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id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Obsah</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">presunúť do postranného panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skryť</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Začiatok</div> </a> </li> <li id="toc-Definícia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definícia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definícia</span> </div> </a> <ul id="toc-Definícia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vlastnosti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vlastnosti"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Vlastnosti</span> </div> </a> <button aria-controls="toc-Vlastnosti-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Přepnout podsekci Vlastnosti</span> </button> <ul id="toc-Vlastnosti-sublist" class="vector-toc-list"> <li id="toc-Ťažisko_zjednotenia_niekoľkých_sústav" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ťažisko_zjednotenia_niekoľkých_sústav"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Ťažisko zjednotenia niekoľkých sústav</span> </div> </a> <ul id="toc-Ťažisko_zjednotenia_niekoľkých_sústav-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pohyb_ťažiska" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pohyb_ťažiska"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Pohyb ťažiska</span> </div> </a> <ul id="toc-Pohyb_ťažiska-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pôsobisko_gravitačnej_a_zotrvačnej_sily_v_homogénnom_poli" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pôsobisko_gravitačnej_a_zotrvačnej_sily_v_homogénnom_poli"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Pôsobisko gravitačnej a zotrvačnej sily v homogénnom poli</span> </div> </a> <ul id="toc-Pôsobisko_gravitačnej_a_zotrvačnej_sily_v_homogénnom_poli-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Referencie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referencie"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Referencie</span> </div> </a> <ul id="toc-Referencie-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Obsah" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Prepnúť obsah" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Prepnúť obsah</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Ťažisko (fyzika)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Prejsť na článok v inom jazyku. Je dostupný v 32 jazykoch" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-32" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">32 jazykov</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Te%C5%BEi%C5%A1te" title="Težište – bosniačtina" lang="bs" hreflang="bs" data-title="Težište" data-language-autonym="Bosanski" data-language-local-name="bosniačtina" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Centre_de_gravetat" title="Centre de gravetat – katalánčina" lang="ca" hreflang="ca" data-title="Centre de gravetat" data-language-autonym="Català" data-language-local-name="katalánčina" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%82%D9%88%D8%B1%D8%B3%D8%A7%DB%8C%DB%8C%DA%AF%DB%95" title="قورساییگە – kurdčina (sorání)" lang="ckb" hreflang="ckb" data-title="قورساییگە" data-language-autonym="کوردی" data-language-local-name="kurdčina (sorání)" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/T%C4%9B%C5%BEi%C5%A1t%C4%9B" title="Těžiště – čeština" lang="cs" hreflang="cs" data-title="Těžiště" data-language-autonym="Čeština" data-language-local-name="čeština" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Tyngdepunkt" title="Tyngdepunkt – dánčina" lang="da" hreflang="da" data-title="Tyngdepunkt" data-language-autonym="Dansk" data-language-local-name="dánčina" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gravizentrum" title="Gravizentrum – nemčina" lang="de" hreflang="de" data-title="Gravizentrum" data-language-autonym="Deutsch" data-language-local-name="nemčina" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%AD%CE%BD%CF%84%CF%81%CE%BF_%CE%BC%CE%AC%CE%B6%CE%B1%CF%82" title="Κέντρο μάζας – gréčtina" lang="el" hreflang="el" data-title="Κέντρο μάζας" data-language-autonym="Ελληνικά" data-language-local-name="gréčtina" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q70893996 mw-list-item" title=""><a href="https://en.wikipedia.org/wiki/Center_of_gravity" title="Center of gravity – angličtina" lang="en" hreflang="en" data-title="Center of gravity" data-language-autonym="English" data-language-local-name="angličtina" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Centro_de_gravedad" title="Centro de gravedad – španielčina" lang="es" hreflang="es" data-title="Centro de gravedad" data-language-autonym="Español" data-language-local-name="španielčina" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Raskuskese" title="Raskuskese – estónčina" lang="et" hreflang="et" data-title="Raskuskese" data-language-autonym="Eesti" data-language-local-name="estónčina" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Grabitate-zentro" title="Grabitate-zentro – baskičtina" lang="eu" hreflang="eu" data-title="Grabitate-zentro" data-language-autonym="Euskara" data-language-local-name="baskičtina" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa badge-Q70893996 mw-list-item" title=""><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D8%A7%D9%86%DB%8C%E2%80%8C%DA%AF%D8%A7%D9%87" title="گرانی‌گاه – perzština" lang="fa" hreflang="fa" data-title="گرانی‌گاه" data-language-autonym="فارسی" data-language-local-name="perzština" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Painopiste" title="Painopiste – fínčina" lang="fi" hreflang="fi" data-title="Painopiste" data-language-autonym="Suomi" data-language-local-name="fínčina" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Centre_de_gravit%C3%A9" title="Centre de gravité – francúzština" lang="fr" hreflang="fr" data-title="Centre de gravité" data-language-autonym="Français" data-language-local-name="francúzština" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Centro_de_gravidade" title="Centro de gravidade – galícijčina" lang="gl" hreflang="gl" data-title="Centro de gravidade" data-language-autonym="Galego" data-language-local-name="galícijčina" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A5%81%E0%A4%B0%E0%A5%81%E0%A4%A4%E0%A5%8D%E0%A4%B5_%E0%A4%95%E0%A5%87%E0%A4%A8%E0%A5%8D%E0%A4%A6%E0%A5%8D%E0%A4%B0" title="गुरुत्व केन्द्र – hindčina" lang="hi" hreflang="hi" data-title="गुरुत्व केन्द्र" data-language-autonym="हिन्दी" data-language-local-name="hindčina" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Te%C5%BEi%C5%A1te" title="Težište – chorvátčina" lang="hr" hreflang="hr" data-title="Težište" data-language-autonym="Hrvatski" data-language-local-name="chorvátčina" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BE%D5%A1%D5%B6%D6%80%D5%B8%D6%82%D5%A9%D5%B5%D5%A1%D5%B6_%D5%AF%D5%A5%D5%B6%D5%BF%D6%80%D5%B8%D5%B6" title="Ծանրության կենտրոն – arménčina" lang="hy" hreflang="hy" data-title="Ծանրության կենտրոն" data-language-autonym="Հայերեն" data-language-local-name="arménčina" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Baricentro" title="Baricentro – ido" lang="io" hreflang="io" data-title="Baricentro" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%83%D1%8B%D1%80%D0%BB%D1%8B%D2%9B_%D1%86%D0%B5%D0%BD%D1%82%D1%80%D1%96" title="Ауырлық центрі – kazaština" lang="kk" hreflang="kk" data-title="Ауырлық центрі" data-language-autonym="Қазақша" data-language-local-name="kazaština" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B3%81%E0%B2%B0%E0%B3%81%E0%B2%A4%E0%B3%8D%E0%B2%B5_%E0%B2%95%E0%B3%87%E0%B2%82%E0%B2%A6%E0%B3%8D%E0%B2%B0" title="ಗುರುತ್ವ ಕೇಂದ್ರ – kannadčina" lang="kn" hreflang="kn" data-title="ಗುರುತ್ವ ಕೇಂದ್ರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannadčina" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Navenda_giraniy%C3%AA" title="Navenda giraniyê – kurdčina" lang="ku" hreflang="ku" data-title="Navenda giraniyê" data-language-autonym="Kurdî" data-language-local-name="kurdčina" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Baricentras" title="Baricentras – litovčina" lang="lt" hreflang="lt" data-title="Baricentras" data-language-autonym="Lietuvių" data-language-local-name="litovčina" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/%C5%9Arodek_ci%C4%99%C5%BCko%C5%9Bci" title="Środek ciężkości – poľština" lang="pl" hreflang="pl" data-title="Środek ciężkości" data-language-autonym="Polski" data-language-local-name="poľština" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Bariss%C3%A8nter" title="Barissènter – piemonština" lang="pms" hreflang="pms" data-title="Barissènter" data-language-autonym="Piemontèis" data-language-local-name="piemonština" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Centro_de_gravidade" title="Centro de gravidade – portugalčina" lang="pt" hreflang="pt" data-title="Centro de gravidade" data-language-autonym="Português" data-language-local-name="portugalčina" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Te%C5%BEi%C5%A1te" title="Težište – srbochorvátčina" lang="sh" hreflang="sh" data-title="Težište" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srbochorvátčina" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Te%C5%BEi%C5%A1%C4%8De" title="Težišče – slovinčina" lang="sl" hreflang="sl" data-title="Težišče" data-language-autonym="Slovenščina" data-language-local-name="slovinčina" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%8A%E0%AE%A4%E0%AF%81_%E0%AE%A8%E0%AE%BF%E0%AE%B1%E0%AF%88_%E0%AE%AE%E0%AF%88%E0%AE%AF%E0%AE%AE%E0%AF%8D" title="பொது நிறை மையம் – tamilčina" lang="ta" hreflang="ta" data-title="பொது நிறை மையம்" data-language-autonym="தமிழ்" data-language-local-name="tamilčina" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/A%C4%9F%C4%B1rl%C4%B1k_merkezi" title="Ağırlık merkezi – turečtina" lang="tr" hreflang="tr" data-title="Ağırlık merkezi" data-language-autonym="Türkçe" data-language-local-name="turečtina" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%90%D0%B2%D1%8B%D1%80%D0%BB%D1%8B%D0%BA_%D2%AF%D0%B7%D3%99%D0%B3%D0%B5" title="Авырлык үзәге – tatárčina" lang="tt" hreflang="tt" data-title="Авырлык үзәге" data-language-autonym="Татарча / tatarça" data-language-local-name="tatárčina" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Og%CA%BBirlik_markazi" title="Ogʻirlik markazi – uzbečtina" lang="uz" hreflang="uz" data-title="Ogʻirlik markazi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečtina" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q114084#sitelinks-wikipedia" title="Upraviť medzijazykové odkazy" class="wbc-editpage">Upraviť odkazy</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Menné priestory"> <div id="p-associated-pages" 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<div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Vzhľad"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Vzhľad</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">presunúť do postranného panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skryť</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">z Wikipédie, slobodnej encyklopédie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="sk" dir="ltr"><p><b>Ťažisko</b> alebo <b>hmotný stred</b> je pôsobisko tiažovej sily pôsobiacej na teleso. Distribúcie hmotnosti v priestore je pre túto distribúciu unikátny <a href="/wiki/Bod" class="mw-disambig" title="Bod">bod</a>, v ktorom je vážený priemer relatívnych pozícií distribuovanej hmotnosti rovný nule, alebo bod, do ktorého ak umiestnime pôsobisko sily, sa bude pohybovať len translačným pohybom po vektorovej priamke tejto sily. V zjednodušenej predstave je to bod, ktorý sa pohybuje, ako keby v ňom bola sústredená celá <a href="/wiki/Hmotnos%C5%A5" title="Hmotnosť">hmotnosť</a> sústavy a pôsobili v ňom všetky <a href="/wiki/Sila" title="Sila">sily</a> pôsobiace na sústavu. V <a href="/wiki/Homog%C3%A9nnos%C5%A5_priestoru" title="Homogénnosť priestoru">homogénnom</a> gravitačnom alebo zotrvačnom poli môžeme všetky <a href="/wiki/Gravita%C4%8Dn%C3%A1_sila" title="Gravitačná sila">gravitačné</a> a <a href="/wiki/Zotrva%C4%8Dn%C3%A1_sila" title="Zotrvačná sila">zotrvačné sily</a> pôsobiace na <a href="/wiki/Tuh%C3%A9_teleso" title="Tuhé teleso">dokonale tuhé teleso</a> nahradiť jedinou silou pôsobