mathbf {\hat {t}} _{i}\right)+\omega \mathbf {V} \cdot \left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {t}} _{i}\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} \cdot \mathbf {V} .\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> K </mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mi> ω </mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> t </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ω </mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> t </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <msup> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> t </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> t </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> t </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}E_{\text{K}}&={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i},\\&={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right)\cdot \left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right),\\&={\frac {1}{2}}\omega ^{2}\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\mathbf {\hat {t}} _{i}\cdot \mathbf {\hat {t}} _{i}\right)+\omega \mathbf {V} \cdot \left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {t}} _{i}\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} \cdot \mathbf {V} .\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2857fa9af9745d1cd6110f7c5781d32ec9d55b98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:79.4ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}E_{\text{K}}&={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i},\\&={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right)\cdot \left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right),\\&={\frac {1}{2}}\omega ^{2}\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\mathbf {\hat {t}} _{i}\cdot \mathbf {\hat {t}} _{i}\right)+\omega \mathbf {V} \cdot \left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {t}} _{i}\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} \cdot \mathbf {V} .\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 79.4ex;height: 21.843ex;vertical-align: -10.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2857fa9af9745d1cd6110f7c5781d32ec9d55b98" data-alt="{\displaystyle {\begin{aligned}E_{\text{K}}&={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i},\\&={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right)\cdot \left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right),\\&={\frac {1}{2}}\omega ^{2}\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\mathbf {\hat {t}} _{i}\cdot \mathbf {\hat {t}} _{i}\right)+\omega \mathbf {V} \cdot \left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {t}} _{i}\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} \cdot \mathbf {V} .\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Нека референтната точка е центар на масата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.931ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" data-alt="{\displaystyle \mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на системот, па вториот мандат станува нула, и ќе го воведеме моментот на инертност <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathbf {C} }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{\mathbf {C} }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed60543babbd8e72ba50197ad76e78856500230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.621ex; height:2.509ex;" alt="{\displaystyle I_{\mathbf {C} }}"> </noscript><span class="lazy-image-placeholder" style="width: 2.621ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed60543babbd8e72ba50197ad76e78856500230" data-alt="{\displaystyle I_{\mathbf {C} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , така што кинетичката енергија ќе биде дадена со<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Beer-18">[18]</a>:1084 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{K}}={\frac {1}{2}}I_{\mathbf {C} }\omega ^{2}+{\frac {1}{2}}M\mathbf {V} \cdot \mathbf {V} .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> K </mtext> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <msup> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mi> M </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle E_{\text{K}}={\frac {1}{2}}I_{\mathbf {C} }\omega ^{2}+{\frac {1}{2}}M\mathbf {V} \cdot \mathbf {V} .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4679b9b32ae721e111ca1734fa8b31aa1bd189c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.09ex; height:5.176ex;" alt="{\displaystyle E_{\text{K}}={\frac {1}{2}}I_{\mathbf {C} }\omega ^{2}+{\frac {1}{2}}M\mathbf {V} \cdot \mathbf {V} .}"> </noscript><span class="lazy-image-placeholder" style="width: 27.09ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4679b9b32ae721e111ca1734fa8b31aa1bd189c6" data-alt="{\displaystyle E_{\text{K}}={\frac {1}{2}}I_{\mathbf {C} }\omega ^{2}+{\frac {1}{2}}M\mathbf {V} \cdot \mathbf {V} .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Моментот на инерција <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathbf {C} }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{\mathbf {C} }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed60543babbd8e72ba50197ad76e78856500230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.621ex; height:2.509ex;" alt="{\displaystyle I_{\mathbf {C} }}"> </noscript><span class="lazy-image-placeholder" style="width: 2.621ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed60543babbd8e72ba50197ad76e78856500230" data-alt="{\displaystyle I_{\mathbf {C} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е поларниот момент на инерција на телото. <div class="mw-heading mw-heading4"> <h4 id="Њутнови_закони">Њутнови закони</h4> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=14&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Њутнови закони“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <figure typeof="mw:File/Thumb"> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Johndeered.jpg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Johndeered.jpg/229px-Johndeered.jpg" decoding="async" width="229" height="172" class="mw-file-element" data-file-width="1024" data-file-height="768"> </noscript><span class="lazy-image-placeholder" style="width: 229px;height: 172px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Johndeered.jpg/229px-Johndeered.jpg" data-width="229" data-height="172" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Johndeered.jpg/344px-Johndeered.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Johndeered.jpg/458px-Johndeered.jpg 2x" data-class="mw-file-element"> </a> <figcaption> Тракторот од Џон Џејмс од 1920 година со спојуваниот <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%97%D0%B0%D0%BC%D0%B0%D0%B5%D1%86&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Замаец (страницата не постои)">замаец</a> на моторот. Големиот момент на инерција на замаецот ја олеснува работата на тракторот. </figcaption> </figure> Њутнови закони за крут систем на n честички, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i},i=1,...,n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mo> . </mo> <mo> . </mo> <mo> . </mo> <mo> , </mo> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{i},i=1,...,n} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda9b5e40922a7dc3ee07ea0859c6391fcdbd4ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.954ex; height:2.509ex;" alt="{\displaystyle P_{i},i=1,...,n}"> </noscript><span class="lazy-image-placeholder" style="width: 14.954ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda9b5e40922a7dc3ee07ea0859c6391fcdbd4ee" data-alt="{\displaystyle P_{i},i=1,...,n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , може да се напише во однос на <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A0%D0%B5%D0%B7%D1%83%D0%BB%D1%82%D0%B0%D0%BD%D1%82%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Резултанта (страницата не постои)">резултантната сила</a> и вртежен момент во референтната точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , за да се добие<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-B-Paul-14">[14]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Uicker-17">[17]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {F} &=\sum _{i=1}^{n}m_{i}\mathbf {A} _{i},\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}\Delta \mathbf {r} _{i}\times m_{i}\mathbf {A} _{i},\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> F </mi> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> τ </mi> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\mathbf {F} &=\sum _{i=1}^{n}m_{i}\mathbf {A} _{i},\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}\Delta \mathbf {r} _{i}\times m_{i}\mathbf {A} _{i},\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d878e318dbc371b1e6fa920e9af0db290bc5356d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:22.259ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {F} &=\sum _{i=1}^{n}m_{i}\mathbf {A} _{i},\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}\Delta \mathbf {r} _{i}\times m_{i}\mathbf {A} _{i},\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 22.259ex;height: 14.176ex;vertical-align: -6.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d878e318dbc371b1e6fa920e9af0db290bc5356d" data-alt="{\displaystyle {\begin{aligned}\mathbf {F} &=\sum _{i=1}^{n}m_{i}\mathbf {A} _{i},\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}\Delta \mathbf {r} _{i}\times m_{i}\mathbf {A} _{i},\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде ri означува траекторија на секоја честичка. Кинематиката на круто тело ја дава формулата за забрзување на честичката <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.292ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3" data-alt="{\displaystyle P_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  во однос на положбата R и забрзувањето A на референтната честичка, како и аголниот вектор на брзина <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\omega }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb8af7a2f64af348e559652b6b1f0d2415ba444" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\omega }}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.669ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb8af7a2f64af348e559652b6b1f0d2415ba444" data-alt="{\displaystyle {\boldsymbol {\omega }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и векторот на аголно забрзување α од крутиот систем на честички како, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} _{i}={\boldsymbol {\alpha }}\times \Delta \mathbf {r} _{i}+{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {A} .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {A} _{i}={\boldsymbol {\alpha }}\times \Delta \mathbf {r} _{i}+{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {A} .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e689c5f3f580c43874d5a86ca01005643bedc633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.567ex; height:2.509ex;" alt="{\displaystyle \mathbf {A} _{i}={\boldsymbol {\alpha }}\times \Delta \mathbf {r} _{i}+{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {A} .}"> </noscript><span class="lazy-image-placeholder" style="width: 35.567ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e689c5f3f580c43874d5a86ca01005643bedc633" data-alt="{\displaystyle \mathbf {A} _{i}={\boldsymbol {\alpha }}\times \Delta \mathbf {r} _{i}+{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {A} .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  За системи кои се ограничени на рамно движење, аголната брзина и векторите на аголното забрзување се насочуваат долж <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {k}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {k}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.843ex;" alt="{\displaystyle \mathbf {\hat {k}} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" data-alt="{\displaystyle \mathbf {\hat {k}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  нормално на рамнината на движење, што ја поедноставува оваа равенка за забрзување. Во овој случај, забрзувачките вектори може да се поедностават со внесување на единечните вектори <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {e}} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {e}} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f01cfee86a741d3d25c6510aea329ab584019060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.136ex; height:2.676ex;" alt="{\displaystyle \mathbf {\hat {e}} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.136ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f01cfee86a741d3d25c6510aea329ab584019060" data-alt="{\displaystyle \mathbf {\hat {e}} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  од референтната точка R во точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {r} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.902ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" data-alt="{\displaystyle \mathbf {r} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и единечните вектори <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> t </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8be043bb3ba35151a191103bb68449759673fb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.622ex; height:3.176ex;" alt="{\displaystyle \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.622ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8be043bb3ba35151a191103bb68449759673fb3" data-alt="{\displaystyle \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , па така <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {A} _{i}&=\alpha \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}-\omega \mathbf {\hat {k}} \times \omega \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} \\&=\alpha \Delta r_{i}\mathbf {\hat {t}} _{i}-\omega ^{2}\Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} .\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mi> α </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mi> α </mi> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> t </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <msup> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\mathbf {A} _{i}&=\alpha \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}-\omega \mathbf {\hat {k}} \times \omega \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} \\&=\alpha \Delta r_{i}\mathbf {\hat {t}} _{i}-\omega ^{2}\Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} .\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a914cf61ff0a1ca9fefa05e4bf3ac1a4f13340c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.344ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {A} _{i}&=\alpha \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}-\omega \mathbf {\hat {k}} \times \omega \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} \\&=\alpha \Delta r_{i}\mathbf {\hat {t}} _{i}-\omega ^{2}\Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} .\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 43.344ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a914cf61ff0a1ca9fefa05e4bf3ac1a4f13340c5" data-alt="{\displaystyle {\begin{aligned}\mathbf {A} _{i}&=\alpha \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}-\omega \mathbf {\hat {k}} \times \omega \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} \\&=\alpha \Delta r_{i}\mathbf {\hat {t}} _{i}-\omega ^{2}\Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} .\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Ова дава резултат на вртежен момент на системот како <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\times \left(\alpha \Delta r_{i}\mathbf {\hat {t}} _{i}-\omega ^{2}\Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} \right)\\&=\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\right)\alpha \mathbf {\hat {k}} +\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\right)\times \mathbf {A} ,\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> τ </mi> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> t </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <msup> <mi> ω </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> <mi> α </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> <mo> , </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\times \left(\alpha \Delta r_{i}\mathbf {\hat {t}} _{i}-\omega ^{2}\Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} \right)\\&=\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\right)\alpha \mathbf {\hat {k}} +\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\right)\times \mathbf {A} ,\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06402c0d2f280037bc077adef38e371ef4ade20d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:48.161ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\times \left(\alpha \Delta r_{i}\mathbf {\hat {t}} _{i}-\omega ^{2}\Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} \right)\\&=\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\right)\alpha \mathbf {\hat {k}} +\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\right)\times \mathbf {A} ,\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 48.161ex;height: 14.509ex;vertical-align: -6.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06402c0d2f280037bc077adef38e371ef4ade20d" data-alt="{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\times \left(\alpha \Delta r_{i}\mathbf {\hat {t}} _{i}-\omega ^{2}\Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} \right)\\&=\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\right)\alpha \mathbf {\hat {k}} +\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\right)\times \mathbf {A} ,\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {e}} _{i}\times \mathbf {\hat {e}} _{i}=\mathbf {0} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold"> 0 </mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {e}} _{i}\times \mathbf {\hat {e}} _{i}=\mathbf {0} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d5871aade530539c6fde126bd6d2212e356c48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.548ex; height:2.676ex;" alt="{\displaystyle \mathbf {\hat {e}} _{i}\times \mathbf {\hat {e}} _{i}=\mathbf {0} }"> </noscript><span class="lazy-image-placeholder" style="width: 11.548ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d5871aade530539c6fde126bd6d2212e356c48" data-alt="{\displaystyle \mathbf {\hat {e}} _{i}\times \mathbf {\hat {e}} _{i}=\mathbf {0} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , и <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {e}} _{i}\times \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> e </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> t </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {e}} _{i}\times \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c88af56868802fdc05b30d048d9fda9fb22ec99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.622ex; height:3.176ex;" alt="{\displaystyle \mathbf {\hat {e}} _{i}\times \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} }"> </noscript><span class="lazy-image-placeholder" style="width: 11.622ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c88af56868802fdc05b30d048d9fda9fb22ec99" data-alt="{\displaystyle \mathbf {\hat {e}} _{i}\times \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е единечен вектор нормален на рамнината за сите честички <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.292ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3" data-alt="{\displaystyle P_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Користете го <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A2%D0%B5%D0%B6%D0%B8%D1%88%D1%82%D0%B5?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Тежиште">центарот на маса</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.931ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" data-alt="{\displaystyle \mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  како референтна точка и дефинирајте го моментот на инерција во однос на центарот на масата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathbf {C} }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{\mathbf {C} }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed60543babbd8e72ba50197ad76e78856500230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.621ex; height:2.509ex;" alt="{\displaystyle I_{\mathbf {C} }}"> </noscript><span class="lazy-image-placeholder" style="width: 2.621ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed60543babbd8e72ba50197ad76e78856500230" data-alt="{\displaystyle I_{\mathbf {C} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , тогаш равенката за добиениот вртежен момент се поедноставува<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Beer-18">[18]</a>:1029 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}=I_{\mathbf {C} }\alpha \mathbf {\hat {k}} .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> τ </mi> </mrow> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mi> α </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\tau }}=I_{\mathbf {C} }\alpha \mathbf {\hat {k}} .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2050af209e999c252e5be940b7d1eba7ff935acd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.683ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {\tau }}=I_{\mathbf {C} }\alpha \mathbf {\hat {k}} .}"> </noscript><span class="lazy-image-placeholder" style="width: 10.683ex;height: 3.176ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2050af209e999c252e5be940b7d1eba7ff935acd" data-alt="{\displaystyle {\boldsymbol {\tau }}=I_{\mathbf {C} }\alpha \mathbf {\hat {k}} .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <h2 id="Движење_во_просторот_на_цврсто_тело_и_инерцијална_матрица"><span id=".D0.94.D0.B2.D0.B8.D0.B6.D0.B5.D1.9A.D0.B5_.D0.B2.D0.BE_.D0.BF.D1.80.D0.BE.D1.81.D1.82.D0.BE.D1.80.D0.BE.D1.82_.D0.BD.D0.B0_.D1.86.D0.B2.D1.80.D1.81.D1.82.D0.BE_.D1.82.D0.B5.D0.BB.D0.BE_.D0.B8_.D0.B8.D0.BD.D0.B5.D1.80.D1.86.D0.B8.D1.98.D0.B0.D0.BB.D0.BD.D0.B0_.D0.BC.D0.B0.D1.82.D1.80.D0.B8.D1.86.D0.B0">Движење во просторот на цврсто тело и инерцијална матрица</h2> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=15&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Движење во просторот на цврсто тело и инерцијална матрица“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> Скаларните моменти на инерција се појавуваат како елементи во матрица кога системот на честички е составен во цврсто тело кое се движи во тридимензионален простор. Оваа инерцијална матрица се појавува при пресметувањето на аголниот момент, кинетичката енергија и резултирачкиот вртежен момент на крутиот систем на честички.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Marion_1995-3">[3]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Symon_1971-4">[4]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Tenenbaum_2004-5">[5]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kane-6">[6]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Tsai-25">[25]</a> <div class="dablink"> За анализа на вртливо тело, видете <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9F%D1%80%D0%B5%D1%86%D0%B5%D1%81%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#%D0%9A%D0%BB%D0%B0%D1%81%D0%B8%D1%87%D0%BD%D0%B0_(%D0%8A%D1%83%D1%82%D0%BD%D0%BE%D0%B2%D0%B0)" title="Прецесија">Прецесија#Класична (Њутнова)</a> и <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9E%D1%98%D0%BB%D0%B5%D1%80%D0%BE%D0%B2%D0%B8_%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D0%BA%D0%B8_(%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0_%D0%BD%D0%B0_%D1%86%D0%B2%D1%80%D1%81%D1%82%D0%BE_%D1%82%D0%B5%D0%BB%D0%BE)&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Ојлерови равенки (динамика на цврсто тело) (страницата не постои)">Ојлерови равенки (динамика на цврсто тело)</a>. </div> Нека системот на честички <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  честички, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i},i=1,...,n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mo> . </mo> <mo> . </mo> <mo> . </mo> <mo> , </mo> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{i},i=1,...,n} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda9b5e40922a7dc3ee07ea0859c6391fcdbd4ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.954ex; height:2.509ex;" alt="{\displaystyle P_{i},i=1,...,n}"> </noscript><span class="lazy-image-placeholder" style="width: 14.954ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda9b5e40922a7dc3ee07ea0859c6391fcdbd4ee" data-alt="{\displaystyle P_{i},i=1,...,n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  се наоѓаат во координатите <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {r} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.902ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" data-alt="{\displaystyle \mathbf {r} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  со брзина <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.211ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" data-alt="{\displaystyle \mathbf {v} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  во однос на фиксната референтна рамка. За (по можност движечка) референтна точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , релативните позиции се <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9b70b1347bdd9d18f140f45ea7e4db3666d38a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.681ex; height:2.509ex;" alt="{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 13.681ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9b70b1347bdd9d18f140f45ea7e4db3666d38a" data-alt="{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и (апсолутните) брзини се <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{i}={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {R} }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{i}={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {R} }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f8e4a3717df9525ccdad8aeac3a14f57f3d37b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.165ex; height:2.509ex;" alt="{\displaystyle \mathbf {v} _{i}={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {R} }}"> </noscript><span class="lazy-image-placeholder" style="width: 20.165ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f8e4a3717df9525ccdad8aeac3a14f57f3d37b" data-alt="{\displaystyle \mathbf {v} _{i}={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {R} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\omega }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb8af7a2f64af348e559652b6b1f0d2415ba444" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\omega }}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.669ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb8af7a2f64af348e559652b6b1f0d2415ba444" data-alt="{\displaystyle {\boldsymbol {\omega }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е аголната брзина на системот, а <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {V_{R}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold"> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {V_{R}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1f71d137ed03b340503b5e49c353f1f67aff27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.668ex; height:2.509ex;" alt="{\displaystyle \mathbf {V_{R}} }"> </noscript><span class="lazy-image-placeholder" style="width: 3.668ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1f71d137ed03b340503b5e49c353f1f67aff27" data-alt="{\displaystyle \mathbf {V_{R}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е брзината на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . <div class="mw-heading mw-heading3"> <h3 id="Момент_на_импулсот">Момент на импулсот</h3> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=16&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Момент на импулсот“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> Забележете дека <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%92%D0%BA%D1%80%D1%81%D1%82%D0%B5%D0%BD_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Вкрстен производ (страницата не постои)">вкрстениот производ може да биде еквивалентно напишан како матрично множење</a> со комбинирање на првиот операнд и операторот во коска-симетрична, матрица, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\mathbf {b} \right]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> b </mi> </mrow> <mo> ] </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left[\mathbf {b} \right]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce9643510bd433e0ed612edd5d3a9ed7faecfec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.779ex; height:2.843ex;" alt="{\displaystyle \left[\mathbf {b} \right]}"> </noscript><span class="lazy-image-placeholder" style="width: 2.779ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce9643510bd433e0ed612edd5d3a9ed7faecfec" data-alt="{\displaystyle \left[\mathbf {b} \right]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , конструирана од компонентите на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {b} =(b_{x},b_{y},b_{z})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> b </mi> </mrow> <mo> = </mo> <mo stretchy="false"> ( </mo> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {b} =(b_{x},b_{y},b_{z})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5684411608489b3650c74432ef2c035f76bc411c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.677ex; height:3.009ex;" alt="{\displaystyle \mathbf {b} =(b_{x},b_{y},b_{z})}"> </noscript><span class="lazy-image-placeholder" style="width: 14.677ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5684411608489b3650c74432ef2c035f76bc411c" data-alt="{\displaystyle \mathbf {b} =(b_{x},b_{y},b_{z})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {b} \times \mathbf {y} &\equiv \left[\mathbf {b} \right]\mathbf {y} \\\left[\mathbf {b} \right]&\equiv {\begin{bmatrix}0&-b_{z}&b_{y}\\b_{z}&0&-b_{x}\\-b_{y}&b_{x}&0\end{bmatrix}}.\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> b </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> y </mi> </mrow> </mtd> <mtd> <mi></mi> <mo> ≡ </mo> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> b </mi> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> y </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> b </mi> </mrow> <mo> ] </mo> </mrow> </mtd> <mtd> <mi></mi> <mo> ≡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> − </mo> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> </mrow> </msub> </mtd> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> − </mo> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> − </mo> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> </mrow> </msub> </mtd> <mtd> <mn> 0 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\mathbf {b} \times \mathbf {y} &\equiv \left[\mathbf {b} \right]\mathbf {y} \\\left[\mathbf {b} \right]&\equiv {\begin{bmatrix}0&-b_{z}&b_{y}\\b_{z}&0&-b_{x}\\-b_{y}&b_{x}&0\end{bmatrix}}.\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4782eb664e98a6d680dd96b77601ff9bc12ab4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:30.371ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {b} \times \mathbf {y} &\equiv \left[\mathbf {b} \right]\mathbf {y} \\\left[\mathbf {b} \right]&\equiv {\begin{bmatrix}0&-b_{z}&b_{y}\\b_{z}&0&-b_{x}\\-b_{y}&b_{x}&0\end{bmatrix}}.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 30.371ex;height: 12.843ex;vertical-align: -5.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4782eb664e98a6d680dd96b77601ff9bc12ab4a" data-alt="{\displaystyle {\begin{aligned}\mathbf {b} \times \mathbf {y} &\equiv \left[\mathbf {b} \right]\mathbf {y} \\\left[\mathbf {b} \right]&\equiv {\begin{bmatrix}0&-b_{z}&b_{y}\\b_{z}&0&-b_{x}\\-b_{y}&b_{x}&0\end{bmatrix}}.\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Инертната матрица е конструирана со разгледување на аголниот момент, со референтната точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на телото избрано да биде центар на маса <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.931ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" data-alt="{\displaystyle \mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \mathbf {v} _{i}\\&=\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {R} }\right)\\&=\left(-\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \left(\Delta \mathbf {r} _{i}\times {\boldsymbol {\omega }}\right)\right)+\left(\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \mathbf {V} _{\mathbf {R} }\right),\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> L </mi> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \mathbf {v} _{i}\\&=\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {R} }\right)\\&=\left(-\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \left(\Delta \mathbf {r} _{i}\times {\boldsymbol {\omega }}\right)\right)+\left(\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \mathbf {V} _{\mathbf {R} }\right),\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24a7ac3c1aa4822b810baba705b84177c6994928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:60.009ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \mathbf {v} _{i}\\&=\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {R} }\right)\\&=\left(-\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \left(\Delta \mathbf {r} _{i}\times {\boldsymbol {\omega }}\right)\right)+\left(\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \mathbf {V} _{\mathbf {R} }\right),\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 60.009ex;height: 21.843ex;vertical-align: -10.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24a7ac3c1aa4822b810baba705b84177c6994928" data-alt="{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \mathbf {v} _{i}\\&=\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {R} }\right)\\&=\left(-\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \left(\Delta \mathbf {r} _{i}\times {\boldsymbol {\omega }}\right)\right)+\left(\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \mathbf {V} _{\mathbf {R} }\right),\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде што условите кои содржат <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {V_{R}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold"> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {V_{R}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1f71d137ed03b340503b5e49c353f1f67aff27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.668ex; height:2.509ex;" alt="{\displaystyle \mathbf {V_{R}} }"> </noscript><span class="lazy-image-placeholder" style="width: 3.668ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1f71d137ed03b340503b5e49c353f1f67aff27" data-alt="{\displaystyle \mathbf {V_{R}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ( <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle =\mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d7a4dee205ebb34cdcd994634c778207154cc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.385ex; height:2.176ex;" alt="{\displaystyle =\mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 4.385ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d7a4dee205ebb34cdcd994634c778207154cc0" data-alt="{\displaystyle =\mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) се изразува на нула со дефиницијата за центар на масата. Потоа, косисиметричната матрица <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\Delta \mathbf {r} _{i}]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> [ </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle [\Delta \mathbf {r} _{i}]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee5960e77611f673ab2f2050f356dbf0e403132" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.131ex; height:2.843ex;" alt="{\displaystyle [\Delta \mathbf {r} _{i}]}"> </noscript><span class="lazy-image-placeholder" style="width: 5.131ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee5960e77611f673ab2f2050f356dbf0e403132" data-alt="{\displaystyle [\Delta \mathbf {r} _{i}]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  добиена од векторот на релативната позиција <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/690eb2a602890211d9c1cd09e601f81b3f8bc9df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.609ex; height:2.509ex;" alt="{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 13.609ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/690eb2a602890211d9c1cd09e601f81b3f8bc9df" data-alt="{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , може да се користи за дефинирање, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}=\mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> L </mi> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {L} =\left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}=\mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ff4f4ab9aa3fff11085b8828fc71a4af7de700" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.433ex; height:7.509ex;" alt="{\displaystyle \mathbf {L} =\left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}=\mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }},}"> </noscript><span class="lazy-image-placeholder" style="width: 33.433ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ff4f4ab9aa3fff11085b8828fc71a4af7de700" data-alt="{\displaystyle \mathbf {L} =\left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}=\mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I_{C}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold"> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I_{C}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d3bdba483ab8231b911ab724380a9ff07d66e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.612ex; height:2.509ex;" alt="{\displaystyle \mathbf {I_{C}} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.612ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d3bdba483ab8231b911ab724380a9ff07d66e2" data-alt="{\displaystyle \mathbf {I_{C}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  се дефинира како <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} _{\mathbf {C} }=-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} _{\mathbf {C} }=-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6766e8a9ce07ff8f32ec86aaa929f1e5a3ce1a21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.32ex; height:6.843ex;" alt="{\displaystyle \mathbf {I} _{\mathbf {C} }=-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2},}"> </noscript><span class="lazy-image-placeholder" style="width: 21.32ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6766e8a9ce07ff8f32ec86aaa929f1e5a3ce1a21" data-alt="{\displaystyle \mathbf {I} _{\mathbf {C} }=-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е симетрична матрица на инерција на цврстиот систем на честички измерен во однос на центарот на масата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.931ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" data-alt="{\displaystyle \mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . <div class="mw-heading mw-heading3"> <h3 id="Кинетичка_енергија_2">Кинетичка енергија</h3> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=17&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Кинетичка енергија“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> Кинетичката енергија на крут систем на честички може да се формулира во однос на центарот на масата и матрицата на масовните моменти на инерција на системот. Нека систем на честички <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i},i=1,...,n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mo> . </mo> <mo> . </mo> <mo> . </mo> <mo> , </mo> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{i},i=1,...,n} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda9b5e40922a7dc3ee07ea0859c6391fcdbd4ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.954ex; height:2.509ex;" alt="{\displaystyle P_{i},i=1,...,n}"> </noscript><span class="lazy-image-placeholder" style="width: 14.954ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda9b5e40922a7dc3ee07ea0859c6391fcdbd4ee" data-alt="{\displaystyle P_{i},i=1,...,n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  се наоѓа во координатите <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {r} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.902ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" data-alt="{\displaystyle \mathbf {r} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  со брзина <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.211ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" data-alt="{\displaystyle \mathbf {v} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , тогаш кинетичката енергија е <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{K}}={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {C} }\right)\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {C} }\right),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> K </mtext> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle E_{\text{K}}={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {C} }\right)\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {C} }\right),} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f56981fba8d261103bebdb3250107b39273d9b9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:69.787ex; height:6.843ex;" alt="{\displaystyle E_{\text{K}}={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {C} }\right)\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {C} }\right),}"> </noscript><span class="lazy-image-placeholder" style="width: 69.787ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f56981fba8d261103bebdb3250107b39273d9b9c" data-alt="{\displaystyle E_{\text{K}}={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {C} }\right)\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {C} }\right),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/690eb2a602890211d9c1cd09e601f81b3f8bc9df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.609ex; height:2.509ex;" alt="{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 13.609ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/690eb2a602890211d9c1cd09e601f81b3f8bc9df" data-alt="{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е положбениот вектор на честичка во однос на центарот на масата. Оваа равенка се проширува за да добие три термини <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{K}}={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+\left(\sum _{i=1}^{n}m_{i}\mathbf {V} _{\mathbf {C} }\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\right).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> K </mtext> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle E_{\text{K}}={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+\left(\sum _{i=1}^{n}m_{i}\mathbf {V} _{\mathbf {C} }\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\right).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2206bca6f4db61956bc2fcf0b5eafc78afac83a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:95.346ex; height:7.509ex;" alt="{\displaystyle E_{\text{K}}={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+\left(\sum _{i=1}^{n}m_{i}\mathbf {V} _{\mathbf {C} }\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\right).}"> </noscript><span class="lazy-image-placeholder" style="width: 95.346ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2206bca6f4db61956bc2fcf0b5eafc78afac83a" data-alt="{\displaystyle E_{\text{K}}={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+\left(\sum _{i=1}^{n}m_{i}\mathbf {V} _{\mathbf {C} }\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\right).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> Вториот израз во оваа равенка е нула, бидејќи <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.931ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" data-alt="{\displaystyle \mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> е центар на масата. Воведување на коска-симетрична матрица <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\Delta \mathbf {r} _{i}]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> [ </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle [\Delta \mathbf {r} _{i}]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee5960e77611f673ab2f2050f356dbf0e403132" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.131ex; height:2.843ex;" alt="{\displaystyle [\Delta \mathbf {r} _{i}]}"> </noscript><span class="lazy-image-placeholder" style="width: 5.131ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee5960e77611f673ab2f2050f356dbf0e403132" data-alt="{\displaystyle [\Delta \mathbf {r} _{i}]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , така што кинетичката енергија станува <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E_{\text{K}}&={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left(\left[\Delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\cdot \left(\left[\Delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\\&={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}^{\mathsf {T}}\left[\Delta \mathbf {r} _{i}\right]^{\mathsf {T}}\left[\Delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\\&={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }.\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> K </mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <msup> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}E_{\text{K}}&={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left(\left[\Delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\cdot \left(\left[\Delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\\&={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}^{\mathsf {T}}\left[\Delta \mathbf {r} _{i}\right]^{\mathsf {T}}\left[\Delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\\&={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }.\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c246c5646e149b67a600084c164f154ec7dda89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.838ex; width:64.755ex; height:22.843ex;" alt="{\displaystyle {\begin{aligned}E_{\text{K}}&={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left(\left[\Delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\cdot \left(\left[\Delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\\&={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}^{\mathsf {T}}\left[\Delta \mathbf {r} _{i}\right]^{\mathsf {T}}\left[\Delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\\&={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 64.755ex;height: 22.843ex;vertical-align: -10.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c246c5646e149b67a600084c164f154ec7dda89" data-alt="{\displaystyle {\begin{aligned}E_{\text{K}}&={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left(\left[\Delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\cdot \left(\left[\Delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\\&={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}^{\mathsf {T}}\left[\Delta \mathbf {r} _{i}\right]^{\mathsf {T}}\left[\Delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\\&={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }.\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Така, кинетичката енергија на крутиот систем на честички е дадена со <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{K}}={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }}+{\frac {1}{2}}M\mathbf {V} _{\mathbf {C} }^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> K </mtext> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mi> M </mi> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> V </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle E_{\text{K}}={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }}+{\frac {1}{2}}M\mathbf {V} _{\mathbf {C} }^{2}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4ef82b66824a7c74937a1db0d8b3923d2fb6bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.497ex; height:5.176ex;" alt="{\displaystyle E_{\text{K}}={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }}+{\frac {1}{2}}M\mathbf {V} _{\mathbf {C} }^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 27.497ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4ef82b66824a7c74937a1db0d8b3923d2fb6bc" data-alt="{\displaystyle E_{\text{K}}={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }}+{\frac {1}{2}}M\mathbf {V} _{\mathbf {C} }^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I_{C}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold"> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I_{C}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d3bdba483ab8231b911ab724380a9ff07d66e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.612ex; height:2.509ex;" alt="{\displaystyle \mathbf {I_{C}} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.612ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d3bdba483ab8231b911ab724380a9ff07d66e2" data-alt="{\displaystyle \mathbf {I_{C}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е инерцијалната матрица во однос на центарот на масата и <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> M </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle M} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> </noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е вкупната маса. <div class="mw-heading mw-heading3"> <h3 id="Резултантен_вртежен_момент">Резултантен вртежен момент</h3> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=18&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Резултантен вртежен момент“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> Инертната матрица се појавува при примената на вториот закон на Њутн на круто собрание на честички. Реалниот вртежен момент на овој систем е<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Marion_1995-3">[3]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kane-6">[6]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}=\sum _{i=1}^{n}\left(\mathbf {r_{i}} -\mathbf {R} \right)\times m_{i}\mathbf {a} _{i},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> τ </mi> </mrow> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold"> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> i </mi> </mrow> </msub> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> × </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\tau }}=\sum _{i=1}^{n}\left(\mathbf {r_{i}} -\mathbf {R} \right)\times m_{i}\mathbf {a} _{i},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93ab161ceb777c444827961be31c4ae628dc3c03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.197ex; height:6.843ex;" alt="{\displaystyle {\boldsymbol {\tau }}=\sum _{i=1}^{n}\left(\mathbf {r_{i}} -\mathbf {R} \right)\times m_{i}\mathbf {a} _{i},}"> </noscript><span class="lazy-image-placeholder" style="width: 25.197ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93ab161ceb777c444827961be31c4ae628dc3c03" data-alt="{\displaystyle {\boldsymbol {\tau }}=\sum _{i=1}^{n}\left(\mathbf {r_{i}} -\mathbf {R} \right)\times m_{i}\mathbf {a} _{i},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {a} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a01879ce830ef8790aa7dc9f3665d6727f3af3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.099ex; height:2.009ex;" alt="{\displaystyle \mathbf {a} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.099ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a01879ce830ef8790aa7dc9f3665d6727f3af3a" data-alt="{\displaystyle \mathbf {a} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е забрзување на честичката <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.292ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3" data-alt="{\displaystyle P_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9A%D0%B8%D0%BD%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Кинематика">Кинематиката</a> на круто тело ја дава формулата за забрзување на честичката <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.292ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3" data-alt="{\displaystyle P_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  во однос на положбата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и забрзувањето <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} _{\mathbf {R} }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {A} _{\mathbf {R} }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c521614a9d5e3931efc3e3787e74e75c6ae88b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.668ex; height:2.509ex;" alt="{\displaystyle \mathbf {A} _{\mathbf {R} }}"> </noscript><span class="lazy-image-placeholder" style="width: 3.668ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c521614a9d5e3931efc3e3787e74e75c6ae88b3" data-alt="{\displaystyle \mathbf {A} _{\mathbf {R} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на референтната точка, како и аголниот вектор на брзина <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\omega }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb8af7a2f64af348e559652b6b1f0d2415ba444" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\omega }}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.669ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb8af7a2f64af348e559652b6b1f0d2415ba444" data-alt="{\displaystyle {\boldsymbol {\omega }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и векторот на аголно забрзување <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\alpha }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\alpha }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a585d2bb19071162720ea56a7b087dab3ec17156" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.769ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\alpha }}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.769ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a585d2bb19071162720ea56a7b087dab3ec17156" data-alt="{\displaystyle {\boldsymbol {\alpha }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> на крутиот систем како, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{i}={\boldsymbol {\alpha }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {A} _{\mathbf {R} }.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {a} _{i}={\boldsymbol {\alpha }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {A} _{\mathbf {R} }.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bed2cf766d12e829544f33f557e646924dfaed8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.93ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} _{i}={\boldsymbol {\alpha }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {A} _{\mathbf {R} }.}"> </noscript><span class="lazy-image-placeholder" style="width: 45.93ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bed2cf766d12e829544f33f557e646924dfaed8" data-alt="{\displaystyle \mathbf {a} _{i}={\boldsymbol {\alpha }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {A} _{\mathbf {R} }.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Користете го центарот на маса <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.931ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" data-alt="{\displaystyle \mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  како референтна точка и внесете ја коси-симетричната матрица <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\Delta \mathbf {r} _{i}\right]=\left[\mathbf {r} _{i}-\mathbf {C} \right]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mo> = </mo> <mrow> <mo> [ </mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> <mo> ] </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left[\Delta \mathbf {r} _{i}\right]=\left[\mathbf {r} _{i}-\mathbf {C} \right]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b83ab9af0efe0f59204a493b10b5eb453594c19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.197ex; height:2.843ex;" alt="{\displaystyle \left[\Delta \mathbf {r} _{i}\right]=\left[\mathbf {r} _{i}-\mathbf {C} \right]}"> </noscript><span class="lazy-image-placeholder" style="width: 16.197ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b83ab9af0efe0f59204a493b10b5eb453594c19" data-alt="{\displaystyle \left[\Delta \mathbf {r} _{i}\right]=\left[\mathbf {r} _{i}-\mathbf {C} \right]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  за да го претставува крстот производ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {r} _{i}-\mathbf {C} )\times }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (\mathbf {r} _{i}-\mathbf {C} )\times } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51f6499f20fd09caa9defcf2a02e5c0226ceab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.291ex; height:2.843ex;" alt="{\displaystyle (\mathbf {r} _{i}-\mathbf {C} )\times }"> </noscript><span class="lazy-image-placeholder" style="width: 10.291ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51f6499f20fd09caa9defcf2a02e5c0226ceab7" data-alt="{\displaystyle (\mathbf {r} _{i}-\mathbf {C} )\times }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , за да се добие <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}=\left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> τ </mi> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\tau }}=\left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b93d7ec560fb15fbaaba36a2fd0869da807fd92e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.366ex; height:7.509ex;" alt="{\displaystyle {\boldsymbol {\tau }}=\left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}}"> </noscript><span class="lazy-image-placeholder" style="width: 53.366ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b93d7ec560fb15fbaaba36a2fd0869da807fd92e" data-alt="{\displaystyle {\boldsymbol {\tau }}=\left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(-\sum _{i=1}^{n}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Пресметката го користи идентитетот <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {r} _{i}\times \left({\boldsymbol {\omega }}\times \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+{\boldsymbol {\omega }}\times \left(\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\times \Delta \mathbf {r} _{i}\right)=0,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Delta \mathbf {r} _{i}\times \left({\boldsymbol {\omega }}\times \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+{\boldsymbol {\omega }}\times \left(\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\times \Delta \mathbf {r} _{i}\right)=0,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff50ef07ccacc8425a7948fc373deac860775a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.053ex; height:2.843ex;" alt="{\displaystyle \Delta \mathbf {r} _{i}\times \left({\boldsymbol {\omega }}\times \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+{\boldsymbol {\omega }}\times \left(\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\times \Delta \mathbf {r} _{i}\right)=0,}"> </noscript><span class="lazy-image-placeholder" style="width: 54.053ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff50ef07ccacc8425a7948fc373deac860775a9" data-alt="{\displaystyle \Delta \mathbf {r} _{i}\times \left({\boldsymbol {\omega }}\times \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+{\boldsymbol {\omega }}\times \left(\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\times \Delta \mathbf {r} _{i}\right)=0,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  добиени од идентитетот на <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%88%D0%B0%D0%BA%D0%BE%D0%B1%D0%B8%D0%B5%D0%B2_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%82%D0%B5%D1%82&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Јакобиев идентитет (страницата не постои)">Јакоби</a> за троен <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%92%D0%BA%D1%80%D1%81%D1%82%D0%B5%D0%BD_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Вкрстен производ (страницата не постои)">вкрстен производ</a> како што е прикажано во доказ подолу: <table class="toccolours collapsible collapsed" style="text-align:left" width="100%"> <tbody> <tr> <th>Доказ</th> </tr> <tr> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}(\mathbf {r_{i}} -\mathbf {R} )\times (m_{i}\mathbf {a} _{i})\\&=\sum _{i=1}^{n}{\boldsymbol {\Delta }}\mathbf {r} _{i}\times (m_{i}\mathbf {a} _{i})\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {a} _{i}]\;\ldots {\text{ векторско скаларно множење }}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times (\mathbf {a} _{{\text{тангенцијално}},i}+\mathbf {a} _{{\text{центрипетално}},i}+\mathbf {A} _{\mathbf {R} })]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times (\mathbf {a} _{{\text{тангенцијално}},i}+\mathbf {a} _{{\text{центрипетално}},i}+0)]\\&\;\;\;\;\;\ldots \;\mathbf {R} {\text{ или е во мирување или се движи со постојана брзина не е забрзано, или }}\\&\;\;\;\;\;\;\;\;\;\;\;{\text{ почетокот на неподвижниот координатен систем е сместен во центарот на масата }}\mathbf {C} \\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {a} _{{\text{тангенцијално}},i}+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {a} _{{\text{центрипетално}},i}]\;\ldots {\text{распределба на векторски производ наспроти собирање}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times \mathbf {v} _{{\text{тангенцијално}},i})]\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\\\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> τ </mi> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold"> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> i </mi> </mrow> </msub> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  векторско скаларно множење  </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> тангенцијално </mtext> </mrow> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> центрипетално </mtext> </mrow> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> тангенцијално </mtext> </mrow> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> центрипетално </mtext> </mrow> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mn> 0 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>  или е во мирување или се движи со постојана брзина не е забрзано, или  </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>  почетокот на неподвижниот координатен систем е сместен во центарот на масата  </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> тангенцијално </mtext> </mrow> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> a </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> центрипетално </mtext> </mrow> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> распределба на векторски производ наспроти собирање </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext> тангенцијално </mtext> </mrow> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> τ </mi> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}(\mathbf {r_{i}} -\mathbf {R} )\times (m_{i}\mathbf {a} _{i})\\&=\sum _{i=1}^{n}{\boldsymbol {\Delta }}\mathbf {r} _{i}\times (m_{i}\mathbf {a} _{i})\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {a} _{i}]\;\ldots {\text{ векторско скаларно множење }}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times (\mathbf {a} _{{\text{тангенцијално}},i}+\mathbf {a} _{{\text{центрипетално}},i}+\mathbf {A} _{\mathbf {R} })]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times (\mathbf {a} _{{\text{тангенцијално}},i}+\mathbf {a} _{{\text{центрипетално}},i}+0)]\\&\;\;\;\;\;\ldots \;\mathbf {R} {\text{ или е во мирување или се движи со постојана брзина не е забрзано, или }}\\&\;\;\;\;\;\;\;\;\;\;\;{\text{ почетокот на неподвижниот координатен систем е сместен во центарот на масата }}\mathbf {C} \\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {a} _{{\text{тангенцијално}},i}+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {a} _{{\text{центрипетално}},i}]\;\ldots {\text{распределба на векторски производ наспроти собирање}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times \mathbf {v} _{{\text{тангенцијално}},i})]\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\\\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936cafe320db33d8134cc71733f9945d31bc7e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -31.222ex; margin-bottom: -0.283ex; width:133.292ex; height:64.176ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}(\mathbf {r_{i}} -\mathbf {R} )\times (m_{i}\mathbf {a} _{i})\\&=\sum _{i=1}^{n}{\boldsymbol {\Delta }}\mathbf {r} _{i}\times (m_{i}\mathbf {a} _{i})\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {a} _{i}]\;\ldots {\text{ векторско скаларно множење }}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times (\mathbf {a} _{{\text{тангенцијално}},i}+\mathbf {a} _{{\text{центрипетално}},i}+\mathbf {A} _{\mathbf {R} })]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times (\mathbf {a} _{{\text{тангенцијално}},i}+\mathbf {a} _{{\text{центрипетално}},i}+0)]\\&\;\;\;\;\;\ldots \;\mathbf {R} {\text{ или е во мирување или се движи со постојана брзина не е забрзано, или }}\\&\;\;\;\;\;\;\;\;\;\;\;{\text{ почетокот на неподвижниот координатен систем е сместен во центарот на масата }}\mathbf {C} \\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {a} _{{\text{тангенцијално}},i}+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {a} _{{\text{центрипетално}},i}]\;\ldots {\text{распределба на векторски производ наспроти собирање}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times \mathbf {v} _{{\text{тангенцијално}},i})]\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\\\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 133.292ex;height: 64.176ex;vertical-align: -31.222ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/936cafe320db33d8134cc71733f9945d31bc7e81" data-alt="{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}(\mathbf {r_{i}} -\mathbf {R} )\times (m_{i}\mathbf {a} _{i})\\&=\sum _{i=1}^{n}{\boldsymbol {\Delta }}\mathbf {r} _{i}\times (m_{i}\mathbf {a} _{i})\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {a} _{i}]\;\ldots {\text{ векторско скаларно множење }}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times (\mathbf {a} _{{\text{тангенцијално}},i}+\mathbf {a} _{{\text{центрипетално}},i}+\mathbf {A} _{\mathbf {R} })]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times (\mathbf {a} _{{\text{тангенцијално}},i}+\mathbf {a} _{{\text{центрипетално}},i}+0)]\\&\;\;\;\;\;\ldots \;\mathbf {R} {\text{ или е во мирување или се движи со постојана брзина не е забрзано, или }}\\&\;\;\;\;\;\;\;\;\;\;\;{\text{ почетокот на неподвижниот координатен систем е сместен во центарот на масата }}\mathbf {C} \\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {a} _{{\text{тангенцијално}},i}+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {a} _{{\text{центрипетално}},i}]\;\ldots {\text{распределба на векторски производ наспроти собирање}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times \mathbf {v} _{{\text{тангенцијално}},i})]\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\\\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Тогаш,нсе користи во последниот израз следниов <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%88%D0%B0%D0%BA%D0%BE%D0%B1%D0%B8%D0%B5%D0%B2_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%82%D0%B5%D1%82&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Јакобиев идентитет (страницата не постои)">Јакобиев идентитет</a>: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}0&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\\&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times -({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\;\ldots {\text{ векторска антикомутативност }}\\&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})]\;\ldots {\text{ скаларано множњење на векторски производ}}\\&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+-[0]\;\ldots {\text{ векторски производ }}\\0&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo> − </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  векторска антикомутативност  </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mo> − </mo> <mo stretchy="false"> [ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  скаларано множњење на векторски производ </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mo> − </mo> <mo stretchy="false"> [ </mo> <mn> 0 </mn> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  векторски производ  </mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}0&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\\&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times -({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\;\ldots {\text{ векторска антикомутативност }}\\&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})]\;\ldots {\text{ скаларано множњење на векторски производ}}\\&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+-[0]\;\ldots {\text{ векторски производ }}\\0&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4b9199f8e544fd69b4dd0db6d732bd5b43eecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:143.041ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}0&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\\&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times -({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\;\ldots {\text{ векторска антикомутативност }}\\&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})]\;\ldots {\text{ скаларано множњење на векторски производ}}\\&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+-[0]\;\ldots {\text{ векторски производ }}\\0&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 143.041ex;height: 17.176ex;vertical-align: -8.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4b9199f8e544fd69b4dd0db6d732bd5b43eecd" data-alt="{\displaystyle {\begin{aligned}0&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\\&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times -({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\;\ldots {\text{ векторска антикомутативност }}\\&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})]\;\ldots {\text{ скаларано множњење на векторски производ}}\\&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+-[0]\;\ldots {\text{ векторски производ }}\\0&={\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl>Резултатот на применетиот <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%88%D0%B0%D0%BA%D0%BE%D0%B1%D0%B8%D0%B5%D0%B2_%D0%B8%D0%B4%D0%B5%D0%BD%D1%82%D0%B8%D1%82%D0%B5%D1%82&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Јакобиев идентитет (страницата не постои)">Јакобиев идентитет</a> може да се запише: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))&=-[{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})]\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\omega }}\cdot ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\;\ldots {\text{ троен векторски производ }}\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot ({\boldsymbol {\omega }}\times {\boldsymbol {\omega }}))]\;\ldots {\text{ троен скаларен производ }}\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot (0))]\;\ldots {\text{ векторски производ}}\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})]\\&=-[{\boldsymbol {\omega }}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\;\ldots {\text{ векторско скаларно множење }}\\&={\boldsymbol {\omega }}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i}))\;\ldots {\text{ векторско скаларно множење }}\\{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))&={\boldsymbol {\omega }}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }}))\;\ldots {\text{ комутативен скаларен производ }}\\\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> </mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <mo stretchy="false"> [ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  троен векторски производ  </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <mo stretchy="false"> [ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  троен скаларен производ  </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <mo stretchy="false"> [ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mo stretchy="false"> ( </mo> <mn> 0 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  векторски производ </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <mo stretchy="false"> [ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  векторско скаларно множење  </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo> − </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  векторско скаларно множење  </mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo> − </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  комутативен скаларен производ  </mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))&=-[{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})]\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\omega }}\cdot ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\;\ldots {\text{ троен векторски производ }}\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot ({\boldsymbol {\omega }}\times {\boldsymbol {\omega }}))]\;\ldots {\text{ троен скаларен производ }}\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot (0))]\;\ldots {\text{ векторски производ}}\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})]\\&=-[{\boldsymbol {\omega }}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\;\ldots {\text{ векторско скаларно множење }}\\&={\boldsymbol {\omega }}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i}))\;\ldots {\text{ векторско скаларно множење }}\\{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))&={\boldsymbol {\omega }}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }}))\;\ldots {\text{ комутативен скаларен производ }}\\\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f28c150fd105a814fcbe762eb77ba8a317196ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.505ex; width:110.62ex; height:28.176ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))&=-[{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})]\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\omega }}\cdot ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\;\ldots {\text{ троен векторски производ }}\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot ({\boldsymbol {\omega }}\times {\boldsymbol {\omega }}))]\;\ldots {\text{ троен скаларен производ }}\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot (0))]\;\ldots {\text{ векторски производ}}\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})]\\&=-[{\boldsymbol {\omega }}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\;\ldots {\text{ векторско скаларно множење }}\\&={\boldsymbol {\omega }}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i}))\;\ldots {\text{ векторско скаларно множење }}\\{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))&={\boldsymbol {\omega }}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }}))\;\ldots {\text{ комутативен скаларен производ }}\\\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 110.62ex;height: 28.176ex;vertical-align: -13.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f28c150fd105a814fcbe762eb77ba8a317196ab" data-alt="{\displaystyle {\begin{aligned}{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))&=-[{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\times {\boldsymbol {\Delta }}\mathbf {r} _{i})]\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\omega }}\cdot ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\;\ldots {\text{ троен векторски производ }}\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot ({\boldsymbol {\omega }}\times {\boldsymbol {\omega }}))]\;\ldots {\text{ троен скаларен производ }}\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot (0))]\;\ldots {\text{ векторски производ}}\\&=-[({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})]\\&=-[{\boldsymbol {\omega }}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\;\ldots {\text{ векторско скаларно множење }}\\&={\boldsymbol {\omega }}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\omega }}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i}))\;\ldots {\text{ векторско скаларно множење }}\\{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))&={\boldsymbol {\omega }}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }}))\;\ldots {\text{ комутативен скаларен производ }}\\\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl>Конечниот резултат може да се замени во главниот доказ како што следи: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }}))]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{0-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{[{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})]-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\;\ldots \;{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})=0\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{[{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})]-{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})\}]\;\ldots {\text{ асоцијативност на собирање }}\\\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> τ </mi> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo> − </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo fence="false" stretchy="false"> { </mo> <mn> 0 </mn> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> <mo stretchy="false"> ] </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo fence="false" stretchy="false"> { </mo> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo fence="false" stretchy="false"> { </mo> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  асоцијативност на собирање  </mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }}))]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{0-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{[{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})]-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\;\ldots \;{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})=0\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{[{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})]-{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})\}]\;\ldots {\text{ асоцијативност на собирање }}\\\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf27c0c711c2504c66a4926d801081ac67d2518" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -17.171ex; width:128.012ex; height:35.509ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }}))]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{0-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{[{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})]-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\;\ldots \;{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})=0\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{[{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})]-{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})\}]\;\ldots {\text{ асоцијативност на собирање }}\\\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 128.012ex;height: 35.509ex;vertical-align: -17.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf27c0c711c2504c66a4926d801081ac67d2518" data-alt="{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i}))]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }}))]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{0-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{[{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})]-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\;\ldots \;{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})=0\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{[{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})]-{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})\}]\;\ldots {\text{ асоцијативност на собирање }}\\\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\quad \quad &=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})]\;\ldots {\text{ распределба на векторски производ наспроти собирање}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\times {\boldsymbol {\omega }})]\;\ldots {\text{ векторско скаларно множење}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})(0)]\;\ldots {\text{ векторски производ}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\}]\;\ldots {\text{ троен векторски производ}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ векторска антикомутитативност }}\\&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ векторско скаларно множење }}\\&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ збирна распределба}}\\{\boldsymbol {\tau }}&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+{\boldsymbol {\omega }}\times -\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})]\;\ldots \;{\boldsymbol {\omega }}{\text{ не е карактеристичен за }}P_{i}\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mspace width="1em"></mspace> <mspace width="1em"></mspace> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo fence="false" stretchy="false"> { </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  распределба на векторски производ наспроти собирање </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo fence="false" stretchy="false"> { </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> <mo> − </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  векторско скаларно множење </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo fence="false" stretchy="false"> { </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> <mo> − </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mn> 0 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  векторски производ </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo fence="false" stretchy="false"> { </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> <mo stretchy="false"> ] </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo fence="false" stretchy="false"> { </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  троен векторски производ </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo> − </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo fence="false" stretchy="false"> { </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo> − </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  векторска антикомутитативност  </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo fence="false" stretchy="false"> { </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  векторско скаларно множење  </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mo> + </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo fence="false" stretchy="false"> { </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  збирна распределба </mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> τ </mi> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>  не е карактеристичен за  </mtext> </mrow> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\quad \quad &=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})]\;\ldots {\text{ распределба на векторски производ наспроти собирање}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\times {\boldsymbol {\omega }})]\;\ldots {\text{ векторско скаларно множење}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})(0)]\;\ldots {\text{ векторски производ}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\}]\;\ldots {\text{ троен векторски производ}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ векторска антикомутитативност }}\\&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ векторско скаларно множење }}\\&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ збирна распределба}}\\{\boldsymbol {\tau }}&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+{\boldsymbol {\omega }}\times -\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})]\;\ldots \;{\boldsymbol {\omega }}{\text{ не е карактеристичен за }}P_{i}\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25103b5d65469de74a3ddff685aaff5639f94e44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -31.082ex; margin-bottom: -0.256ex; width:166.782ex; height:63.843ex;" alt="{\displaystyle {\begin{aligned}\quad \quad &=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})]\;\ldots {\text{ распределба на векторски производ наспроти собирање}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\times {\boldsymbol {\omega }})]\;\ldots {\text{ векторско скаларно множење}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})(0)]\;\ldots {\text{ векторски производ}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\}]\;\ldots {\text{ троен векторски производ}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ векторска антикомутитативност }}\\&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ векторско скаларно множење }}\\&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ збирна распределба}}\\{\boldsymbol {\tau }}&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+{\boldsymbol {\omega }}\times -\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})]\;\ldots \;{\boldsymbol {\omega }}{\text{ не е карактеристичен за }}P_{i}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 166.782ex;height: 63.843ex;vertical-align: -31.082ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25103b5d65469de74a3ddff685aaff5639f94e44" data-alt="{\displaystyle {\begin{aligned}\quad \quad &=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})]\;\ldots {\text{ распределба на векторски производ наспроти собирање}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})({\boldsymbol {\omega }}\times {\boldsymbol {\omega }})]\;\ldots {\text{ векторско скаларно множење}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})(0)]\;\ldots {\text{ векторски производ}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\Delta }}\mathbf {r} _{i})-{\boldsymbol {\Delta }}\mathbf {r} _{i}({\boldsymbol {\Delta }}\mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times {\boldsymbol {\Delta }}\mathbf {r} _{i})\}]\;\ldots {\text{ троен векторски производ}}\\&=\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times -({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ векторска антикомутитативност }}\\&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ векторско скаларно множење }}\\&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\omega }}\times \{{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ збирна распределба}}\\{\boldsymbol {\tau }}&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+{\boldsymbol {\omega }}\times -\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})]\;\ldots \;{\boldsymbol {\omega }}{\text{ не е карактеристичен за }}P_{i}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> Забележете дека за секој вектор <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} \,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c284d16dab51c280ba2288270cbcb7701cd8da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.872ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} \,}"> </noscript><span class="lazy-image-placeholder" style="width: 1.872ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c284d16dab51c280ba2288270cbcb7701cd8da4" data-alt="{\displaystyle \mathbf {u} \,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , ќе важи следново: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {u} )]&=-\sum _{i=1}^{n}m_{i}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}\right)\right)\;\ldots {\text{ векторски производ од множење на матрици}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}{\begin{bmatrix}-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3}\\+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3}\\-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2}\end{bmatrix}}\right)\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-\Delta r_{3,i}(+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3})+\Delta r_{2,i}(-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2})\\+\Delta r_{3,i}(-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3})-\Delta r_{1,i}(-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2})\\-\Delta r_{2,i}(-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3})+\Delta r_{1,i}(+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3})\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-\Delta r_{3,i}^{2}\,u_{1}+\Delta r_{1,i}\Delta r_{3,i}\,u_{3}-\Delta r_{2,i}^{2}\,u_{1}+\Delta r_{1,i}\Delta r_{2,i}\,u_{2}\\-\Delta r_{3,i}^{2}\,u_{2}+\Delta r_{2,i}\Delta r_{3,i}\,u_{3}+\Delta r_{2,i}\Delta r_{1,i}\,u_{1}-\Delta r_{1,i}^{2}\,u_{2}\\+\Delta r_{3,i}\Delta r_{2,i}\,u_{2}-\Delta r_{2,i}^{2}\,u_{3}+\Delta r_{3,i}\Delta r_{1,i}\,u_{1}-\Delta r_{1,i}^{2}\,u_{3}\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-(\Delta r_{2,i}^{2}+\Delta r_{3,i}^{2})\,u_{1}+\Delta r_{1,i}\Delta r_{2,i}\,u_{2}+\Delta r_{1,i}\Delta r_{3,i}\,u_{3}\\+\Delta r_{2,i}\Delta r_{1,i}\,u_{1}-(\Delta r_{1,i}^{2}+\Delta r_{3,i}^{2})\,u_{2}+\Delta r_{2,i}\Delta r_{3,i}\,u_{3}\\+\Delta r_{3,i}\Delta r_{1,i}\,u_{1}+\Delta r_{3,i}\Delta r_{2,i}\,u_{2}-(\Delta r_{1,i}^{2}+\Delta r_{2,i}^{2})\,u_{3}\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-(\Delta r_{2,i}^{2}+\Delta r_{3,i}^{2})&\Delta r_{1,i}\Delta r_{2,i}&\Delta r_{1,i}\Delta r_{3,i}\\\Delta r_{2,i}\Delta r_{1,i}&-(\Delta r_{1,i}^{2}+\Delta r_{3,i}^{2})&\Delta r_{2,i}\Delta r_{3,i}\\\Delta r_{3,i}\Delta r_{1,i}&\Delta r_{3,i}\Delta r_{2,i}&-(\Delta r_{1,i}^{2}+\Delta r_{2,i}^{2})\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}\\&=-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\mathbf {u} \\[6pt]-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {u} )]&=\left(-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right)\mathbf {u} \;\ldots \;\mathbf {u} {\text{ не е карактеристичен за }}P_{i}\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.9em 0.9em 0.9em 0.3em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mn> 0 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mn> 0 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>  векторски производ од множење на матрици </mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mn> 0 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo stretchy="false"> ) </mo> <mspace width="thinmathspace"></mspace> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo stretchy="false"> ) </mo> </mtd> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo stretchy="false"> ) </mo> </mtd> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> </msub> </mtd> <mtd> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <mi mathvariant="normal"> Δ </mi> <msubsup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> <mo> , </mo> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo stretchy="false"> ) </mo> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mspace width="thickmathspace"></mspace> <mo> … </mo> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>  не е карактеристичен за  </mtext> </mrow> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {u} )]&=-\sum _{i=1}^{n}m_{i}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}\right)\right)\;\ldots {\text{ векторски производ од множење на матрици}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}{\begin{bmatrix}-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3}\\+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3}\\-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2}\end{bmatrix}}\right)\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-\Delta r_{3,i}(+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3})+\Delta r_{2,i}(-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2})\\+\Delta r_{3,i}(-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3})-\Delta r_{1,i}(-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2})\\-\Delta r_{2,i}(-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3})+\Delta r_{1,i}(+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3})\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-\Delta r_{3,i}^{2}\,u_{1}+\Delta r_{1,i}\Delta r_{3,i}\,u_{3}-\Delta r_{2,i}^{2}\,u_{1}+\Delta r_{1,i}\Delta r_{2,i}\,u_{2}\\-\Delta r_{3,i}^{2}\,u_{2}+\Delta r_{2,i}\Delta r_{3,i}\,u_{3}+\Delta r_{2,i}\Delta r_{1,i}\,u_{1}-\Delta r_{1,i}^{2}\,u_{2}\\+\Delta r_{3,i}\Delta r_{2,i}\,u_{2}-\Delta r_{2,i}^{2}\,u_{3}+\Delta r_{3,i}\Delta r_{1,i}\,u_{1}-\Delta r_{1,i}^{2}\,u_{3}\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-(\Delta r_{2,i}^{2}+\Delta r_{3,i}^{2})\,u_{1}+\Delta r_{1,i}\Delta r_{2,i}\,u_{2}+\Delta r_{1,i}\Delta r_{3,i}\,u_{3}\\+\Delta r_{2,i}\Delta r_{1,i}\,u_{1}-(\Delta r_{1,i}^{2}+\Delta r_{3,i}^{2})\,u_{2}+\Delta r_{2,i}\Delta r_{3,i}\,u_{3}\\+\Delta r_{3,i}\Delta r_{1,i}\,u_{1}+\Delta r_{3,i}\Delta r_{2,i}\,u_{2}-(\Delta r_{1,i}^{2}+\Delta r_{2,i}^{2})\,u_{3}\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-(\Delta r_{2,i}^{2}+\Delta r_{3,i}^{2})&\Delta r_{1,i}\Delta r_{2,i}&\Delta r_{1,i}\Delta r_{3,i}\\\Delta r_{2,i}\Delta r_{1,i}&-(\Delta r_{1,i}^{2}+\Delta r_{3,i}^{2})&\Delta r_{2,i}\Delta r_{3,i}\\\Delta r_{3,i}\Delta r_{1,i}&\Delta r_{3,i}\Delta r_{2,i}&-(\Delta r_{1,i}^{2}+\Delta r_{2,i}^{2})\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}\\&=-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\mathbf {u} \\[6pt]-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {u} )]&=\left(-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right)\mathbf {u} \;\ldots \;\mathbf {u} {\text{ не е карактеристичен за }}P_{i}\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baf7e9903525ebf48b6977b5b81bfe683e1c73d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -43.171ex; width:170.508ex; height:87.509ex;" alt="{\displaystyle {\begin{aligned}-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {u} )]&=-\sum _{i=1}^{n}m_{i}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}\right)\right)\;\ldots {\text{ векторски производ од множење на матрици}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}{\begin{bmatrix}-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3}\\+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3}\\-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2}\end{bmatrix}}\right)\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-\Delta r_{3,i}(+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3})+\Delta r_{2,i}(-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2})\\+\Delta r_{3,i}(-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3})-\Delta r_{1,i}(-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2})\\-\Delta r_{2,i}(-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3})+\Delta r_{1,i}(+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3})\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-\Delta r_{3,i}^{2}\,u_{1}+\Delta r_{1,i}\Delta r_{3,i}\,u_{3}-\Delta r_{2,i}^{2}\,u_{1}+\Delta r_{1,i}\Delta r_{2,i}\,u_{2}\\-\Delta r_{3,i}^{2}\,u_{2}+\Delta r_{2,i}\Delta r_{3,i}\,u_{3}+\Delta r_{2,i}\Delta r_{1,i}\,u_{1}-\Delta r_{1,i}^{2}\,u_{2}\\+\Delta r_{3,i}\Delta r_{2,i}\,u_{2}-\Delta r_{2,i}^{2}\,u_{3}+\Delta r_{3,i}\Delta r_{1,i}\,u_{1}-\Delta r_{1,i}^{2}\,u_{3}\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-(\Delta r_{2,i}^{2}+\Delta r_{3,i}^{2})\,u_{1}+\Delta r_{1,i}\Delta r_{2,i}\,u_{2}+\Delta r_{1,i}\Delta r_{3,i}\,u_{3}\\+\Delta r_{2,i}\Delta r_{1,i}\,u_{1}-(\Delta r_{1,i}^{2}+\Delta r_{3,i}^{2})\,u_{2}+\Delta r_{2,i}\Delta r_{3,i}\,u_{3}\\+\Delta r_{3,i}\Delta r_{1,i}\,u_{1}+\Delta r_{3,i}\Delta r_{2,i}\,u_{2}-(\Delta r_{1,i}^{2}+\Delta r_{2,i}^{2})\,u_{3}\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-(\Delta r_{2,i}^{2}+\Delta r_{3,i}^{2})&\Delta r_{1,i}\Delta r_{2,i}&\Delta r_{1,i}\Delta r_{3,i}\\\Delta r_{2,i}\Delta r_{1,i}&-(\Delta r_{1,i}^{2}+\Delta r_{3,i}^{2})&\Delta r_{2,i}\Delta r_{3,i}\\\Delta r_{3,i}\Delta r_{1,i}&\Delta r_{3,i}\Delta r_{2,i}&-(\Delta r_{1,i}^{2}+\Delta r_{2,i}^{2})\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}\\&=-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\mathbf {u} \\[6pt]-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {u} )]&=\left(-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right)\mathbf {u} \;\ldots \;\mathbf {u} {\text{ не е карактеристичен за }}P_{i}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 170.508ex;height: 87.509ex;vertical-align: -43.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baf7e9903525ebf48b6977b5b81bfe683e1c73d1" data-alt="{\displaystyle {\begin{aligned}-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {u} )]&=-\sum _{i=1}^{n}m_{i}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}\right)\right)\;\ldots {\text{ векторски производ од множење на матрици}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}{\begin{bmatrix}-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3}\\+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3}\\-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2}\end{bmatrix}}\right)\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-\Delta r_{3,i}(+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3})+\Delta r_{2,i}(-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2})\\+\Delta r_{3,i}(-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3})-\Delta r_{1,i}(-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2})\\-\Delta r_{2,i}(-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3})+\Delta r_{1,i}(+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3})\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-\Delta r_{3,i}^{2}\,u_{1}+\Delta r_{1,i}\Delta r_{3,i}\,u_{3}-\Delta r_{2,i}^{2}\,u_{1}+\Delta r_{1,i}\Delta r_{2,i}\,u_{2}\\-\Delta r_{3,i}^{2}\,u_{2}+\Delta r_{2,i}\Delta r_{3,i}\,u_{3}+\Delta r_{2,i}\Delta r_{1,i}\,u_{1}-\Delta r_{1,i}^{2}\,u_{2}\\+\Delta r_{3,i}\Delta r_{2,i}\,u_{2}-\Delta r_{2,i}^{2}\,u_{3}+\Delta r_{3,i}\Delta r_{1,i}\,u_{1}-\Delta r_{1,i}^{2}\,u_{3}\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-(\Delta r_{2,i}^{2}+\Delta r_{3,i}^{2})\,u_{1}+\Delta r_{1,i}\Delta r_{2,i}\,u_{2}+\Delta r_{1,i}\Delta r_{3,i}\,u_{3}\\+\Delta r_{2,i}\Delta r_{1,i}\,u_{1}-(\Delta r_{1,i}^{2}+\Delta r_{3,i}^{2})\,u_{2}+\Delta r_{2,i}\Delta r_{3,i}\,u_{3}\\+\Delta r_{3,i}\Delta r_{1,i}\,u_{1}+\Delta r_{3,i}\Delta r_{2,i}\,u_{2}-(\Delta r_{1,i}^{2}+\Delta r_{2,i}^{2})\,u_{3}\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-(\Delta r_{2,i}^{2}+\Delta r_{3,i}^{2})&\Delta r_{1,i}\Delta r_{2,i}&\Delta r_{1,i}\Delta r_{3,i}\\\Delta r_{2,i}\Delta r_{1,i}&-(\Delta r_{1,i}^{2}+\Delta r_{3,i}^{2})&\Delta r_{2,i}\Delta r_{3,i}\\\Delta r_{3,i}\Delta r_{1,i}&\Delta r_{3,i}\Delta r_{2,i}&-(\Delta r_{1,i}^{2}+\Delta r_{2,i}^{2})\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}\\&=-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\mathbf {u} \\[6pt]-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times \mathbf {u} )]&=\left(-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right)\mathbf {u} \;\ldots \;\mathbf {u} {\text{ не е карактеристичен за }}P_{i}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl>Конечно, резултатот се користи за комплетирање на главниот доказ во продолжение: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+{\boldsymbol {\omega }}\times -\sum _{i=1}^{n}m_{i}{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})]\\&=\left(-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right){\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right){\boldsymbol {\omega }}\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> τ </mi> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Δ </mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ] </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+{\boldsymbol {\omega }}\times -\sum _{i=1}^{n}m_{i}{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})]\\&=\left(-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right){\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right){\boldsymbol {\omega }}\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0971bbc71e7ceb3c810afa1043b08aca54608486" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:67.042ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+{\boldsymbol {\omega }}\times -\sum _{i=1}^{n}m_{i}{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})]\\&=\left(-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right){\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right){\boldsymbol {\omega }}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 67.042ex;height: 14.509ex;vertical-align: -6.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0971bbc71e7ceb3c810afa1043b08aca54608486" data-alt="{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+{\boldsymbol {\omega }}\times -\sum _{i=1}^{n}m_{i}{\boldsymbol {\Delta }}\mathbf {r} _{i}\times ({\boldsymbol {\Delta }}\mathbf {r} _{i}\times {\boldsymbol {\omega }})]\\&=\left(-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right){\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(-\sum _{i=1}^{n}m_{i}[\Delta r_{i}]^{2}\right){\boldsymbol {\omega }}\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl></td> </tr> </tbody> </table> Така, резултирачкиот вртежен момент на цврстиот систем на честички е даден од <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}=\mathbf {I} _{\mathbf {C} }{\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> τ </mi> </mrow> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> α </mi> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> × </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic"> ω </mi> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\tau }}=\mathbf {I} _{\mathbf {C} }{\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6817a680bf02967f5ab5788badea07e5b205cb42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.174ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\tau }}=\mathbf {I} _{\mathbf {C} }{\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }},}"> </noscript><span class="lazy-image-placeholder" style="width: 21.174ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6817a680bf02967f5ab5788badea07e5b205cb42" data-alt="{\displaystyle {\boldsymbol {\tau }}=\mathbf {I} _{\mathbf {C} }{\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I_{C}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold"> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I_{C}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d3bdba483ab8231b911ab724380a9ff07d66e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.612ex; height:2.509ex;" alt="{\displaystyle \mathbf {I_{C}} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.612ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d3bdba483ab8231b911ab724380a9ff07d66e2" data-alt="{\displaystyle \mathbf {I_{C}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е инерцијалната матрица во однос на центарот на масата. <div class="mw-heading mw-heading3"> <h3 id="Теорема_на_паралелна_оска">Теорема на паралелна оска</h3> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=19&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Теорема на паралелна оска“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r4650192"> <div role="note" class="hatnote navigation-not-searchable"> Главна статија: <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BD%D0%B0_%D0%BF%D0%B0%D1%80%D0%B0%D0%BB%D0%B5%D0%BB%D0%BD%D0%B0_%D0%BE%D1%81%D0%BA%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Теорема на паралелна оска (страницата не постои)">Теорема на паралелна оска</a> </div> Инертната матрица на телото зависи од изборот на референтната точка. Постои корисна врска помеѓу матрицата на инерција во однос на центарот на масата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.931ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" data-alt="{\displaystyle \mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и матрицата на инерција во однос на друга точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Оваа врска се нарекува теорема за паралелна оска.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Marion_1995-3">[3]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kane-6">[6]</a> Размислете за инерцијалната матрица <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I_{R}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold"> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I_{R}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6d262f5376a22cca507b03419910ddd33a5fc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.663ex; height:2.509ex;" alt="{\displaystyle \mathbf {I_{R}} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.663ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6d262f5376a22cca507b03419910ddd33a5fc7" data-alt="{\displaystyle \mathbf {I_{R}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  добиена за крут систем на честички измерен во однос на референтната точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , дадена од <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} _{\mathbf {R} }=-\sum _{i=1}^{n}m_{i}\left[\mathbf {r} _{i}-\mathbf {R} \right]^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mrow> <mo> [ </mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} _{\mathbf {R} }=-\sum _{i=1}^{n}m_{i}\left[\mathbf {r} _{i}-\mathbf {R} \right]^{2}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93e62233b3acff4fc919fc24e8d6241551543087" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.279ex; height:6.843ex;" alt="{\displaystyle \mathbf {I} _{\mathbf {R} }=-\sum _{i=1}^{n}m_{i}\left[\mathbf {r} _{i}-\mathbf {R} \right]^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 24.279ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93e62233b3acff4fc919fc24e8d6241551543087" data-alt="{\displaystyle \mathbf {I} _{\mathbf {R} }=-\sum _{i=1}^{n}m_{i}\left[\mathbf {r} _{i}-\mathbf {R} \right]^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Потоа, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.931ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" data-alt="{\displaystyle \mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  нека биде центар на масата на крутиот систем <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} =(\mathbf {R} -\mathbf {C} )+\mathbf {C} =\mathbf {d} +\mathbf {C} ,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> <mo> = </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} =(\mathbf {R} -\mathbf {C} )+\mathbf {C} =\mathbf {d} +\mathbf {C} ,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3acacbb57b62840187254b0c59ca86dd7baa40fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.46ex; height:2.843ex;" alt="{\displaystyle \mathbf {R} =(\mathbf {R} -\mathbf {C} )+\mathbf {C} =\mathbf {d} +\mathbf {C} ,}"> </noscript><span class="lazy-image-placeholder" style="width: 28.46ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3acacbb57b62840187254b0c59ca86dd7baa40fe" data-alt="{\displaystyle \mathbf {R} =(\mathbf {R} -\mathbf {C} )+\mathbf {C} =\mathbf {d} +\mathbf {C} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {d} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {d} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec3b626fc045b6ff579316e29978fccfed884c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {d} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec3b626fc045b6ff579316e29978fccfed884c2" data-alt="{\displaystyle \mathbf {d} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е векторот од центарот на масата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.931ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" data-alt="{\displaystyle \mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  до референтната точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Користете ја оваа равенка за да ја пресметате матрицата на инерција, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} _{\mathbf {R} }=-\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\left(\mathbf {C} +\mathbf {d} \right)]^{2}=-\sum _{i=1}^{n}m_{i}[\left(\mathbf {r} _{i}-\mathbf {C} \right)-\mathbf {d} ]^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} _{\mathbf {R} }=-\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\left(\mathbf {C} +\mathbf {d} \right)]^{2}=-\sum _{i=1}^{n}m_{i}[\left(\mathbf {r} _{i}-\mathbf {C} \right)-\mathbf {d} ]^{2}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0063409d605348215928372e62f294287f06fb98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.374ex; height:6.843ex;" alt="{\displaystyle \mathbf {I} _{\mathbf {R} }=-\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\left(\mathbf {C} +\mathbf {d} \right)]^{2}=-\sum _{i=1}^{n}m_{i}[\left(\mathbf {r} _{i}-\mathbf {C} \right)-\mathbf {d} ]^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 57.374ex;height: 6.843ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0063409d605348215928372e62f294287f06fb98" data-alt="{\displaystyle \mathbf {I} _{\mathbf {R} }=-\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\left(\mathbf {C} +\mathbf {d} \right)]^{2}=-\sum _{i=1}^{n}m_{i}[\left(\mathbf {r} _{i}-\mathbf {C} \right)-\mathbf {d} ]^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Дистрибуирајте преку вкрстениот производ за да се добие <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} _{\mathbf {R} }=-\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {C} ]^{2}\right)+\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {C} ]\right)[\mathbf {d} ]+[\mathbf {d} ]\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {C} ]\right)-\left(\sum _{i=1}^{n}m_{i}\right)[\mathbf {d} ]^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> <mo> = </mo> <mo> − </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> <mo stretchy="false"> ] </mo> </mrow> <mo> ) </mo> </mrow> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> <mo stretchy="false"> ] </mo> <mo> + </mo> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> <mo stretchy="false"> ] </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> [ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> <mo stretchy="false"> ] </mo> </mrow> <mo> ) </mo> </mrow> <mo> − </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} _{\mathbf {R} }=-\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {C} ]^{2}\right)+\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {C} ]\right)[\mathbf {d} ]+[\mathbf {d} ]\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {C} ]\right)-\left(\sum _{i=1}^{n}m_{i}\right)[\mathbf {d} ]^{2}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68eee7fefe0e716823fd5d80384303b8aef58b99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:93.686ex; height:7.509ex;" alt="{\displaystyle \mathbf {I} _{\mathbf {R} }=-\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {C} ]^{2}\right)+\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {C} ]\right)[\mathbf {d} ]+[\mathbf {d} ]\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {C} ]\right)-\left(\sum _{i=1}^{n}m_{i}\right)[\mathbf {d} ]^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 93.686ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68eee7fefe0e716823fd5d80384303b8aef58b99" data-alt="{\displaystyle \mathbf {I} _{\mathbf {R} }=-\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {C} ]^{2}\right)+\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {C} ]\right)[\mathbf {d} ]+[\mathbf {d} ]\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {C} ]\right)-\left(\sum _{i=1}^{n}m_{i}\right)[\mathbf {d} ]^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Првиот термин е инертната матрица <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I_{C}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold"> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I_{C}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d3bdba483ab8231b911ab724380a9ff07d66e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.612ex; height:2.509ex;" alt="{\displaystyle \mathbf {I_{C}} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.612ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d3bdba483ab8231b911ab724380a9ff07d66e2" data-alt="{\displaystyle \mathbf {I_{C}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  во однос на центарот на масата. Вториот и третиот термин се нула по дефиниција на центарот на масата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.931ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" data-alt="{\displaystyle \mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . И последниот термин е вкупната маса на системот помножена со квадратот на кососиметричната матрица <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbf {d} ]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle [\mathbf {d} ]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5785dcf9605fb6ace7c1a8c8b4d6b365af48366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.779ex; height:2.843ex;" alt="{\displaystyle [\mathbf {d} ]}"> </noscript><span class="lazy-image-placeholder" style="width: 2.779ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5785dcf9605fb6ace7c1a8c8b4d6b365af48366" data-alt="{\displaystyle [\mathbf {d} ]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  конструирана од <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {d} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {d} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec3b626fc045b6ff579316e29978fccfed884c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {d} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec3b626fc045b6ff579316e29978fccfed884c2" data-alt="{\displaystyle \mathbf {d} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Резултатот е теоремата за паралелна оска, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} _{\mathbf {R} }=\mathbf {I} _{\mathbf {C} }-M[\mathbf {d} ]^{2},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mo> − </mo> <mi> M </mi> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} _{\mathbf {R} }=\mathbf {I} _{\mathbf {C} }-M[\mathbf {d} ]^{2},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c59b0848c4b1d69fff75c4257005225cd42c74d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.135ex; height:3.176ex;" alt="{\displaystyle \mathbf {I} _{\mathbf {R} }=\mathbf {I} _{\mathbf {C} }-M[\mathbf {d} ]^{2},}"> </noscript><span class="lazy-image-placeholder" style="width: 18.135ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c59b0848c4b1d69fff75c4257005225cd42c74d" data-alt="{\displaystyle \mathbf {I} _{\mathbf {R} }=\mathbf {I} _{\mathbf {C} }-M[\mathbf {d} ]^{2},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {d} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> d </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {d} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec3b626fc045b6ff579316e29978fccfed884c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {d} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec3b626fc045b6ff579316e29978fccfed884c2" data-alt="{\displaystyle \mathbf {d} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е векторот од центарот на масата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {C} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.931ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" data-alt="{\displaystyle \mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  до референтната точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Забелешка за знакот минус: Со користење на симетричната матрица на пресек на положбени места во однос на референтната точка, матрицата на инерција на секоја честичка има облик <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -m\left[\mathbf {r} \right]^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <mi> m </mi> <msup> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -m\left[\mathbf {r} \right]^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b3ed54432828b43d6ae23e9a2e9c009c7336024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.299ex; height:3.343ex;" alt="{\displaystyle -m\left[\mathbf {r} \right]^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 7.299ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b3ed54432828b43d6ae23e9a2e9c009c7336024" data-alt="{\displaystyle -m\left[\mathbf {r} \right]^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , кој е сличен на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mr^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle mr^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd9d0ea2911509b014b72a7b536acb7376cb455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.143ex; height:2.676ex;" alt="{\displaystyle mr^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.143ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd9d0ea2911509b014b72a7b536acb7376cb455" data-alt="{\displaystyle mr^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  кој се појавува при рамно движење. Сепак, за да се направи ова правилно, потребно е знак минус. Овој знак за минус може да се апсорбира во терминот <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\left[\mathbf {r} \right]^{\mathsf {T}}\left[\mathbf {r} \right]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <msup> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo> ] </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m\left[\mathbf {r} \right]^{\mathsf {T}}\left[\mathbf {r} \right]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27fed5e06e85cd3d74847c333b7a32cbd6a31fe7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.57ex; height:3.343ex;" alt="{\displaystyle m\left[\mathbf {r} \right]^{\mathsf {T}}\left[\mathbf {r} \right]}"> </noscript><span class="lazy-image-placeholder" style="width: 8.57ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27fed5e06e85cd3d74847c333b7a32cbd6a31fe7" data-alt="{\displaystyle m\left[\mathbf {r} \right]^{\mathsf {T}}\left[\mathbf {r} \right]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , по желба, со користење на својството на skew-symmetry на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbf {r} ]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle [\mathbf {r} ]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba4c4e0b7a62e35392117d1331b60c9e3276b5c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.396ex; height:2.843ex;" alt="{\displaystyle [\mathbf {r} ]}"> </noscript><span class="lazy-image-placeholder" style="width: 2.396ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba4c4e0b7a62e35392117d1331b60c9e3276b5c9" data-alt="{\displaystyle [\mathbf {r} ]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . <div class="mw-heading mw-heading3"> <h3 id="Скаларен_момент_на_инерција_во_рамнина"><span id=".D0.A1.D0.BA.D0.B0.D0.BB.D0.B0.D1.80.D0.B5.D0.BD_.D0.BC.D0.BE.D0.BC.D0.B5.D0.BD.D1.82_.D0.BD.D0.B0_.D0.B8.D0.BD.D0.B5.D1.80.D1.86.D0.B8.D1.98.D0.B0_.D0.B2.D0.BE_.D1.80.D0.B0.D0.BC.D0.BD.D0.B8.D0.BD.D0.B0">Скаларен момент на инерција во рамнина</h3> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=20&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Скаларен момент на инерција во рамнина“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> Скаларен момент на инерција, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{L}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> L </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{L}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f1b57d141754c4d3852d6d0bd08d2e16166630c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.375ex; height:2.509ex;" alt="{\displaystyle I_{L}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.375ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f1b57d141754c4d3852d6d0bd08d2e16166630c" data-alt="{\displaystyle I_{L}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , на тело околу одредена оска чија насока е одредена од единечниот вектор <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {k}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {k}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.843ex;" alt="{\displaystyle \mathbf {\hat {k}} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" data-alt="{\displaystyle \mathbf {\hat {k}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и поминува низ телото во точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е како што следува:<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kane-6">[6]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{L}=\mathbf {\hat {k}} \cdot \left(-\sum _{i=1}^{N}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right)\mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} =\mathbf {\hat {k}} ^{\mathsf {T}}\mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} ,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> L </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{L}=\mathbf {\hat {k}} \cdot \left(-\sum _{i=1}^{N}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right)\mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} =\mathbf {\hat {k}} ^{\mathsf {T}}\mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} ,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78bb7c062959fd615347939341b32bf66de221c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:49.849ex; height:7.509ex;" alt="{\displaystyle I_{L}=\mathbf {\hat {k}} \cdot \left(-\sum _{i=1}^{N}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right)\mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} =\mathbf {\hat {k}} ^{\mathsf {T}}\mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} ,}"> </noscript><span class="lazy-image-placeholder" style="width: 49.849ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78bb7c062959fd615347939341b32bf66de221c6" data-alt="{\displaystyle I_{L}=\mathbf {\hat {k}} \cdot \left(-\sum _{i=1}^{N}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right)\mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} =\mathbf {\hat {k}} ^{\mathsf {T}}\mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I_{R}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold"> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I_{R}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6d262f5376a22cca507b03419910ddd33a5fc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.663ex; height:2.509ex;" alt="{\displaystyle \mathbf {I_{R}} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.663ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6d262f5376a22cca507b03419910ddd33a5fc7" data-alt="{\displaystyle \mathbf {I_{R}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е момент на инерција матрица на системот во однос на референтната точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , и <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\Delta \mathbf {r} _{i}]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> [ </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo stretchy="false"> ] </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle [\Delta \mathbf {r} _{i}]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee5960e77611f673ab2f2050f356dbf0e403132" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.131ex; height:2.843ex;" alt="{\displaystyle [\Delta \mathbf {r} _{i}]}"> </noscript><span class="lazy-image-placeholder" style="width: 5.131ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee5960e77611f673ab2f2050f356dbf0e403132" data-alt="{\displaystyle [\Delta \mathbf {r} _{i}]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е косо симетрична матрица добиена од векторот <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9b70b1347bdd9d18f140f45ea7e4db3666d38a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.681ex; height:2.509ex;" alt="{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 13.681ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9b70b1347bdd9d18f140f45ea7e4db3666d38a" data-alt="{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Ова е изведено на следниов начин. Нека круто собрание на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  честички, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i},i=1,...,n}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mo> . </mo> <mo> . </mo> <mo> . </mo> <mo> , </mo> <mi> n </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{i},i=1,...,n} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda9b5e40922a7dc3ee07ea0859c6391fcdbd4ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.954ex; height:2.509ex;" alt="{\displaystyle P_{i},i=1,...,n}"> </noscript><span class="lazy-image-placeholder" style="width: 14.954ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda9b5e40922a7dc3ee07ea0859c6391fcdbd4ee" data-alt="{\displaystyle P_{i},i=1,...,n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  , имаат координати <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {r} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.902ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" data-alt="{\displaystyle \mathbf {r} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Изберете <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  како референтна точка и пресметајте го моментот на инерција околу линијата L дефинирана од единечниот вектор k преку референтната точка R, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} (t)=\mathbf {R} +t\mathbf {\hat {k}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> L </mi> </mrow> <mo stretchy="false"> ( </mo> <mi> t </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> <mo> + </mo> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {L} (t)=\mathbf {R} +t\mathbf {\hat {k}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7d7d2c2d4f498f8ec004f0a3efa2f82f833893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.45ex; height:3.343ex;" alt="{\displaystyle \mathbf {L} (t)=\mathbf {R} +t\mathbf {\hat {k}} }"> </noscript><span class="lazy-image-placeholder" style="width: 14.45ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7d7d2c2d4f498f8ec004f0a3efa2f82f833893" data-alt="{\displaystyle \mathbf {L} (t)=\mathbf {R} +t\mathbf {\hat {k}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Вертикалниот вектор од оваа линија до честичката <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.292ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3" data-alt="{\displaystyle P_{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е добиен од <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {r} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Delta \mathbf {r} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ab5ca8f4f58653e5a4ec0e9571b9c5e1d76b74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.838ex; height:2.509ex;" alt="{\displaystyle \Delta \mathbf {r} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.838ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ab5ca8f4f58653e5a4ec0e9571b9c5e1d76b74" data-alt="{\displaystyle \Delta \mathbf {r} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  со отстранување на компонентата која проектира врз <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {k}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {k}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.843ex;" alt="{\displaystyle \mathbf {\hat {k}} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" data-alt="{\displaystyle \mathbf {\hat {k}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {r} _{i}^{\perp }=\Delta \mathbf {r} _{i}-\left(\mathbf {\hat {k}} \cdot \Delta \mathbf {r} _{i}\right)\mathbf {\hat {k}} =\left(\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}\right)\Delta \mathbf {r} _{i},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Δ </mi> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> ⊥ </mo> </mrow> </msubsup> <mo> = </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> − </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Delta \mathbf {r} _{i}^{\perp }=\Delta \mathbf {r} _{i}-\left(\mathbf {\hat {k}} \cdot \Delta \mathbf {r} _{i}\right)\mathbf {\hat {k}} =\left(\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}\right)\Delta \mathbf {r} _{i},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a67ee90abc7d893471fed7c8d37bdb9b73c8a967" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.343ex; height:4.843ex;" alt="{\displaystyle \Delta \mathbf {r} _{i}^{\perp }=\Delta \mathbf {r} _{i}-\left(\mathbf {\hat {k}} \cdot \Delta \mathbf {r} _{i}\right)\mathbf {\hat {k}} =\left(\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}\right)\Delta \mathbf {r} _{i},}"> </noscript><span class="lazy-image-placeholder" style="width: 45.343ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a67ee90abc7d893471fed7c8d37bdb9b73c8a967" data-alt="{\displaystyle \Delta \mathbf {r} _{i}^{\perp }=\Delta \mathbf {r} _{i}-\left(\mathbf {\hat {k}} \cdot \Delta \mathbf {r} _{i}\right)\mathbf {\hat {k}} =\left(\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}\right)\Delta \mathbf {r} _{i},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {E} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.757ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" data-alt="{\displaystyle \mathbf {E} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> е идентитет матрица, со цел да се избегне забуна со инерција матрица, и <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96d0817ff96d9ee9243ddc37f688e5385c0b406e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.173ex; height:3.343ex;" alt="{\displaystyle \mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.173ex;height: 3.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96d0817ff96d9ee9243ddc37f688e5385c0b406e" data-alt="{\displaystyle \mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> е матрица на надворешен производ формирана од единечниот вектор <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {k}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {k}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.843ex;" alt="{\displaystyle \mathbf {\hat {k}} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" data-alt="{\displaystyle \mathbf {\hat {k}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  по должината на линијата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> </noscript><span class="lazy-image-placeholder" style="width: 1.583ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" data-alt="{\displaystyle L}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . За да го поврзе овој скаларен момент на инерција со инерцијалната матрица на телото, воведејте ја коси-симетричната матрица <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\mathbf {\hat {k}} \right]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> ] </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left[\mathbf {\hat {k}} \right]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a515c7495954477be07dee4f29d4c4c4ec50cf98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.606ex; height:4.843ex;" alt="{\displaystyle \left[\mathbf {\hat {k}} \right]}"> </noscript><span class="lazy-image-placeholder" style="width: 3.606ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a515c7495954477be07dee4f29d4c4c4ec50cf98" data-alt="{\displaystyle \left[\mathbf {\hat {k}} \right]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  така што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\mathbf {\hat {k}} \right]\mathbf {y} =\mathbf {\hat {k}} \times \mathbf {y} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> y </mi> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> y </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left[\mathbf {\hat {k}} \right]\mathbf {y} =\mathbf {\hat {k}} \times \mathbf {y} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d1281a399f50084c4d71e516f5d56b1cf5426f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.165ex; height:4.843ex;" alt="{\displaystyle \left[\mathbf {\hat {k}} \right]\mathbf {y} =\mathbf {\hat {k}} \times \mathbf {y} }"> </noscript><span class="lazy-image-placeholder" style="width: 14.165ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d1281a399f50084c4d71e516f5d56b1cf5426f" data-alt="{\displaystyle \left[\mathbf {\hat {k}} \right]\mathbf {y} =\mathbf {\hat {k}} \times \mathbf {y} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , тогаш имаме идентитет <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\left[\mathbf {\hat {k}} \right]^{2}\equiv \left|\mathbf {\hat {k}} \right|^{2}\left(\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}\right)=\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <msup> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> ≡ </mo> <msup> <mrow> <mo> | </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -\left[\mathbf {\hat {k}} \right]^{2}\equiv \left|\mathbf {\hat {k}} \right|^{2}\left(\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}\right)=\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92be0b3adbb50fbe61b31dc0237f0fc102476a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.775ex; height:5.176ex;" alt="{\displaystyle -\left[\mathbf {\hat {k}} \right]^{2}\equiv \left|\mathbf {\hat {k}} \right|^{2}\left(\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}\right)=\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}},}"> </noscript><span class="lazy-image-placeholder" style="width: 37.775ex;height: 5.176ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92be0b3adbb50fbe61b31dc0237f0fc102476a0" data-alt="{\displaystyle -\left[\mathbf {\hat {k}} \right]^{2}\equiv \left|\mathbf {\hat {k}} \right|^{2}\left(\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}\right)=\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  истакнувајќи дека <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {k}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {k}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.843ex;" alt="{\displaystyle \mathbf {\hat {k}} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" data-alt="{\displaystyle \mathbf {\hat {k}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е единица вектор. Квадрираната големина од нормалниот вектор е <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left|\Delta \mathbf {r} _{i}^{\perp }\right|^{2}&=\left(-\left[\mathbf {\hat {k}} \right]^{2}\Delta \mathbf {r} _{i}\right)\cdot \left(-\left[\mathbf {\hat {k}} \right]^{2}\Delta \mathbf {r} _{i}\right)\\&=\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow> <mo> | </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> ⊥ </mo> </mrow> </msubsup> </mrow> <mo> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <msup> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <msup> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\left|\Delta \mathbf {r} _{i}^{\perp }\right|^{2}&=\left(-\left[\mathbf {\hat {k}} \right]^{2}\Delta \mathbf {r} _{i}\right)\cdot \left(-\left[\mathbf {\hat {k}} \right]^{2}\Delta \mathbf {r} _{i}\right)\\&=\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d205a6da935ecfa8f5e13feb1a930f1d032022a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:48.208ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\left|\Delta \mathbf {r} _{i}^{\perp }\right|^{2}&=\left(-\left[\mathbf {\hat {k}} \right]^{2}\Delta \mathbf {r} _{i}\right)\cdot \left(-\left[\mathbf {\hat {k}} \right]^{2}\Delta \mathbf {r} _{i}\right)\\&=\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 48.208ex;height: 10.843ex;vertical-align: -4.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d205a6da935ecfa8f5e13feb1a930f1d032022a" data-alt="{\displaystyle {\begin{aligned}\left|\Delta \mathbf {r} _{i}^{\perp }\right|^{2}&=\left(-\left[\mathbf {\hat {k}} \right]^{2}\Delta \mathbf {r} _{i}\right)\cdot \left(-\left[\mathbf {\hat {k}} \right]^{2}\Delta \mathbf {r} _{i}\right)\\&=\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Поедноставувањето на оваа равенка го користиме идентитетот на тројниот скаларен производ <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\equiv \left(\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\times \mathbf {\hat {k}} \right)\cdot \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ≡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\equiv \left(\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\times \mathbf {\hat {k}} \right)\cdot \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right),} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8db6d7de03a2fc68aaec686046142b6d435307" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:79.056ex; height:4.843ex;" alt="{\displaystyle \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\equiv \left(\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\times \mathbf {\hat {k}} \right)\cdot \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right),}"> </noscript><span class="lazy-image-placeholder" style="width: 79.056ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8db6d7de03a2fc68aaec686046142b6d435307" data-alt="{\displaystyle \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\equiv \left(\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\times \mathbf {\hat {k}} \right)\cdot \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> каде што точки и крстот производи се разменуваат. Разменување на производи и поедноставување со забележување дека <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {r} _{i}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Delta \mathbf {r} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ab5ca8f4f58653e5a4ec0e9571b9c5e1d76b74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.838ex; height:2.509ex;" alt="{\displaystyle \Delta \mathbf {r} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.838ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ab5ca8f4f58653e5a4ec0e9571b9c5e1d76b74" data-alt="{\displaystyle \Delta \mathbf {r} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {k}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {k}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.843ex;" alt="{\displaystyle \mathbf {\hat {k}} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" data-alt="{\displaystyle \mathbf {\hat {k}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  се ортогонални: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\\={}&\left(\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\times \mathbf {\hat {k}} \right)\cdot \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\\={}&\left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\cdot \left(-\Delta \mathbf {r} _{i}\times \mathbf {\hat {k}} \right)\\={}&-\mathbf {\hat {k}} \cdot \left(\Delta \mathbf {r} _{i}\times \Delta \mathbf {r} _{i}\times \mathbf {\hat {k}} \right)\\={}&-\mathbf {\hat {k}} \cdot \left[\Delta \mathbf {r} _{i}\right]^{2}\mathbf {\hat {k}} .\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd></mtd> <mtd> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi></mi> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mi></mi> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <msup> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}&\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\\={}&\left(\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\times \mathbf {\hat {k}} \right)\cdot \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\\={}&\left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\cdot \left(-\Delta \mathbf {r} _{i}\times \mathbf {\hat {k}} \right)\\={}&-\mathbf {\hat {k}} \cdot \left(\Delta \mathbf {r} _{i}\times \Delta \mathbf {r} _{i}\times \mathbf {\hat {k}} \right)\\={}&-\mathbf {\hat {k}} \cdot \left[\Delta \mathbf {r} _{i}\right]^{2}\mathbf {\hat {k}} .\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62111ff66a6922e2705a7eb60478088077b7d9bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.838ex; width:40.667ex; height:22.843ex;" alt="{\displaystyle {\begin{aligned}&\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\\={}&\left(\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\times \mathbf {\hat {k}} \right)\cdot \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\\={}&\left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\cdot \left(-\Delta \mathbf {r} _{i}\times \mathbf {\hat {k}} \right)\\={}&-\mathbf {\hat {k}} \cdot \left(\Delta \mathbf {r} _{i}\times \Delta \mathbf {r} _{i}\times \mathbf {\hat {k}} \right)\\={}&-\mathbf {\hat {k}} \cdot \left[\Delta \mathbf {r} _{i}\right]^{2}\mathbf {\hat {k}} .\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 40.667ex;height: 22.843ex;vertical-align: -10.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62111ff66a6922e2705a7eb60478088077b7d9bf" data-alt="{\displaystyle {\begin{aligned}&\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\\={}&\left(\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\times \mathbf {\hat {k}} \right)\cdot \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\\={}&\left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\cdot \left(-\Delta \mathbf {r} _{i}\times \mathbf {\hat {k}} \right)\\={}&-\mathbf {\hat {k}} \cdot \left(\Delta \mathbf {r} _{i}\times \Delta \mathbf {r} _{i}\times \mathbf {\hat {k}} \right)\\={}&-\mathbf {\hat {k}} \cdot \left[\Delta \mathbf {r} _{i}\right]^{2}\mathbf {\hat {k}} .\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Така, моментот на инерција околу линијата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> </noscript><span class="lazy-image-placeholder" style="width: 1.583ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" data-alt="{\displaystyle L}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  преку <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  во правецот <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {k}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {\hat {k}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.843ex;" alt="{\displaystyle \mathbf {\hat {k}} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 2.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" data-alt="{\displaystyle \mathbf {\hat {k}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  се добива од пресметката <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}I_{L}&=\sum _{i=1}^{N}m_{i}\left|\Delta \mathbf {r} _{i}^{\perp }\right|^{2}\\&=-\sum _{i=1}^{N}m_{i}\mathbf {\hat {k}} \cdot \left[\Delta \mathbf {r} _{i}\right]^{2}\mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \left(-\sum _{i=1}^{N}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right)\mathbf {\hat {k}} \\&=\mathbf {\hat {k}} \cdot \mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} =\mathbf {\hat {k}} ^{\mathsf {T}}\mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} ,\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> L </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mrow> <mo> | </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> ⊥ </mo> </mrow> </msubsup> </mrow> <mo> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <msup> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <msup> <mrow> <mo> [ </mo> <mrow> <mi mathvariant="normal"> Δ </mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> </mrow> <mo> ] </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> = </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold"> k </mi> <mo mathvariant="bold" stretchy="false"> ^ </mo> </mover> </mrow> </mrow> <mo> , </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}I_{L}&=\sum _{i=1}^{N}m_{i}\left|\Delta \mathbf {r} _{i}^{\perp }\right|^{2}\\&=-\sum _{i=1}^{N}m_{i}\mathbf {\hat {k}} \cdot \left[\Delta \mathbf {r} _{i}\right]^{2}\mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \left(-\sum _{i=1}^{N}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right)\mathbf {\hat {k}} \\&=\mathbf {\hat {k}} \cdot \mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} =\mathbf {\hat {k}} ^{\mathsf {T}}\mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} ,\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4a4a61db87467a4b76294e68bba221a2810cdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:52.319ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}I_{L}&=\sum _{i=1}^{N}m_{i}\left|\Delta \mathbf {r} _{i}^{\perp }\right|^{2}\\&=-\sum _{i=1}^{N}m_{i}\mathbf {\hat {k}} \cdot \left[\Delta \mathbf {r} _{i}\right]^{2}\mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \left(-\sum _{i=1}^{N}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right)\mathbf {\hat {k}} \\&=\mathbf {\hat {k}} \cdot \mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} =\mathbf {\hat {k}} ^{\mathsf {T}}\mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} ,\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 52.319ex;height: 18.843ex;vertical-align: -8.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4a4a61db87467a4b76294e68bba221a2810cdf" data-alt="{\displaystyle {\begin{aligned}I_{L}&=\sum _{i=1}^{N}m_{i}\left|\Delta \mathbf {r} _{i}^{\perp }\right|^{2}\\&=-\sum _{i=1}^{N}m_{i}\mathbf {\hat {k}} \cdot \left[\Delta \mathbf {r} _{i}\right]^{2}\mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \left(-\sum _{i=1}^{N}m_{i}\left[\Delta \mathbf {r} _{i}\right]^{2}\right)\mathbf {\hat {k}} \\&=\mathbf {\hat {k}} \cdot \mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} =\mathbf {\hat {k}} ^{\mathsf {T}}\mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} ,\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I_{R}} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold"> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I_{R}} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6d262f5376a22cca507b03419910ddd33a5fc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.663ex; height:2.509ex;" alt="{\displaystyle \mathbf {I_{R}} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.663ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6d262f5376a22cca507b03419910ddd33a5fc7" data-alt="{\displaystyle \mathbf {I_{R}} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е момент на инерција матрица на системот во однос на референтната точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> R </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {R} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Ова покажува дека матрицата на инерција може да се користи за пресметување на моментот на инерција на телото околу одредената оска на вртење во телото. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <h2 id="Инертен_тензор">Инертен тензор</h2> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=21&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Инертен тензор“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> Инертната матрица често се опишува како инертен тензор, кој се состои од истите моменти на инерција и инерцијални производи за трите координатни оски.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kane-6">[6]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Goldstein-23">[23]</a> Тензорот на инерција е конструиран од тензорите од девет компоненти, (симболот ⊗ е <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A2%D0%B5%D0%BD%D0%B7%D0%BE%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Тензорски производ (страницата не постои)">тензорски производ</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{i}\otimes \mathbf {e} _{j},\quad i,j=1,2,3,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⊗ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> , </mo> <mspace width="1em"></mspace> <mi> i </mi> <mo> , </mo> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {e} _{i}\otimes \mathbf {e} _{j},\quad i,j=1,2,3,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abedc58b6d80fdaedc4cd96484ba32992f406fc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.452ex; height:2.843ex;" alt="{\displaystyle \mathbf {e} _{i}\otimes \mathbf {e} _{j},\quad i,j=1,2,3,}"> </noscript><span class="lazy-image-placeholder" style="width: 22.452ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abedc58b6d80fdaedc4cd96484ba32992f406fc8" data-alt="{\displaystyle \mathbf {e} _{i}\otimes \mathbf {e} _{j},\quad i,j=1,2,3,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{i},i=1,2,3}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> , </mo> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mn> 3 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {e} _{i},i=1,2,3} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed1a7885826cf42b57c957d7fd0eded82001a12f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.515ex; height:2.509ex;" alt="{\displaystyle \mathbf {e} _{i},i=1,2,3}"> </noscript><span class="lazy-image-placeholder" style="width: 12.515ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed1a7885826cf42b57c957d7fd0eded82001a12f" data-alt="{\displaystyle \mathbf {e} _{i},i=1,2,3}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  се трите ортогонални <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%95%D0%B4%D0%B8%D0%BD%D0%B5%D1%87%D0%B5%D0%BD_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Единечен вектор (страницата не постои)">единечни вектори</a> кои ја дефинираат инерцијалната рамка во која телото се движи. Користејќи ја оваа основата на тензорот на инерција е даден од <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} =\sum _{i=1}^{3}\sum _{j=1}^{3}I_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </munderover> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mi> j </mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> ⊗ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} =\sum _{i=1}^{3}\sum _{j=1}^{3}I_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54f95e3ec2b2dcb269cf41160c289586ccd8b7d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:21.744ex; height:7.509ex;" alt="{\displaystyle \mathbf {I} =\sum _{i=1}^{3}\sum _{j=1}^{3}I_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}.}"> </noscript><span class="lazy-image-placeholder" style="width: 21.744ex;height: 7.509ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54f95e3ec2b2dcb269cf41160c289586ccd8b7d7" data-alt="{\displaystyle \mathbf {I} =\sum _{i=1}^{3}\sum _{j=1}^{3}I_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Овој тензор е од степен два, бидејќи тензорите на компонентите се конструирани од два базични вектори. Во оваа форма тензијата на инерција е исто така наречена инерцијален бинор. За крут систем на честички <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{k},k=1,...,N}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> P </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> , </mo> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mo> . </mo> <mo> . </mo> <mo> . </mo> <mo> , </mo> <mi> N </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle P_{k},k=1,...,N} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/796766aba31837e10b56b98ff9696f9fc155f651" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.32ex; height:2.509ex;" alt="{\displaystyle P_{k},k=1,...,N}"> </noscript><span class="lazy-image-placeholder" style="width: 16.32ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/796766aba31837e10b56b98ff9696f9fc155f651" data-alt="{\displaystyle P_{k},k=1,...,N}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  секоја маса <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m_{k}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5e3b1ba705c5adda8381cb22580d26d64968f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.129ex; height:2.009ex;" alt="{\displaystyle m_{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.129ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5e3b1ba705c5adda8381cb22580d26d64968f8" data-alt="{\displaystyle m_{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  со координати на положбата <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{k}=(x_{k},y_{k},z_{k})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> = </mo> <mo stretchy="false"> ( </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> , </mo> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {r} _{k}=(x_{k},y_{k},z_{k})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8bac0427cf943e3aa14cf19d70496b00389c654" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.983ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} _{k}=(x_{k},y_{k},z_{k})}"> </noscript><span class="lazy-image-placeholder" style="width: 15.983ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8bac0427cf943e3aa14cf19d70496b00389c654" data-alt="{\displaystyle \mathbf {r} _{k}=(x_{k},y_{k},z_{k})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , инертен тензорот е даден со <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} =\sum _{k=1}^{N}m_{k}\left(\left(\mathbf {r} _{k}\cdot \mathbf {r} _{k}\right)\mathbf {E} -\mathbf {r} _{k}\otimes \mathbf {r} _{k}\right),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> <mo> − </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> ⊗ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} =\sum _{k=1}^{N}m_{k}\left(\left(\mathbf {r} _{k}\cdot \mathbf {r} _{k}\right)\mathbf {E} -\mathbf {r} _{k}\otimes \mathbf {r} _{k}\right),} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d30664ba2571d8121f16d0debdf0c27e97253d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.29ex; height:7.343ex;" alt="{\displaystyle \mathbf {I} =\sum _{k=1}^{N}m_{k}\left(\left(\mathbf {r} _{k}\cdot \mathbf {r} _{k}\right)\mathbf {E} -\mathbf {r} _{k}\otimes \mathbf {r} _{k}\right),}"> </noscript><span class="lazy-image-placeholder" style="width: 34.29ex;height: 7.343ex;vertical-align: -3.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d30664ba2571d8121f16d0debdf0c27e97253d" data-alt="{\displaystyle \mathbf {I} =\sum _{k=1}^{N}m_{k}\left(\left(\mathbf {r} _{k}\cdot \mathbf {r} _{k}\right)\mathbf {E} -\mathbf {r} _{k}\otimes \mathbf {r} _{k}\right),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {E} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.757ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" data-alt="{\displaystyle \mathbf {E} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> е тензор на идентитетот <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} =\mathbf {e} _{1}\otimes \mathbf {e} _{1}+\mathbf {e} _{2}\otimes \mathbf {e} _{2}+\mathbf {e} _{3}\otimes \mathbf {e} _{3}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> ⊗ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> ⊗ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> + </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> ⊗ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {E} =\mathbf {e} _{1}\otimes \mathbf {e} _{1}+\mathbf {e} _{2}\otimes \mathbf {e} _{2}+\mathbf {e} _{3}\otimes \mathbf {e} _{3}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62bc98498bb4135b7886953bc34f162140934984" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.381ex; height:2.509ex;" alt="{\displaystyle \mathbf {E} =\mathbf {e} _{1}\otimes \mathbf {e} _{1}+\mathbf {e} _{2}\otimes \mathbf {e} _{2}+\mathbf {e} _{3}\otimes \mathbf {e} _{3}.}"> </noscript><span class="lazy-image-placeholder" style="width: 33.381ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62bc98498bb4135b7886953bc34f162140934984" data-alt="{\displaystyle \mathbf {E} =\mathbf {e} _{1}\otimes \mathbf {e} _{1}+\mathbf {e} _{2}\otimes \mathbf {e} _{2}+\mathbf {e} _{3}\otimes \mathbf {e} _{3}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Во овој случај, компонентите на инертниот тензор се дадени со <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}I_{11}=I_{xx}&=\sum _{k=1}^{N}m_{k}\left(y_{k}^{2}+z_{k}^{2}\right),\\I_{22}=I_{yy}&=\sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+z_{k}^{2}\right),\\I_{33}=I_{zz}&=\sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+y_{k}^{2}\right),\\I_{12}=I_{21}=I_{xy}&=-\sum _{k=1}^{N}m_{k}x_{k}y_{k},\\I_{13}=I_{31}=I_{xz}&=-\sum _{k=1}^{N}m_{k}x_{k}z_{k},\\I_{23}=I_{32}=I_{yz}&=-\sum _{k=1}^{N}m_{k}y_{k}z_{k}.\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 11 </mn> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> x </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <msubsup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 22 </mn> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> y </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <msubsup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 33 </mn> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> <mi> z </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> <mo> + </mo> <msubsup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 12 </mn> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 21 </mn> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> y </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 13 </mn> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 31 </mn> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> z </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 23 </mn> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 32 </mn> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> z </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mo> − </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> N </mi> </mrow> </munderover> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <msub> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <msub> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}I_{11}=I_{xx}&=\sum _{k=1}^{N}m_{k}\left(y_{k}^{2}+z_{k}^{2}\right),\\I_{22}=I_{yy}&=\sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+z_{k}^{2}\right),\\I_{33}=I_{zz}&=\sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+y_{k}^{2}\right),\\I_{12}=I_{21}=I_{xy}&=-\sum _{k=1}^{N}m_{k}x_{k}y_{k},\\I_{13}=I_{31}=I_{xz}&=-\sum _{k=1}^{N}m_{k}x_{k}z_{k},\\I_{23}=I_{32}=I_{yz}&=-\sum _{k=1}^{N}m_{k}y_{k}z_{k}.\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3098f142898d70ad0f240f22ffae6dbae2ea78b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -22.005ex; width:36.767ex; height:45.176ex;" alt="{\displaystyle {\begin{aligned}I_{11}=I_{xx}&=\sum _{k=1}^{N}m_{k}\left(y_{k}^{2}+z_{k}^{2}\right),\\I_{22}=I_{yy}&=\sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+z_{k}^{2}\right),\\I_{33}=I_{zz}&=\sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+y_{k}^{2}\right),\\I_{12}=I_{21}=I_{xy}&=-\sum _{k=1}^{N}m_{k}x_{k}y_{k},\\I_{13}=I_{31}=I_{xz}&=-\sum _{k=1}^{N}m_{k}x_{k}z_{k},\\I_{23}=I_{32}=I_{yz}&=-\sum _{k=1}^{N}m_{k}y_{k}z_{k}.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 36.767ex;height: 45.176ex;vertical-align: -22.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3098f142898d70ad0f240f22ffae6dbae2ea78b0" data-alt="{\displaystyle {\begin{aligned}I_{11}=I_{xx}&=\sum _{k=1}^{N}m_{k}\left(y_{k}^{2}+z_{k}^{2}\right),\\I_{22}=I_{yy}&=\sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+z_{k}^{2}\right),\\I_{33}=I_{zz}&=\sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+y_{k}^{2}\right),\\I_{12}=I_{21}=I_{xy}&=-\sum _{k=1}^{N}m_{k}x_{k}y_{k},\\I_{13}=I_{31}=I_{xz}&=-\sum _{k=1}^{N}m_{k}x_{k}z_{k},\\I_{23}=I_{32}=I_{yz}&=-\sum _{k=1}^{N}m_{k}y_{k}z_{k}.\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Инерцијалниот тензор за континуирано тело е даден со <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} =\iiint \limits _{Q}\rho (\mathbf {r} )\left(\left(\mathbf {r} \cdot \mathbf {r} \right)\mathbf {E} -\mathbf {r} \otimes \mathbf {r} \right)\,\mathrm {d} V,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mo> = </mo> <munder> <mo> ∭ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> Q </mi> </mrow> </munder> <mi> ρ </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo stretchy="false"> ) </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> E </mi> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo> ⊗ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} =\iiint \limits _{Q}\rho (\mathbf {r} )\left(\left(\mathbf {r} \cdot \mathbf {r} \right)\mathbf {E} -\mathbf {r} \otimes \mathbf {r} \right)\,\mathrm {d} V,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/459183900efa0001901395bac8073ceef970ec05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:35.631ex; height:7.676ex;" alt="{\displaystyle \mathbf {I} =\iiint \limits _{Q}\rho (\mathbf {r} )\left(\left(\mathbf {r} \cdot \mathbf {r} \right)\mathbf {E} -\mathbf {r} \otimes \mathbf {r} \right)\,\mathrm {d} V,}"> </noscript><span class="lazy-image-placeholder" style="width: 35.631ex;height: 7.676ex;vertical-align: -4.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/459183900efa0001901395bac8073ceef970ec05" data-alt="{\displaystyle \mathbf {I} =\iiint \limits _{Q}\rho (\mathbf {r} )\left(\left(\mathbf {r} \cdot \mathbf {r} \right)\mathbf {E} -\mathbf {r} \otimes \mathbf {r} \right)\,\mathrm {d} V,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {r} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.102ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" data-alt="{\displaystyle \mathbf {r} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  ги дефинира координатите на точката во телото и <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\mathbf {r} )}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ρ </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \rho (\mathbf {r} )} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f477411625125978c0a18946bdfae2c1f13bcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.113ex; height:2.843ex;" alt="{\displaystyle \rho (\mathbf {r} )}"> </noscript><span class="lazy-image-placeholder" style="width: 4.113ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f477411625125978c0a18946bdfae2c1f13bcb" data-alt="{\displaystyle \rho (\mathbf {r} )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е масата на таа точка. Интегралот се презема преку волуменот <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> </noscript><span class="lazy-image-placeholder" style="width: 1.787ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" data-alt="{\displaystyle V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на телото. Инерцијалниот тензор е симетричен, бидејќи <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{ij}=I_{ji}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mi> j </mi> </mrow> </msub> <mo> = </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{ij}=I_{ji}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb20246ff5088b705f96db0452e7e365bec1313a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.099ex; height:2.843ex;" alt="{\displaystyle I_{ij}=I_{ji}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.099ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb20246ff5088b705f96db0452e7e365bec1313a" data-alt="{\displaystyle I_{ij}=I_{ji}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Алтернативно, исто така, може да се напише во однос на <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%92%D0%BA%D1%80%D1%81%D1%82%D0%B5%D0%BD_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Вкрстен производ (страницата не постои)">операторот на аголен импулс</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbf {r} ]\mathbf {x} =\mathbf {r} \times \mathbf {x} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo> × </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle [\mathbf {r} ]\mathbf {x} =\mathbf {r} \times \mathbf {x} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f32ba1c18fa5966e4dcd2275a7f7e3bc21f6dc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.259ex; height:2.843ex;" alt="{\displaystyle [\mathbf {r} ]\mathbf {x} =\mathbf {r} \times \mathbf {x} }"> </noscript><span class="lazy-image-placeholder" style="width: 12.259ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f32ba1c18fa5966e4dcd2275a7f7e3bc21f6dc0" data-alt="{\displaystyle [\mathbf {r} ]\mathbf {x} =\mathbf {r} \times \mathbf {x} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} =\iiint \limits _{Q}\rho (\mathbf {r} )[\mathbf {r} ]^{T}[\mathbf {r} ]\,\mathrm {d} V=-\iiint \limits _{Q}\rho (\mathbf {r} )[\mathbf {r} ]^{2}\,\mathrm {d} V}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mo> = </mo> <munder> <mo> ∭ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> Q </mi> </mrow> </munder> <mi> ρ </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> T </mi> </mrow> </msup> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo stretchy="false"> ] </mo> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> <mo> = </mo> <mo> − </mo> <munder> <mo> ∭ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> Q </mi> </mrow> </munder> <mi> ρ </mi> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo stretchy="false"> ) </mo> <mo stretchy="false"> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <msup> <mo stretchy="false"> ] </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> d </mi> </mrow> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} =\iiint \limits _{Q}\rho (\mathbf {r} )[\mathbf {r} ]^{T}[\mathbf {r} ]\,\mathrm {d} V=-\iiint \limits _{Q}\rho (\mathbf {r} )[\mathbf {r} ]^{2}\,\mathrm {d} V} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30802ee347b075d63c5345bff73e38d2a7965be1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:44.171ex; height:7.676ex;" alt="{\displaystyle \mathbf {I} =\iiint \limits _{Q}\rho (\mathbf {r} )[\mathbf {r} ]^{T}[\mathbf {r} ]\,\mathrm {d} V=-\iiint \limits _{Q}\rho (\mathbf {r} )[\mathbf {r} ]^{2}\,\mathrm {d} V}"> </noscript><span class="lazy-image-placeholder" style="width: 44.171ex;height: 7.676ex;vertical-align: -4.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30802ee347b075d63c5345bff73e38d2a7965be1" data-alt="{\displaystyle \mathbf {I} =\iiint \limits _{Q}\rho (\mathbf {r} )[\mathbf {r} ]^{T}[\mathbf {r} ]\,\mathrm {d} V=-\iiint \limits _{Q}\rho (\mathbf {r} )[\mathbf {r} ]^{2}\,\mathrm {d} V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Инерцијалниот тензор може да се користи на ист начин како инерцијалната матрица за пресметување на скаларниот момент на инерција околу произволна оска во насока <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {n} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" data-alt="{\displaystyle \mathbf {n} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{n}=\mathbf {n} \cdot \mathbf {I} \cdot \mathbf {n} ,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{n}=\mathbf {n} \cdot \mathbf {I} \cdot \mathbf {n} ,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4665b70c28077a426fbd1f1af3f17b6d97ced4e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.329ex; height:2.509ex;" alt="{\displaystyle I_{n}=\mathbf {n} \cdot \mathbf {I} \cdot \mathbf {n} ,}"> </noscript><span class="lazy-image-placeholder" style="width: 13.329ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4665b70c28077a426fbd1f1af3f17b6d97ced4e7" data-alt="{\displaystyle I_{n}=\mathbf {n} \cdot \mathbf {I} \cdot \mathbf {n} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде точниот производ се зема со соодветните елементи во тензорите на компонентата. Производ на термин инерција како што е <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{12}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 12 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{12}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d87c9988135efda49f4cdfc0ffced4872a6497" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.899ex; height:2.509ex;" alt="{\displaystyle I_{12}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.899ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d87c9988135efda49f4cdfc0ffced4872a6497" data-alt="{\displaystyle I_{12}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  се добива со пресметка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{12}=\mathbf {e} _{1}\cdot \mathbf {I} \cdot \mathbf {e} _{2},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 12 </mn> </mrow> </msub> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{12}=\mathbf {e} _{1}\cdot \mathbf {I} \cdot \mathbf {e} _{2},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8121bd2cb30b298966841de1aaa32a05199d4372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.575ex; height:2.509ex;" alt="{\displaystyle I_{12}=\mathbf {e} _{1}\cdot \mathbf {I} \cdot \mathbf {e} _{2},}"> </noscript><span class="lazy-image-placeholder" style="width: 15.575ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8121bd2cb30b298966841de1aaa32a05199d4372" data-alt="{\displaystyle I_{12}=\mathbf {e} _{1}\cdot \mathbf {I} \cdot \mathbf {e} _{2},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и може да се толкува како момент на инерција околу x-оската кога предметот ротира околу y-оската. Компонентите на тензорите од степен два можат да се соберат во матрица. За тензорот на инерција оваа матрица е дадена со, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} ={\begin{bmatrix}I_{11}&I_{12}&I_{13}\\I_{21}&I_{22}&I_{23}\\I_{31}&I_{32}&I_{33}\end{bmatrix}}={\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{bmatrix}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 11 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 12 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 13 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 21 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 22 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 23 </mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 31 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 32 </mn> </mrow> </msub> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 33 </mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> x </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> y </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> z </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> x </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> y </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> y </mi> <mi> z </mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> <mi> x </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> <mi> y </mi> </mrow> </msub> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> z </mi> <mi> z </mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} ={\begin{bmatrix}I_{11}&I_{12}&I_{13}\\I_{21}&I_{22}&I_{23}\\I_{31}&I_{32}&I_{33}\end{bmatrix}}={\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{bmatrix}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9abbf96477fe8eedeff4e1e43471d89f7804056c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:42.664ex; height:9.843ex;" alt="{\displaystyle \mathbf {I} ={\begin{bmatrix}I_{11}&I_{12}&I_{13}\\I_{21}&I_{22}&I_{23}\\I_{31}&I_{32}&I_{33}\end{bmatrix}}={\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{bmatrix}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 42.664ex;height: 9.843ex;vertical-align: -4.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9abbf96477fe8eedeff4e1e43471d89f7804056c" data-alt="{\displaystyle \mathbf {I} ={\begin{bmatrix}I_{11}&I_{12}&I_{13}\\I_{21}&I_{22}&I_{23}\\I_{31}&I_{32}&I_{33}\end{bmatrix}}={\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{bmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Често во ригидните механичари на телото е да се користи нотација која експлицитно го идентификува x, y и z-оски, како што се <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{xx}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> x </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{xx}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23d06dfedcfe6aa332c557ec72c913472732de65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.136ex; height:2.509ex;" alt="{\displaystyle I_{xx}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.136ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23d06dfedcfe6aa332c557ec72c913472732de65" data-alt="{\displaystyle I_{xx}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> и <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{xy}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mi> y </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{xy}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600241c202cafa4422df4a4106d081f21bafe545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.013ex; height:2.843ex;" alt="{\displaystyle I_{xy}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.013ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600241c202cafa4422df4a4106d081f21bafe545" data-alt="{\displaystyle I_{xy}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , за компонентите на тензорот на инерција. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <h2 id="Инерцијална_матрица_во_различни_појдовни_системи"><span id=".D0.98.D0.BD.D0.B5.D1.80.D1.86.D0.B8.D1.98.D0.B0.D0.BB.D0.BD.D0.B0_.D0.BC.D0.B0.D1.82.D1.80.D0.B8.D1.86.D0.B0_.D0.B2.D0.BE_.D1.80.D0.B0.D0.B7.D0.BB.D0.B8.D1.87.D0.BD.D0.B8_.D0.BF.D0.BE.D1.98.D0.B4.D0.BE.D0.B2.D0.BD.D0.B8_.D1.81.D0.B8.D1.81.D1.82.D0.B5.D0.BC.D0.B8">Инерцијална матрица во различни појдовни системи</h2> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=22&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Инерцијална матрица во различни појдовни системи“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> Употребата на матрицата на инерција во вториот закон на Њутн претпоставува дека нејзините компоненти се пресметуваат во однос на оските паралелни на инерцијалната рамка, а не во однос на фиксирана референтна рамка на телото.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kane-6">[6]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Goldstein-23">[23]</a> Ова значи дека додека телото ги поместува компонентите на матрицата на инерција, со текот на времето се менува. Спротивно на тоа, компонентите на матрицата на инерција, измерени во фиксирана рамка на телото, се константни. <div class="mw-heading mw-heading3"> <h3 id="Рамка_на_телото">Рамка на телото</h3> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=23&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Рамка на телото“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> Нека инерцијалната матрица на рамката на телото во однос на центарот на масата се означува со <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} _{\mathbf {C} }^{B}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> B </mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} _{\mathbf {C} }^{B}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5375b412570a299355aebd42dc45bc0c46cd3a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.612ex; height:3.176ex;" alt="{\displaystyle \mathbf {I} _{\mathbf {C} }^{B}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.612ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5375b412570a299355aebd42dc45bc0c46cd3a20" data-alt="{\displaystyle \mathbf {I} _{\mathbf {C} }^{B}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и ја дефинира ориентацијата на телото рамка во однос на инерцијалната рамка од матрицата на ротација <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {A} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.019ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" data-alt="{\displaystyle \mathbf {A} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , така што, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =\mathbf {A} \mathbf {y} ,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> y </mi> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {x} =\mathbf {A} \mathbf {y} ,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f199d57a899bdf9d2d69ade12203d8c11d56c851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.587ex; height:2.509ex;" alt="{\displaystyle \mathbf {x} =\mathbf {A} \mathbf {y} ,}"> </noscript><span class="lazy-image-placeholder" style="width: 8.587ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f199d57a899bdf9d2d69ade12203d8c11d56c851" data-alt="{\displaystyle \mathbf {x} =\mathbf {A} \mathbf {y} ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде што векторите y во фиксираниот координатен рамка на телото имаат координати x во инерцијалната рамка. Потоа, матрицата на инерција на телото измерена во инерцијалната рамка е дадена со <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} _{\mathbf {C} }=\mathbf {A} \mathbf {I} _{\mathbf {C} }^{B}\mathbf {A} ^{\mathsf {T}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> B </mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} _{\mathbf {C} }=\mathbf {A} \mathbf {I} _{\mathbf {C} }^{B}\mathbf {A} ^{\mathsf {T}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/932ee4752bbfe56b40201f525520bdbb5e368df1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.359ex; height:3.176ex;" alt="{\displaystyle \mathbf {I} _{\mathbf {C} }=\mathbf {A} \mathbf {I} _{\mathbf {C} }^{B}\mathbf {A} ^{\mathsf {T}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 14.359ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/932ee4752bbfe56b40201f525520bdbb5e368df1" data-alt="{\displaystyle \mathbf {I} _{\mathbf {C} }=\mathbf {A} \mathbf {I} _{\mathbf {C} }^{B}\mathbf {A} ^{\mathsf {T}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Забележете дека <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> A </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {A} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.019ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" data-alt="{\displaystyle \mathbf {A} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  се менува додека телото се движи, додека <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} _{\mathbf {C} }^{B}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> B </mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} _{\mathbf {C} }^{B}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5375b412570a299355aebd42dc45bc0c46cd3a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.612ex; height:3.176ex;" alt="{\displaystyle \mathbf {I} _{\mathbf {C} }^{B}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.612ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5375b412570a299355aebd42dc45bc0c46cd3a20" data-alt="{\displaystyle \mathbf {I} _{\mathbf {C} }^{B}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  останува константен. <div class="mw-heading mw-heading3"> <h3 id="Главни_оски">Главни оски</h3> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=24&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Главни оски“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> Измерен во телото рамка инерција матрица е константна реална симетрична матрица. Вистинската симетрична матрица има <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A1%D0%BF%D0%B5%D0%BA%D1%82%D1%80%D0%B0%D0%BB%D0%BD%D0%BE_%D1%80%D0%B0%D1%81%D0%BF%D0%B0%D1%93%D0%B0%D1%9A%D0%B5_%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0%D1%82%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Спектрално распаѓање на матрицата (страницата не постои)">спектрално распаѓање на матрицата</a> во производот на вртежна матрица <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Q </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {Q} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132d0144479d6f47c30ad82a65d458966ccbe928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} }"> </noscript><span class="lazy-image-placeholder" style="width: 2.008ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132d0144479d6f47c30ad82a65d458966ccbe928" data-alt="{\displaystyle \mathbf {Q} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  и дијагонална матрица <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Lambda }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Λ </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\Lambda }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0dd88f811f247adeb58a4fd0174e95f11eefca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.873ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\Lambda }}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.873ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0dd88f811f247adeb58a4fd0174e95f11eefca2" data-alt="{\displaystyle {\boldsymbol {\Lambda }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , дадена од <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} _{\mathbf {C} }^{B}=\mathbf {Q} {\boldsymbol {\Lambda }}\mathbf {Q} ^{\mathsf {T}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> I </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> C </mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> B </mi> </mrow> </msubsup> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Q </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Λ </mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Q </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {I} _{\mathbf {C} }^{B}=\mathbf {Q} {\boldsymbol {\Lambda }}\mathbf {Q} ^{\mathsf {T}},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54e2e55f1753f6bd594ab4eae2793aad38b6bf50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.597ex; height:3.176ex;" alt="{\displaystyle \mathbf {I} _{\mathbf {C} }^{B}=\mathbf {Q} {\boldsymbol {\Lambda }}\mathbf {Q} ^{\mathsf {T}},}"> </noscript><span class="lazy-image-placeholder" style="width: 13.597ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54e2e55f1753f6bd594ab4eae2793aad38b6bf50" data-alt="{\displaystyle \mathbf {I} _{\mathbf {C} }^{B}=\mathbf {Q} {\boldsymbol {\Lambda }}\mathbf {Q} ^{\mathsf {T}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  каде <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Lambda }}={\begin{bmatrix}I_{1}&0&0\\0&I_{2}&0\\0&0&I_{3}\end{bmatrix}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Λ </mi> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mtd> <mtd> <mn> 0 </mn> </mtd> <mtd> <mn> 0 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mtd> <mtd> <mn> 0 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mn> 0 </mn> </mtd> <mtd> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\Lambda }}={\begin{bmatrix}I_{1}&0&0\\0&I_{2}&0\\0&0&I_{3}\end{bmatrix}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7144289c4fcf135450cf15f6957847a8921cb2f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:20.348ex; height:9.176ex;" alt="{\displaystyle {\boldsymbol {\Lambda }}={\begin{bmatrix}I_{1}&0&0\\0&I_{2}&0\\0&0&I_{3}\end{bmatrix}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 20.348ex;height: 9.176ex;vertical-align: -4.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7144289c4fcf135450cf15f6957847a8921cb2f6" data-alt="{\displaystyle {\boldsymbol {\Lambda }}={\begin{bmatrix}I_{1}&0&0\\0&I_{2}&0\\0&0&I_{3}\end{bmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Постојат знаци на телото на телото на телото. Овој резултат првпат го покажал <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%8F%D0%B5%D1%98%D0%BC%D1%81_%D0%8F%D0%BE%D0%B7%D0%B5%D1%84_%D0%A1%D0%B8%D0%BB%D0%B2%D0%B5%D1%81%D1%82%D0%B5%D1%80&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Џејмс Џозеф Силвестер (страницата не постои)">Џ. Силвестер (1852)</a>, и претставува форма на <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A1%D0%B8%D0%BB%D0%B2%D0%B5%D1%81%D1%82%D0%B5%D1%80%D0%BE%D0%B2_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%B5%D0%BD_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Силвестеров инерцијален закон (страницата не постои)">Силвестеровиот инерцијален закон</a>.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-syl852-26">[26]</a><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-norm-27">[27]</a> За тела со постојана оска. <div class="mw-heading mw-heading3"> <h3 id="Елипсоид">Елипсоид</h3> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=25&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Елипсоид“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <figure typeof="mw:File/Thumb"> <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/File:Triaxial_Ellipsoid.jpg"> <noscript> <img alt="" resource="/wiki/%D0%9F%D0%BE%D0%B4%D0%B0%D1%82%D0%BE%D1%82%D0%B5%D0%BA%D0%B0:Triaxial_Ellipsoid.jpg" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Triaxial_Ellipsoid.jpg/196px-Triaxial_Ellipsoid.jpg" decoding="async" width="196" height="110" class="mw-file-element" data-file-width="998" data-file-height="559"> </noscript><span class="lazy-image-placeholder" style="width: 196px;height: 110px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Triaxial_Ellipsoid.jpg/196px-Triaxial_Ellipsoid.jpg" data-alt="" data-width="196" data-height="110" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Triaxial_Ellipsoid.jpg/294px-Triaxial_Ellipsoid.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Triaxial_Ellipsoid.jpg/392px-Triaxial_Ellipsoid.jpg 2x" data-class="mw-file-element"> </a> <figcaption> Елипсоид[<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%98%D0%B0:%D0%9C%D1%80%D1%82%D0%B2%D0%B8_%D0%B2%D1%80%D1%81%D0%BA%D0%B8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Википедија:Мртви врски">мртва врска</a>] со полуглавни пречници етикетирани <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> </noscript><span class="lazy-image-placeholder" style="width: 1.23ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" data-alt="{\displaystyle a}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> b </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle b} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> </noscript><span class="lazy-image-placeholder" style="width: 0.998ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" data-alt="{\displaystyle b}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , и <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> c </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle c} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> </noscript><span class="lazy-image-placeholder" style="width: 1.007ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" data-alt="{\displaystyle c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . </figcaption> </figure> Моментот на инерцијална матрица во координатите на телото-рамка е квадратна форма која ја дефинира површината во телото наречена <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9F%D1%83%D0%B0%D0%BD%D1%81%D0%BE%D0%BD%D0%BE%D0%B2_%D0%B5%D0%BB%D0%B8%D0%BF%D1%81%D0%BE%D0%B8%D0%B4&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Пуансонов елипсоид (страницата не постои)">Пуансонов елипсоид</a>.<a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-28">[28]</a> Нека <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\Lambda }}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Λ </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\boldsymbol {\Lambda }}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0dd88f811f247adeb58a4fd0174e95f11eefca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.873ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\Lambda }}}"> </noscript><span class="lazy-image-placeholder" style="width: 1.873ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0dd88f811f247adeb58a4fd0174e95f11eefca2" data-alt="{\displaystyle {\boldsymbol {\Lambda }}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е инерција матрица во однос на центарот на маса усогласен со главните оски, а потоа на површината <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {x} =1,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Λ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {x} =1,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25e1b955758b9a28b1e5f06b7a3ae54ad78fdb3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.954ex; height:3.009ex;" alt="{\displaystyle \mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {x} =1,}"> </noscript><span class="lazy-image-placeholder" style="width: 10.954ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25e1b955758b9a28b1e5f06b7a3ae54ad78fdb3f" data-alt="{\displaystyle \mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {x} =1,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  или <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{1}x^{2}+I_{2}y^{2}+I_{3}z^{2}=1,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <msup> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <msup> <mi> y </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <msup> <mi> z </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{1}x^{2}+I_{2}y^{2}+I_{3}z^{2}=1,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f77feed11fc0b809a4fd3946eb966b0d41af0ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.564ex; height:3.009ex;" alt="{\displaystyle I_{1}x^{2}+I_{2}y^{2}+I_{3}z^{2}=1,}"> </noscript><span class="lazy-image-placeholder" style="width: 23.564ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f77feed11fc0b809a4fd3946eb966b0d41af0ec" data-alt="{\displaystyle I_{1}x^{2}+I_{2}y^{2}+I_{3}z^{2}=1,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  дефинира <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%95%D0%BB%D0%B8%D0%BF%D1%81%D0%BE%D0%B8%D0%B4%D0%BD%D0%B0_%D1%84%D0%BE%D1%80%D0%BC%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Елипсоидна форма">елипсоид</a> во рамката на телото. Напишете ја оваа равенка во форма, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {x}{1/{\sqrt {I_{1}}}}}\right)^{2}+\left({\frac {y}{1/{\sqrt {I_{2}}}}}\right)^{2}+\left({\frac {z}{1/{\sqrt {I_{3}}}}}\right)^{2}=1,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> x </mi> <mrow> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> y </mi> <mrow> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> z </mi> <mrow> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left({\frac {x}{1/{\sqrt {I_{1}}}}}\right)^{2}+\left({\frac {y}{1/{\sqrt {I_{2}}}}}\right)^{2}+\left({\frac {z}{1/{\sqrt {I_{3}}}}}\right)^{2}=1,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24cb74ab9d03623f89fb144e09f2afe23ca1243c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.706ex; height:8.009ex;" alt="{\displaystyle \left({\frac {x}{1/{\sqrt {I_{1}}}}}\right)^{2}+\left({\frac {y}{1/{\sqrt {I_{2}}}}}\right)^{2}+\left({\frac {z}{1/{\sqrt {I_{3}}}}}\right)^{2}=1,}"> </noscript><span class="lazy-image-placeholder" style="width: 46.706ex;height: 8.009ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24cb74ab9d03623f89fb144e09f2afe23ca1243c" data-alt="{\displaystyle \left({\frac {x}{1/{\sqrt {I_{1}}}}}\right)^{2}+\left({\frac {y}{1/{\sqrt {I_{2}}}}}\right)^{2}+\left({\frac {z}{1/{\sqrt {I_{3}}}}}\right)^{2}=1,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  да се види дека полуглавните пречници на овој елипсоид се дадени со <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={\frac {1}{\sqrt {I_{1}}}},\quad b={\frac {1}{\sqrt {I_{2}}}},\quad c={\frac {1}{\sqrt {I_{3}}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msqrt> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo> , </mo> <mspace width="1em"></mspace> <mi> b </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msqrt> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo> , </mo> <mspace width="1em"></mspace> <mi> c </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msqrt> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a={\frac {1}{\sqrt {I_{1}}}},\quad b={\frac {1}{\sqrt {I_{2}}}},\quad c={\frac {1}{\sqrt {I_{3}}}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/914038d07d02704f45ee854f8f72eb6cc1916320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.825ex; height:6.509ex;" alt="{\displaystyle a={\frac {1}{\sqrt {I_{1}}}},\quad b={\frac {1}{\sqrt {I_{2}}}},\quad c={\frac {1}{\sqrt {I_{3}}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 34.825ex;height: 6.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/914038d07d02704f45ee854f8f72eb6cc1916320" data-alt="{\displaystyle a={\frac {1}{\sqrt {I_{1}}}},\quad b={\frac {1}{\sqrt {I_{2}}}},\quad c={\frac {1}{\sqrt {I_{3}}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Нека точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {x} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" data-alt="{\displaystyle \mathbf {x} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на овој елипсоид се дефинира во однос на неговата големина и насока, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =||\mathbf {x} ||\mathbf {n} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {x} =||\mathbf {x} ||\mathbf {n} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7abec9c256860320c666ced21b5254c1d99d63e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.993ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} =||\mathbf {x} ||\mathbf {n} }"> </noscript><span class="lazy-image-placeholder" style="width: 9.993ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7abec9c256860320c666ced21b5254c1d99d63e1" data-alt="{\displaystyle \mathbf {x} =||\mathbf {x} ||\mathbf {n} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , каде што <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {n} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" data-alt="{\displaystyle \mathbf {n} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  е единичен вектор. Тогаш врската прикажана погоре, помеѓу инерција матрица и скаларен момент на инерција <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\mathbf {n} }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle I_{\mathbf {n} }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72ebb3350fb95aa9983a893859493a811ef446b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.306ex; height:2.509ex;" alt="{\displaystyle I_{\mathbf {n} }}"> </noscript><span class="lazy-image-placeholder" style="width: 2.306ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72ebb3350fb95aa9983a893859493a811ef446b" data-alt="{\displaystyle I_{\mathbf {n} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  околу оската во правец <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {n} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" data-alt="{\displaystyle \mathbf {n} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , дава <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {x} =||\mathbf {x} ||^{2}\mathbf {n} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {n} =||\mathbf {x} ||^{2}I_{\mathbf {n} }=1.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Λ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> Λ </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mrow> </msub> <mo> = </mo> <mn> 1. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {x} =||\mathbf {x} ||^{2}\mathbf {n} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {n} =||\mathbf {x} ||^{2}I_{\mathbf {n} }=1.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01df1dbd3d597e33e19fa25af3f0dd0508f07c5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.757ex; height:3.343ex;" alt="{\displaystyle \mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {x} =||\mathbf {x} ||^{2}\mathbf {n} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {n} =||\mathbf {x} ||^{2}I_{\mathbf {n} }=1.}"> </noscript><span class="lazy-image-placeholder" style="width: 35.757ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01df1dbd3d597e33e19fa25af3f0dd0508f07c5e" data-alt="{\displaystyle \mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {x} =||\mathbf {x} ||^{2}\mathbf {n} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {n} =||\mathbf {x} ||^{2}I_{\mathbf {n} }=1.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Така, големината на точката <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {x} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" data-alt="{\displaystyle \mathbf {x} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  во правец <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {n} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" data-alt="{\displaystyle \mathbf {n} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  на инерцијалниот елипсоид е <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ||\mathbf {x} ||={\frac {1}{\sqrt {I_{\mathbf {n} }}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> x </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msqrt> <msub> <mi> I </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle ||\mathbf {x} ||={\frac {1}{\sqrt {I_{\mathbf {n} }}}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ece1d74342c0787200a13450e06681f2b47745b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.821ex; height:6.176ex;" alt="{\displaystyle ||\mathbf {x} ||={\frac {1}{\sqrt {I_{\mathbf {n} }}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 12.821ex;height: 6.176ex;vertical-align: -2.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ece1d74342c0787200a13450e06681f2b47745b" data-alt="{\displaystyle ||\mathbf {x} ||={\frac {1}{\sqrt {I_{\mathbf {n} }}}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"> <h2 id="Поврзано">Поврзано</h2> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=26&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Поврзано“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <section class="mf-section-9 collapsible-block" id="mf-section-9"> <ul> <li><a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A6%D0%B5%D0%BD%D1%82%D1%80%D0%B0%D0%BB%D0%B5%D0%BD_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Централен момент (страницата не постои)">Централен момент</a></li> <li><a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%A1%D0%BF%D0%B8%D1%81%D0%BE%D0%BA_%D0%BD%D0%B0_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82%D0%B8_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Список на моменти на инерција (страницата не постои)">Список на моменти на инерција</a></li> <li><a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%92%D1%80%D1%82%D0%B5%D0%B6%D0%BD%D0%B0_%D0%B5%D0%BD%D0%B5%D1%80%D0%B3%D0%B8%D1%98%D0%B0&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Вртежна енергија (страницата не постои)">Вртежна енергија</a></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"> <h2 id="Наводи">Наводи</h2> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=27&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Наводи“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <section class="mf-section-10 collapsible-block" id="mf-section-10"> <div class="reflist columns references-column-width" style="-moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-mach-1">↑ <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-mach_1-0">1,0</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-mach_1-1">1,1</a> <cite id="CITEREFMach1919" class="citation book">Mach, Ernst (1919). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/scienceofmechani005860mbp">The Science of Mechanics</a>. стр. 173–187. Посетено на November 21, 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Science+of+Mechanics&rft.pages=173-187.&rft.date=1919&rft.aulast=Mach&rft.aufirst=Ernst&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fscienceofmechani005860mbp&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"><style data-mw-deduplicate="TemplateStyles:r5289462">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background-image:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png");background-image:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg");background-repeat:no-repeat;background-size:9px;background-position:right .1em center}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background-image:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png");background-image:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg");background-repeat:no-repeat;background-size:9px;background-position:right .1em center}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background-image:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png");background-image:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg");background-repeat:no-repeat;background-size:9px;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:var(--color-subtle,#54595d)}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background-image:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png");background-image:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg");background-repeat:no-repeat;background-size:12px;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style></li> <li id="cite_note-Euler1730-2"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Euler1730_2-0">↑</a> <cite id="CITEREFEuler1765" class="citation book">Euler, Leonhard (1765). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3Dzw4OAAAAQAAJ%26pg%3DPA166%23v%3Donepage%26q%26f%3Dfalse">Theoria motus corporum solidorum seu rigidorum: Ex primis nostrae cognitionis principiis stabilita et ad omnes motus, qui in huiusmodi corpora cadere possunt, accommodata [The theory of motion of solid or rigid bodies: established from first principles of our knowledge and appropriate for all motions which can occur in such bodies]</a> (латински). Rostock and Greifswald (Germany): A. F. Röse. стр. 166. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/978-1-4297-4281-8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/978-1-4297-4281-8"><bdi>978-1-4297-4281-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theoria+motus+corporum+solidorum+seu+rigidorum%3A+Ex+primis+nostrae+cognitionis+principiis+stabilita+et+ad+omnes+motus%2C+qui+in+huiusmodi+corpora+cadere+possunt%2C+accommodata+%5BThe+theory+of+motion+of+solid+or+rigid+bodies%3A+established+from+first+principles+of+our+knowledge+and+appropriate+for+all+motions+which+can+occur+in+such+bodies%5D&rft.place=Rostock+and+Greifswald+%28Germany%29&rft.pages=166&rft.pub=A.+F.+R%C3%B6se&rft.date=1765&rft.isbn=978-1-4297-4281-8&rft.aulast=Euler&rft.aufirst=Leonhard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dzw4OAAAAQAAJ%26pg%3DPA166%23v%3Donepage%26q%26f%3Dfalse&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"> From page 166: "Definitio 7. 422. Momentum inertiae corporis respectu eujuspiam axis est summa omnium productorum, quae oriuntur, si singula corporis elementa per quadrata distantiarum suarum ab axe multiplicentur." (Definition 7. 422. A body's moment of inertia with respect to any axis is the sum of all of the products, which arise, if the individual elements of the body are multiplied by the square of their distances from the axis.)</li> <li id="cite_note-Marion_1995-3">↑ <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Marion_1995_3-0">3,0</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Marion_1995_3-1">3,1</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Marion_1995_3-2">3,2</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Marion_1995_3-3">3,3</a> <cite id="CITEREFMarionThornton1995" class="citation book">Marion, JB; Thornton, ST (1995). Classical dynamics of particles & systems (4. изд.). Thomson. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/0-03-097302-3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/0-03-097302-3"><bdi>0-03-097302-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+dynamics+of+particles+%26+systems&rft.edition=4&rft.pub=Thomson&rft.date=1995&rft.isbn=0-03-097302-3&rft.aulast=Marion&rft.aufirst=JB&rft.au=Thornton%2C+ST&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-Symon_1971-4">↑ <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Symon_1971_4-0">4,0</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Symon_1971_4-1">4,1</a> <cite id="CITEREFSymon1971" class="citation book">Symon, KR (1971). Mechanics (3. изд.). Addison-Wesley. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/0-201-07392-7?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/0-201-07392-7"><bdi>0-201-07392-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics&rft.edition=3&rft.pub=Addison-Wesley&rft.date=1971&rft.isbn=0-201-07392-7&rft.aulast=Symon&rft.aufirst=KR&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-Tenenbaum_2004-5">↑ <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Tenenbaum_2004_5-0">5,0</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Tenenbaum_2004_5-1">5,1</a> <cite id="CITEREFTenenbaum2004" class="citation book">Tenenbaum, RA (2004). Fundamentals of Applied Dynamics. Springer. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/0-387-00887-X?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/0-387-00887-X"><bdi>0-387-00887-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Applied+Dynamics&rft.pub=Springer&rft.date=2004&rft.isbn=0-387-00887-X&rft.aulast=Tenenbaum&rft.aufirst=RA&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-Kane-6">↑ <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kane_6-0">6,0</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kane_6-1">6,1</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kane_6-2">6,2</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kane_6-3">6,3</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kane_6-4">6,4</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kane_6-5">6,5</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kane_6-6">6,6</a> <cite id="CITEREFKaneLevinson1985" class="citation book">Kane, T. R.; Levinson, D. A. (1985). Dynamics, Theory and Applications. New York: McGraw-Hill.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamics%2C+Theory+and+Applications&rft.place=New+York&rft.pub=McGraw-Hill&rft.date=1985&rft.aulast=Kane&rft.aufirst=T.+R.&rft.au=Levinson%2C+D.+A.&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-Winn-7">↑ <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Winn_7-0">7,0</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Winn_7-1">7,1</a> <cite id="CITEREFWinn2010" class="citation book">Winn, Will (2010). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DNH8m7j9V0cUC%26pg%3DSA10-PA10%26dq%3D%2522ice%2Bskater%2522%2B%2522moment%2Bof%2Binertia">Introduction to Understandable Physics: Volume I - Mechanics</a>. AuthorHouse. стр. 10.10. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/1449063330?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/1449063330"><bdi>1449063330</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Understandable+Physics%3A+Volume+I+-+Mechanics&rft.pages=10.10&rft.pub=AuthorHouse&rft.date=2010&rft.isbn=1449063330&rft.aulast=Winn&rft.aufirst=Will&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNH8m7j9V0cUC%26pg%3DSA10-PA10%26dq%3D%2522ice%2Bskater%2522%2B%2522moment%2Bof%2Binertia&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-Fullerton-8">↑ <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Fullerton_8-0">8,0</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Fullerton_8-1">8,1</a> <cite id="CITEREFFullerton2011" class="citation book">Fullerton, Dan (2011). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3D8XmF2dy-9YYC%26pg%3DPA143%26dq%3D%2522ice%2Bskater%2522%2B%2522moment%2Bof%2Binertia%23v%3Donepage%26q%3D%2522ice%2520skater%2522%2520%2522moment%2520of%2520inertia%26f%3Dfalse">Honors Physics Essentials</a>. Silly Beagle Productions. стр. 142–143. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/0983563330?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/0983563330"><bdi>0983563330</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Honors+Physics+Essentials&rft.pages=142-143&rft.pub=Silly+Beagle+Productions&rft.date=2011&rft.isbn=0983563330&rft.aulast=Fullerton&rft.aufirst=Dan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D8XmF2dy-9YYC%26pg%3DPA143%26dq%3D%2522ice%2Bskater%2522%2B%2522moment%2Bof%2Binertia%23v%3Donepage%26q%3D%2522ice%2520skater%2522%2520%2522moment%2520of%2520inertia%26f%3Dfalse&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-Wolfram-9"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Wolfram_9-0">↑</a> <cite id="CITEREFWolfram2014" class="citation web">Wolfram, Stephen (2014). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://demonstrations.wolfram.com/SpinningIceSkater/">„Spinning Ice Skater“</a>. Wolfram Demonstrations Project. Mathematica, Inc. Посетено на September 30, 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+Demonstrations+Project&rft.atitle=Spinning+Ice+Skater&rft.date=2014&rft.aulast=Wolfram&rft.aufirst=Stephen&rft_id=http%3A%2F%2Fdemonstrations.wolfram.com%2FSpinningIceSkater%2F&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-Hokin-10"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Hokin_10-0">↑</a> <cite id="CITEREFHokin2014" class="citation web">Hokin, Samuel (2014). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20201126232109/http://bsharp.org/physics/spins">„Figure Skating Spins“</a>. The Physics of Everyday Stuff. Архивирано од <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.bsharp.org/physics/spins">изворникот</a> на 2020-11-26. Посетено на September 30, 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Physics+of+Everyday+Stuff&rft.atitle=Figure+Skating+Spins&rft.date=2014&rft.aulast=Hokin&rft.aufirst=Samuel&rft_id=http%3A%2F%2Fwww.bsharp.org%2Fphysics%2Fspins&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-Breithaupt-11"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Breithaupt_11-0">↑</a> <cite id="CITEREFBreithaupt2000" class="citation book">Breithaupt, Jim (2000). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3Dr8I1gyNNKnoC%26pg%3DPT73%26dq%3D%2522ice%2Bskater%2522%2B%2522moment%2Bof%2Binertia">New Understanding Physics for Advanced Level</a>. Nelson Thomas. стр. 64. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/0748743146?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/0748743146"><bdi>0748743146</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=New+Understanding+Physics+for+Advanced+Level&rft.pages=64&rft.pub=Nelson+Thomas&rft.date=2000&rft.isbn=0748743146&rft.aulast=Breithaupt&rft.aufirst=Jim&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dr8I1gyNNKnoC%26pg%3DPT73%26dq%3D%2522ice%2Bskater%2522%2B%2522moment%2Bof%2Binertia&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-Crowell-12"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Crowell_12-0">↑</a> <cite id="CITEREFCrowell2003" class="citation book">Crowell, Benjamin (2003). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DpVcROBeNJKcC%26pg%3DPA107%26dq%3Dice%2Bskater%2B%2522conservation%2Bof%2Bangular%2Bmomentum">Conservation Laws</a>. Light and Matter. стр. 107. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/0970467028?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/0970467028"><bdi>0970467028</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Conservation+Laws&rft.pages=107&rft.pub=Light+and+Matter&rft.date=2003&rft.isbn=0970467028&rft.aulast=Crowell&rft.aufirst=Benjamin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpVcROBeNJKcC%26pg%3DPA107%26dq%3Dice%2Bskater%2B%2522conservation%2Bof%2Bangular%2Bmomentum&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-Tipler-13"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Tipler_13-0">↑</a> <cite id="CITEREFTipler1999" class="citation book">Tipler, Paul A. (1999). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DU9lkAkTdAosC%26pg%3DPA304%26dq%3Dskater%2B%2522conservation%2Bof%2Bangular%2Bmomentum">Physics for Scientists and Engineers, Vol. 1: Mechanics, Oscillations and Waves, Thermodynamics</a>. Macmillan. стр. 304. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/1572594918?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/1572594918"><bdi>1572594918</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers%2C+Vol.+1%3A+Mechanics%2C+Oscillations+and+Waves%2C+Thermodynamics&rft.pages=304&rft.pub=Macmillan&rft.date=1999&rft.isbn=1572594918&rft.aulast=Tipler&rft.aufirst=Paul+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DU9lkAkTdAosC%26pg%3DPA304%26dq%3Dskater%2B%2522conservation%2Bof%2Bangular%2Bmomentum&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-B-Paul-14">↑ <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-B-Paul_14-0">14,0</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-B-Paul_14-1">14,1</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-B-Paul_14-2">14,2</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-B-Paul_14-3">14,3</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-B-Paul_14-4">14,4</a> <cite id="CITEREFPaul1979" class="citation book">Paul, Burton (June 1979). Kinematics and Dynamics of Planar Machinery. Prentice Hall. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/978-0135160626?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/978-0135160626"><bdi>978-0135160626</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kinematics+and+Dynamics+of+Planar+Machinery&rft.pub=Prentice+Hall&rft.date=1979-06&rft.isbn=978-0135160626&rft.aulast=Paul&rft.aufirst=Burton&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-Resnick-15"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Resnick_15-0">↑</a> <cite id="CITEREFHallidayResnickWalker2005" class="citation book">Halliday, David; Resnick, Robert; Walker, Jearl (2005). Fundamentals of physics (7. изд.). Hoboken, NJ: Wiley. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/9780471216438?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/9780471216438"><bdi>9780471216438</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+physics&rft.place=Hoboken%2C+NJ&rft.edition=7&rft.pub=Wiley&rft.date=2005&rft.isbn=9780471216438&rft.aulast=Halliday&rft.aufirst=David&rft.au=Resnick%2C+Robert&rft.au=Walker%2C+Jearl&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-16"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-16">↑</a> <cite id="CITEREFFrench1971" class="citation book">French, A.P. (1971). Vibrations and waves. Boca Raton, FL: CRC Press. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/9780748744473?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/9780748744473"><bdi>9780748744473</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vibrations+and+waves&rft.place=Boca+Raton%2C+FL&rft.pub=CRC+Press&rft.date=1971&rft.isbn=9780748744473&rft.aulast=French&rft.aufirst=A.P.&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-Uicker-17">↑ <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Uicker_17-0">17,0</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Uicker_17-1">17,1</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Uicker_17-2">17,2</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Uicker_17-3">17,3</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Uicker_17-4">17,4</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Uicker_17-5">17,5</a> <cite id="CITEREFUickerPennockShigley2010" class="citation book">Uicker, John J.; Pennock, Gordon R.; Shigley, Joseph E. (2010). Theory of Machines and Mechanisms (4. изд.). Oxford University Press. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/978-0195371239?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/978-0195371239"><bdi>978-0195371239</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Machines+and+Mechanisms&rft.edition=4&rft.pub=Oxford+University+Press&rft.date=2010&rft.isbn=978-0195371239&rft.aulast=Uicker&rft.aufirst=John+J.&rft.au=Pennock%2C+Gordon+R.&rft.au=Shigley%2C+Joseph+E.&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-Beer-18">↑ <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Beer_18-0">18,00</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Beer_18-1">18,01</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Beer_18-2">18,02</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Beer_18-3">18,03</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Beer_18-4">18,04</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Beer_18-5">18,05</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Beer_18-6">18,06</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Beer_18-7">18,07</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Beer_18-8">18,08</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Beer_18-9">18,09</a> <cite id="CITEREFFerdinand_P._BeerE._Russell_JohnstonJr.,_Phillip_J._Cornwell2010" class="citation book">Ferdinand P. Beer; E. Russell Johnston; Jr., Phillip J. Cornwell (2010). Vector mechanics for engineers: Dynamics (9. изд.). Boston: McGraw-Hill. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/978-0077295493?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/978-0077295493"><bdi>978-0077295493</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vector+mechanics+for+engineers%3A+Dynamics&rft.place=Boston&rft.edition=9&rft.pub=McGraw-Hill&rft.date=2010&rft.isbn=978-0077295493&rft.au=Ferdinand+P.+Beer&rft.au=E.+Russell+Johnston&rft.au=Jr.%2C+Phillip+J.+Cornwell&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-19"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-19">↑</a> H. Williams, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.ima.org.uk/_db/_documents/maths07_williams_huw.pdf">Measuring the inertia tensor</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20150924033745/http://www.ima.org.uk/_db/_documents/maths07_williams_huw.pdf">Архивирано</a> на 24 септември 2015 г., presented at the IMA Mathematics 2007 Conference.</li> <li id="cite_note-20"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-20">↑</a> Gracey, William, The experimental determination of the moments of inertia of airplanes by a simplified compound-pendulum method, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://naca.central.cranfield.ac.uk/reports/1948/naca-tn-1629.pdf">NACA Technical Note No. 1629</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20201231124121/http://naca.central.cranfield.ac.uk/reports/1948/naca-tn-1629.pdf">Архивирано</a> на 31 декември 2020 г., 1948</li> <li id="cite_note-21"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-21">↑</a> In that situation this moment of inertia only describes how a torque applied along that axis causes a rotation about that axis. But, torques not aligned along a principal axis will also cause rotations about other axes.</li> <li id="cite_note-22"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-22">↑</a> Walter D. Pilkey, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3D4hEsqvplmFMC%26pg%3DPA437%26dq%3Dpolar%2Bmoment%2Bof%2Binertia%26hl%3Den%26sa%3DX%26ei%3D1vxkUbj1JIr-rQH-5oC4Bg%26ved%3D0CF4Q6AEwCTgK%23v%3Donepage%26q%3D%2522polar%2520moment%2520of%2520inertia%2522%26f%3Dfalse">Analysis and Design of Elastic Beams: Computational Methods</a>, John Wiley, 2002.</li> <li id="cite_note-Goldstein-23">↑ <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Goldstein_23-0">23,0</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Goldstein_23-1">23,1</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Goldstein_23-2">23,2</a> <cite id="CITEREFGoldstein1980" class="citation book">Goldstein, H. (1980). Classical Mechanics (2. изд.). Addison-Wesley. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/0-201-02918-9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/0-201-02918-9"><bdi>0-201-02918-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics&rft.edition=2&rft.pub=Addison-Wesley&rft.date=1980&rft.isbn=0-201-02918-9&rft.aulast=Goldstein&rft.aufirst=H.&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-24"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-24">↑</a> L. D. Landau and E. M. Lifshitz, <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/Mechanics_541">Mechanics</a>, Vol 1. 2nd Ed., Pergamon Press, 1969.</li> <li id="cite_note-Tsai-25"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Tsai_25-0">↑</a> L. W. Tsai, Robot Analysis: The mechanics of serial and parallel manipulators, John-Wiley, NY, 1999.</li> <li id="cite_note-syl852-26"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-syl852_26-0">↑</a> <cite id="CITEREFSylvester,_J_J1852" class="citation journal">Sylvester, J J (1852). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.maths.ed.ac.uk/~aar/sylv/inertia.pdf">„A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares“</a> (PDF). Philosophical Magazine. 4th Series. 4 (23): 138–142. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/Doi?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1080%252F14786445208647087">10.1080/14786445208647087</a>. Посетено на June 27, 2008.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=A+demonstration+of+the+theorem+that+every+homogeneous+quadratic+polynomial+is+reducible+by+real+orthogonal+substitutions+to+the+form+of+a+sum+of+positive+and+negative+squares&rft.volume=4&rft.issue=23&rft.pages=138-142&rft.date=1852&rft_id=info%3Adoi%2F10.1080%2F14786445208647087&rft.au=Sylvester%2C+J+J&rft_id=http%3A%2F%2Fwww.maths.ed.ac.uk%2F~aar%2Fsylv%2Finertia.pdf&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-norm-27"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-norm_27-0">↑</a> <cite id="CITEREFNorman,_C.W.1986" class="citation book">Norman, C.W. (1986). Undergraduate algebra. Oxford University Press. стр. 360–361. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/0-19-853248-2?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/0-19-853248-2"><bdi>0-19-853248-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Undergraduate+algebra&rft.pages=360-361&rft.pub=Oxford+University+Press&rft.date=1986&rft.isbn=0-19-853248-2&rft.au=Norman%2C+C.W.&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> <li id="cite_note-28"><a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-28">↑</a> <cite id="CITEREFMason2001" class="citation book">Mason, Matthew T. (2001). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DNgdeu3go014C">Mechanics of Robotics Manipulation</a>. MIT Press. <a href="https://mk-m-wikipedia-org.translate.goog/wiki/ISBN?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="ISBN">ISBN</a> <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0:%D0%9F%D0%B5%D1%87%D0%B0%D1%82%D0%B5%D0%BD%D0%98%D0%B7%D0%B2%D0%BE%D1%80/978-0-262-13396-8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Специјална:ПечатенИзвор/978-0-262-13396-8"><bdi>978-0-262-13396-8</bdi></a>. Посетено на November 21, 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Robotics+Manipulation&rft.pub=MIT+Press&rft.date=2001&rft.isbn=978-0-262-13396-8&rft.aulast=Mason&rft.aufirst=Matthew+T.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNgdeu3go014C&rfr_id=info%3Asid%2Fmk.wikipedia.org%3A%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82+%D0%BD%D0%B0+%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0" class="Z3988"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5289462"></li> </ol> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(11)"> <h2 id="Надворешни_врски">Надворешни врски</h2> <a role="button" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=edit&section=28&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Уреди го одделот „Надворешни врски“" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> уреди </a> </div> <section class="mf-section-11 collapsible-block" id="mf-section-11"> <table role="presentation" class="mbox-small plainlinks sistersitebox" style="background-color:var(--background-color-neutral-subtle, #f8f9fa);border:1px solid var(--border-color-base, #a2a9b1);color:inherit"> <tbody> <tr> <td class="mbox-text plainlist">„<a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://commons.wikimedia.org/wiki/Category:Moments_of_inertia?uselang%3Dmk">Момент на инерција</a>“ на Ризницата <a href="https://mk-m-wikipedia-org.translate.goog/wiki/%D0%A0%D0%B8%D0%B7%D0%BD%D0%B8%D1%86%D0%B0_(%D0%92%D0%B8%D0%BA%D0%B8%D0%BC%D0%B5%D0%B4%D0%B8%D1%98%D0%B0)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ризница (Викимедија)">?</a></td> </tr> </tbody> </table> <ul> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.lightandmatter.com/html_books/0sn/ch04/ch04.html">Angular momentum and rigid-body rotation in two and three dimensions</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20100329113035/http://www.lightandmatter.com/html_books/0sn/ch04/ch04.html">Архивирано</a> на 29 март 2010 г.</li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html">Lecture notes on rigid-body rotation and moments of inertia</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://kwon3d.com/theory/moi/iten.html">The moment of inertia tensor</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.phy.hk/wiki/englishhtm/Balance.htm">An introductory lesson on moment of inertia: keeping a vertical pole not falling down (Java simulation)</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://hypertextbook.com/physics/mechanics/rotational-inertia/">Tutorial on finding moments of inertia, with problems and solutions on various basic shapes</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20090814083636/http://hypertextbook.com/physics/mechanics/rotational-inertia/">Архивирано</a> на 14 август 2009 г.</li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.cs.cmu.edu/afs/cs/academic/class/16741-s07/www/">Notes on mechanics of manipulation: the angular inertia tensor</a></li> </ul> <a href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9F%D1%80%D0%B5%D0%B4%D0%BB%D0%BE%D1%88%D0%BA%D0%B0:Tensors&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Предлошка:Tensors (страницата не постои)">Предлошка:Tensors</a> </section> </div> <noscript> <img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=mobile" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div class="printfooter" data-nosnippet=""> Преземено од „<a dir="ltr" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mk.wikipedia.org/w/index.php?title%3D%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0%26oldid%3D5256009">https://mk.wikipedia.org/w/index.php?title=Момент_на_инерција&oldid=5256009</a>“ </div> </div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"><a class="last-modified-bar" href="https://mk-m-wikipedia-org.translate.goog/w/index.php?title=%D0%9C%D0%BE%D0%BC%D0%B5%D0%BD%D1%82_%D0%BD%D0%B0_%D0%B8%D0%BD%D0%B5%D1%80%D1%86%D0%B8%D1%98%D0%B0&action=history&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB"> <div class="post-content last-modified-bar__content"> Последна измена на 29 август 2024, во 22:15 ч. </div></a> <div class="post-content footer-content"> <div id="mw-data-after-content"> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Јазици</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://af.wikipedia.org/wiki/Traagheidsmoment" title="Traagheidsmoment — африканс" lang="af" hreflang="af" data-title="Traagheidsmoment" data-language-autonym="Afrikaans" data-language-local-name="африканс" class="interlanguage-link-target">Afrikaans</a></li> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ar.wikipedia.org/wiki/%25D8%25B9%25D8%25B2%25D9%2585_%25D8%25A7%25D9%2584%25D9%2582%25D8%25B5%25D9%2588%25D8%25B1_%25D8%25A7%25D9%2584%25D8%25B0%25D8%25A7%25D8%25AA%25D9%258A" title="عزم القصور الذاتي — арапски" lang="ar" hreflang="ar" data-title="عزم القصور الذاتي" data-language-autonym="العربية" data-language-local-name="арапски" class="interlanguage-link-target">العربية</a></li> <li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ast.wikipedia.org/wiki/Momentu_d%2527inercia" title="Momentu d'inercia — астурски" lang="ast" hreflang="ast" data-title="Momentu d'inercia" data-language-autonym="Asturianu" data-language-local-name="астурски" class="interlanguage-link-target">Asturianu</a></li> <li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bn.wikipedia.org/wiki/%25E0%25A6%259C%25E0%25A6%25A1%25E0%25A6%25BC%25E0%25A6%25A4%25E0%25A6%25BE%25E0%25A6%25B0_%25E0%25A6%25AD%25E0%25A7%258D%25E0%25A6%25B0%25E0%25A6%25BE%25E0%25A6%25AE%25E0%25A6%2595" title="জড়তার ভ্রামক — бенгалски" lang="bn" hreflang="bn" data-title="জড়তার ভ্রামক" data-language-autonym="বাংলা" data-language-local-name="бенгалски" class="interlanguage-link-target">বাংলা</a></li> <li class="interlanguage-link interwiki-be mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://be.wikipedia.org/wiki/%25D0%259C%25D0%25BE%25D0%25BC%25D0%25B0%25D0%25BD%25D1%2582_%25D1%2596%25D0%25BD%25D0%25B5%25D1%2580%25D1%2586%25D1%258B%25D1%2596" title="Момант інерцыі — белоруски" lang="be" hreflang="be" data-title="Момант інерцыі" data-language-autonym="Беларуская" data-language-local-name="белоруски" class="interlanguage-link-target">Беларуская</a></li> <li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://be-tarask.wikipedia.org/wiki/%25D0%259C%25D0%25BE%25D0%25BC%25D0%25B0%25D0%25BD%25D1%2582_%25D1%2596%25D0%25BD%25D1%258D%25D1%2580%25D1%2586%25D1%258B%25D1%2596" title="Момант інэрцыі — Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Момант інэрцыі" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bg.wikipedia.org/wiki/%25D0%259C%25D0%25B0%25D1%2581%25D0%25BE%25D0%25B2_%25D0%25B8%25D0%25BD%25D0%25B5%25D1%2580%25D1%2586%25D0%25B8%25D0%25BE%25D0%25BD%25D0%25B5%25D0%25BD_%25D0%25BC%25D0%25BE%25D0%25BC%25D0%25B5%25D0%25BD%25D1%2582" title="Масов инерционен момент — бугарски" lang="bg" hreflang="bg" data-title="Масов инерционен момент" data-language-autonym="Български" data-language-local-name="бугарски" class="interlanguage-link-target">Български</a></li> <li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bs.wikipedia.org/wiki/Moment_inercije" title="Moment inercije — босански" lang="bs" hreflang="bs" data-title="Moment inercije" data-language-autonym="Bosanski" data-language-local-name="босански" class="interlanguage-link-target">Bosanski</a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/Moment_d%2527in%25C3%25A8rcia" title="Moment d'inèrcia — каталонски" lang="ca" hreflang="ca" data-title="Moment d'inèrcia" data-language-autonym="Català" data-language-local-name="каталонски" class="interlanguage-link-target">Català</a></li> <li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cv.wikipedia.org/wiki/%25D0%2598%25D0%25BD%25D0%25B5%25D1%2580%25D1%2586%25D0%25B8_%25D1%2581%25D0%25B0%25D0%25BC%25D0%25B0%25D0%25BD%25D1%2587%25C4%2595" title="Инерци саманчĕ — чувашки" lang="cv" hreflang="cv" data-title="Инерци саманчĕ" data-language-autonym="Чӑвашла" data-language-local-name="чувашки" class="interlanguage-link-target">Чӑвашла</a></li> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/wiki/Moment_setrva%25C4%258Dnosti" title="Moment setrvačnosti — чешки" lang="cs" hreflang="cs" data-title="Moment setrvačnosti" data-language-autonym="Čeština" data-language-local-name="чешки" class="interlanguage-link-target">Čeština</a></li> <li class="interlanguage-link interwiki-da mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://da.wikipedia.org/wiki/Inertimoment" title="Inertimoment — дански" lang="da" hreflang="da" data-title="Inertimoment" data-language-autonym="Dansk" data-language-local-name="дански" class="interlanguage-link-target">Dansk</a></li> <li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="добра статија"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/wiki/Tr%25C3%25A4gheitsmoment" title="Trägheitsmoment — германски" lang="de" hreflang="de" data-title="Trägheitsmoment" data-language-autonym="Deutsch" data-language-local-name="германски" class="interlanguage-link-target">Deutsch</a></li> <li class="interlanguage-link interwiki-et mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/wiki/Inertsimoment" title="Inertsimoment — естонски" lang="et" hreflang="et" data-title="Inertsimoment" data-language-autonym="Eesti" data-language-local-name="естонски" class="interlanguage-link-target">Eesti</a></li> <li class="interlanguage-link interwiki-el mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://el.wikipedia.org/wiki/%25CE%25A1%25CE%25BF%25CF%2580%25CE%25AE_%25CE%25B1%25CE%25B4%25CF%2581%25CE%25AC%25CE%25BD%25CE%25B5%25CE%25B9%25CE%25B1%25CF%2582" title="Ροπή αδράνειας — грчки" lang="el" hreflang="el" data-title="Ροπή αδράνειας" data-language-autonym="Ελληνικά" data-language-local-name="грчки" class="interlanguage-link-target">Ελληνικά</a></li> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Moment_of_inertia" title="Moment of inertia — англиски" lang="en" hreflang="en" data-title="Moment of inertia" data-language-autonym="English" data-language-local-name="англиски" class="interlanguage-link-target">English</a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Momento_de_inercia" title="Momento de inercia — шпански" lang="es" hreflang="es" data-title="Momento de inercia" data-language-autonym="Español" data-language-local-name="шпански" class="interlanguage-link-target">Español</a></li> <li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eo.wikipedia.org/wiki/Inercimomanto" title="Inercimomanto — есперанто" lang="eo" hreflang="eo" data-title="Inercimomanto" data-language-autonym="Esperanto" data-language-local-name="есперанто" class="interlanguage-link-target">Esperanto</a></li> <li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eu.wikipedia.org/wiki/Inertzia-momentu" title="Inertzia-momentu — баскиски" lang="eu" hreflang="eu" data-title="Inertzia-momentu" data-language-autonym="Euskara" data-language-local-name="баскиски" class="interlanguage-link-target">Euskara</a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25DA%25AF%25D8%25B4%25D8%25AA%25D8%25A7%25D9%2588%25D8%25B1_%25D9%2584%25D8%25AE%25D8%25AA%25DB%258C" title="گشتاور لختی — персиски" lang="fa" hreflang="fa" data-title="گشتاور لختی" data-language-autonym="فارسی" data-language-local-name="персиски" class="interlanguage-link-target">فارسی</a></li> <li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/Moment_d%2527inertie" title="Moment d'inertie — француски" lang="fr" hreflang="fr" data-title="Moment d'inertie" data-language-autonym="Français" data-language-local-name="француски" class="interlanguage-link-target">Français</a></li> <li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ga.wikipedia.org/wiki/M%25C3%25B3imint_na_t%25C3%25A1imhe" title="Móimint na táimhe — ирски" lang="ga" hreflang="ga" data-title="Móimint na táimhe" data-language-autonym="Gaeilge" data-language-local-name="ирски" class="interlanguage-link-target">Gaeilge</a></li> <li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gl.wikipedia.org/wiki/Momento_de_inercia" title="Momento de inercia — галициски" lang="gl" hreflang="gl" data-title="Momento de inercia" data-language-autonym="Galego" data-language-local-name="галициски" class="interlanguage-link-target">Galego</a></li> <li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ko.wikipedia.org/wiki/%25EA%25B4%2580%25EC%2584%25B1_%25EB%25AA%25A8%25EB%25A9%2598%25ED%258A%25B8" title="관성 모멘트 — корејски" lang="ko" hreflang="ko" data-title="관성 모멘트" data-language-autonym="한국어" data-language-local-name="корејски" class="interlanguage-link-target">한국어</a></li> <li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hy.wikipedia.org/wiki/%25D4%25BB%25D5%25B6%25D5%25A5%25D6%2580%25D6%2581%25D5%25AB%25D5%25A1%25D5%25B5%25D5%25AB_%25D5%25B4%25D5%25B8%25D5%25B4%25D5%25A5%25D5%25B6%25D5%25BF" title="Իներցիայի մոմենտ — ерменски" lang="hy" hreflang="hy" data-title="Իներցիայի մոմենտ" data-language-autonym="Հայերեն" data-language-local-name="ерменски" class="interlanguage-link-target">Հայերեն</a></li> <li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hi.wikipedia.org/wiki/%25E0%25A4%259C%25E0%25A4%25A1%25E0%25A4%25BC%25E0%25A4%25A4%25E0%25A5%258D%25E0%25A4%25B5%25E0%25A4%25BE%25E0%25A4%2598%25E0%25A5%2582%25E0%25A4%25B0%25E0%25A5%258D%25E0%25A4%25A3" title="जड़त्वाघूर्ण — хинди" lang="hi" hreflang="hi" data-title="जड़त्वाघूर्ण" data-language-autonym="हिन्दी" data-language-local-name="хинди" class="interlanguage-link-target">हिन्दी</a></li> <li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hr.wikipedia.org/wiki/Moment_inercije" title="Moment inercije — хрватски" lang="hr" hreflang="hr" data-title="Moment inercije" data-language-autonym="Hrvatski" data-language-local-name="хрватски" class="interlanguage-link-target">Hrvatski</a></li> <li class="interlanguage-link interwiki-id mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://id.wikipedia.org/wiki/Momen_inersia" title="Momen inersia — индонезиски" lang="id" hreflang="id" data-title="Momen inersia" data-language-autonym="Bahasa Indonesia" data-language-local-name="индонезиски" class="interlanguage-link-target">Bahasa Indonesia</a></li> <li class="interlanguage-link interwiki-is mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://is.wikipedia.org/wiki/Hverfitreg%25C3%25B0a" title="Hverfitregða — исландски" lang="is" hreflang="is" data-title="Hverfitregða" data-language-autonym="Íslenska" data-language-local-name="исландски" class="interlanguage-link-target">Íslenska</a></li> <li class="interlanguage-link interwiki-it mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://it.wikipedia.org/wiki/Momento_di_inerzia" title="Momento di inerzia — италијански" lang="it" hreflang="it" data-title="Momento di inerzia" data-language-autonym="Italiano" data-language-local-name="италијански" class="interlanguage-link-target">Italiano</a></li> <li class="interlanguage-link interwiki-he mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%259E%25D7%2595%25D7%259E%25D7%25A0%25D7%2598_%25D7%2594%25D7%25AA%25D7%259E%25D7%2593" title="מומנט התמד — хебрејски" lang="he" hreflang="he" data-title="מומנט התמד" data-language-autonym="עברית" data-language-local-name="хебрејски" class="interlanguage-link-target">עברית</a></li> <li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ka.wikipedia.org/wiki/%25E1%2583%2598%25E1%2583%259C%25E1%2583%2594%25E1%2583%25A0%25E1%2583%25AA%25E1%2583%2598%25E1%2583%2598%25E1%2583%25A1_%25E1%2583%259B%25E1%2583%259D%25E1%2583%259B%25E1%2583%2594%25E1%2583%259C%25E1%2583%25A2%25E1%2583%2598" title="ინერციის მომენტი — грузиски" lang="ka" hreflang="ka" data-title="ინერციის მომენტი" data-language-autonym="ქართული" data-language-local-name="грузиски" class="interlanguage-link-target">ქართული</a></li> <li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://kk.wikipedia.org/wiki/%25D0%2598%25D0%25BD%25D0%25B5%25D1%2580%25D1%2586%25D0%25B8%25D1%258F_%25D0%25BC%25D0%25BE%25D0%25BC%25D0%25B5%25D0%25BD%25D1%2582%25D1%2596" title="Инерция моменті — казашки" lang="kk" hreflang="kk" data-title="Инерция моменті" data-language-autonym="Қазақша" data-language-local-name="казашки" class="interlanguage-link-target">Қазақша</a></li> <li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ht.wikipedia.org/wiki/Moman_in%25C3%25A8si" title="Moman inèsi — хаитски" lang="ht" hreflang="ht" data-title="Moman inèsi" data-language-autonym="Kreyòl ayisyen" data-language-local-name="хаитски" class="interlanguage-link-target">Kreyòl ayisyen</a></li> <li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lv.wikipedia.org/wiki/Inerces_moments" title="Inerces moments — латвиски" lang="lv" hreflang="lv" data-title="Inerces moments" data-language-autonym="Latviešu" data-language-local-name="латвиски" class="interlanguage-link-target">Latviešu</a></li> <li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lt.wikipedia.org/wiki/Inercijos_momentas" title="Inercijos momentas — литвански" lang="lt" hreflang="lt" data-title="Inercijos momentas" data-language-autonym="Lietuvių" data-language-local-name="литвански" class="interlanguage-link-target">Lietuvių</a></li> <li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hu.wikipedia.org/wiki/Tehetetlens%25C3%25A9gi_nyomat%25C3%25A9k" title="Tehetetlenségi nyomaték — унгарски" lang="hu" hreflang="hu" data-title="Tehetetlenségi nyomaték" data-language-autonym="Magyar" data-language-local-name="унгарски" class="interlanguage-link-target">Magyar</a></li> <li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ml.wikipedia.org/wiki/%25E0%25B4%259C%25E0%25B4%25A2%25E0%25B4%25A4%25E0%25B5%258D%25E0%25B4%25B5%25E0%25B4%25BE%25E0%25B4%2598%25E0%25B5%2582%25E0%25B5%25BC%25E0%25B4%25A3%25E0%25B4%2582" title="ജഢത്വാഘൂർണം — малајалски" lang="ml" hreflang="ml" data-title="ജഢത്വാഘൂർണം" data-language-autonym="മലയാളം" data-language-local-name="малајалски" class="interlanguage-link-target">മലയാളം</a></li> <li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ms.wikipedia.org/wiki/Momen_inersia" title="Momen inersia — малајски" lang="ms" hreflang="ms" data-title="Momen inersia" data-language-autonym="Bahasa Melayu" data-language-local-name="малајски" class="interlanguage-link-target">Bahasa Melayu</a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Traagheidsmoment" title="Traagheidsmoment — холандски" lang="nl" hreflang="nl" data-title="Traagheidsmoment" data-language-autonym="Nederlands" data-language-local-name="холандски" class="interlanguage-link-target">Nederlands</a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E6%2585%25A3%25E6%2580%25A7%25E3%2583%25A2%25E3%2583%25BC%25E3%2583%25A1%25E3%2583%25B3%25E3%2583%2588" title="慣性モーメント — јапонски" lang="ja" hreflang="ja" data-title="慣性モーメント" data-language-autonym="日本語" data-language-local-name="јапонски" class="interlanguage-link-target">日本語</a></li> <li class="interlanguage-link interwiki-no mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://no.wikipedia.org/wiki/Treghetsmoment" title="Treghetsmoment — норвешки букмол" lang="nb" hreflang="nb" data-title="Treghetsmoment" data-language-autonym="Norsk bokmål" data-language-local-name="норвешки букмол" class="interlanguage-link-target">Norsk bokmål</a></li> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nn.wikipedia.org/wiki/Tregleiksmoment" title="Tregleiksmoment — норвешки нинорск" lang="nn" hreflang="nn" data-title="Tregleiksmoment" data-language-autonym="Norsk nynorsk" data-language-local-name="норвешки нинорск" class="interlanguage-link-target">Norsk nynorsk</a></li> <li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uz.wikipedia.org/wiki/Inersiya_momenti" title="Inersiya momenti — узбечки" lang="uz" hreflang="uz" data-title="Inersiya momenti" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="узбечки" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li> <li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pl.wikipedia.org/wiki/Moment_bezw%25C5%2582adno%25C5%259Bci" title="Moment bezwładności — полски" lang="pl" hreflang="pl" data-title="Moment bezwładności" data-language-autonym="Polski" data-language-local-name="полски" class="interlanguage-link-target">Polski</a></li> <li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/Momento_de_in%25C3%25A9rcia" title="Momento de inércia — португалски" lang="pt" hreflang="pt" data-title="Momento de inércia" data-language-autonym="Português" data-language-local-name="португалски" class="interlanguage-link-target">Português</a></li> <li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ro.wikipedia.org/wiki/Moment_de_iner%25C8%259Bie" title="Moment de inerție — романски" lang="ro" hreflang="ro" data-title="Moment de inerție" data-language-autonym="Română" data-language-local-name="романски" class="interlanguage-link-target">Română</a></li> <li class="interlanguage-link interwiki-ru badge-Q17559452 badge-recommendedarticle mw-list-item" title="препорачана статија"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%259C%25D0%25BE%25D0%25BC%25D0%25B5%25D0%25BD%25D1%2582_%25D0%25B8%25D0%25BD%25D0%25B5%25D1%2580%25D1%2586%25D0%25B8%25D0%25B8" title="Момент инерции — руски" lang="ru" hreflang="ru" data-title="Момент инерции" data-language-autonym="Русский" data-language-local-name="руски" class="interlanguage-link-target">Русский</a></li> <li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sq.wikipedia.org/wiki/Momenti_i_Inercis%25C3%25AB" title="Momenti i Inercisë — албански" lang="sq" hreflang="sq" data-title="Momenti i Inercisë" data-language-autonym="Shqip" data-language-local-name="албански" class="interlanguage-link-target">Shqip</a></li> <li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://simple.wikipedia.org/wiki/Moment_of_inertia" title="Moment of inertia — Simple English" lang="en-simple" hreflang="en-simple" data-title="Moment of inertia" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li> <li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sk.wikipedia.org/wiki/Moment_zotrva%25C4%258Dnosti" title="Moment zotrvačnosti — словачки" lang="sk" hreflang="sk" data-title="Moment zotrvačnosti" data-language-autonym="Slovenčina" data-language-local-name="словачки" class="interlanguage-link-target">Slovenčina</a></li> <li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sl.wikipedia.org/wiki/Vztrajnostni_moment" title="Vztrajnostni moment — словенечки" lang="sl" hreflang="sl" data-title="Vztrajnostni moment" data-language-autonym="Slovenščina" data-language-local-name="словенечки" class="interlanguage-link-target">Slovenščina</a></li> <li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sr.wikipedia.org/wiki/%25D0%259C%25D0%25BE%25D0%25BC%25D0%25B5%25D0%25BD%25D1%2582_%25D0%25B8%25D0%25BD%25D0%25B5%25D1%2580%25D1%2586%25D0%25B8%25D1%2598%25D0%25B5" title="Момент инерције — српски" lang="sr" hreflang="sr" data-title="Момент инерције" data-language-autonym="Српски / srpski" data-language-local-name="српски" class="interlanguage-link-target">Српски / srpski</a></li> <li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sh.wikipedia.org/wiki/Moment_inercije" title="Moment inercije — српскохрватски" lang="sh" hreflang="sh" data-title="Moment inercije" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="српскохрватски" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li> <li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fi.wikipedia.org/wiki/Hitausmomentti" title="Hitausmomentti — фински" lang="fi" hreflang="fi" data-title="Hitausmomentti" data-language-autonym="Suomi" data-language-local-name="фински" class="interlanguage-link-target">Suomi</a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Tr%25C3%25B6ghetsmoment" title="Tröghetsmoment — шведски" lang="sv" hreflang="sv" data-title="Tröghetsmoment" data-language-autonym="Svenska" data-language-local-name="шведски" class="interlanguage-link-target">Svenska</a></li> <li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ta.wikipedia.org/wiki/%25E0%25AE%25A8%25E0%25AE%25BF%25E0%25AE%25B2%25E0%25AF%2588%25E0%25AE%25AE%25E0%25AE%25A4%25E0%25AF%258D_%25E0%25AE%25A4%25E0%25AE%25BF%25E0%25AE%25B0%25E0%25AF%2581%25E0%25AE%25AA%25E0%25AF%258D%25E0%25AE%25AA%25E0%25AF%2581%25E0%25AE%25A4%25E0%25AF%258D%25E0%25AE%25A4%25E0%25AE%25BF%25E0%25AE%25B1%25E0%25AE%25A9%25E0%25AF%258D" title="நிலைமத் திருப்புத்திறன் — тамилски" lang="ta" hreflang="ta" data-title="நிலைமத் திருப்புத்திறன்" data-language-autonym="தமிழ்" data-language-local-name="тамилски" class="interlanguage-link-target">தமிழ்</a></li> <li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tt.wikipedia.org/wiki/%25C4%25B0nertsi%25C3%25A4_moment%25C4%25B1" title="İnertsiä momentı — татарски" lang="tt" hreflang="tt" data-title="İnertsiä momentı" data-language-autonym="Татарча / tatarça" data-language-local-name="татарски" class="interlanguage-link-target">Татарча / tatarça</a></li> <li class="interlanguage-link interwiki-th mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://th.wikipedia.org/wiki/%25E0%25B9%2582%25E0%25B8%25A1%25E0%25B9%2580%25E0%25B8%25A1%25E0%25B8%2599%25E0%25B8%2595%25E0%25B9%258C%25E0%25B8%2584%25E0%25B8%25A7%25E0%25B8%25B2%25E0%25B8%25A1%25E0%25B9%2580%25E0%25B8%2589%25E0%25B8%25B7%25E0%25B9%2588%25E0%25B8%25AD%25E0%25B8%25A2" title="โมเมนต์ความเฉื่อย — тајландски" lang="th" hreflang="th" data-title="โมเมนต์ความเฉื่อย" data-language-autonym="ไทย" data-language-local-name="тајландски" class="interlanguage-link-target">ไทย</a></li> <li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tg.wikipedia.org/wiki/%25D0%259C%25D0%25BE%25D0%25BC%25D0%25B5%25D0%25BD%25D1%2582%25D0%25B8_%25D0%25B8%25D0%25BD%25D0%25B5%25D1%2580%25D1%2581%25D0%25B8%25D1%258F" title="Моменти инерсия — таџикистански" lang="tg" hreflang="tg" data-title="Моменти инерсия" data-language-autonym="Тоҷикӣ" data-language-local-name="таџикистански" class="interlanguage-link-target">Тоҷикӣ</a></li> <li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/wiki/Eylemsizlik_momenti" title="Eylemsizlik momenti — турски" lang="tr" hreflang="tr" data-title="Eylemsizlik momenti" data-language-autonym="Türkçe" data-language-local-name="турски" class="interlanguage-link-target">Türkçe</a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%259C%25D0%25BE%25D0%25BC%25D0%25B5%25D0%25BD%25D1%2582_%25D1%2596%25D0%25BD%25D0%25B5%25D1%2580%25D1%2586%25D1%2596%25D1%2597" title="Момент інерції — украински" lang="uk" hreflang="uk" data-title="Момент інерції" data-language-autonym="Українська" data-language-local-name="украински" class="interlanguage-link-target">Українська</a></li> <li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ur.wikipedia.org/wiki/%25D8%25AC%25D9%2585%25D9%2588%25D8%25AF_%25DA%25A9%25D8%25A7_%25D9%2585%25D8%25B9%25DB%258C%25D8%25A7%25D8%25B1_%25D8%25A7%25D8%25AB%25D8%25B1" title="جمود کا معیار اثر — урду" lang="ur" hreflang="ur" data-title="جمود کا معیار اثر" data-language-autonym="اردو" data-language-local-name="урду" class="interlanguage-link-target">اردو</a></li> <li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vi.wikipedia.org/wiki/M%25C3%25B4_men_qu%25C3%25A1n_t%25C3%25ADnh" title="Mô men quán tính — виетнамски" lang="vi" hreflang="vi" data-title="Mô men quán tính" data-language-autonym="Tiếng Việt" data-language-local-name="виетнамски" class="interlanguage-link-target">Tiếng Việt</a></li> <li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://wuu.wikipedia.org/wiki/%25E8%25BD%25AC%25E5%258A%25A8%25E6%2583%25AF%25E9%2587%258F" title="转动惯量 — ву" lang="wuu" hreflang="wuu" data-title="转动惯量" data-language-autonym="吴语" data-language-local-name="ву" class="interlanguage-link-target">吴语</a></li> <li class="interlanguage-link interwiki-zh-yue mw-list-item"><a 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Момент на инерција — Википедија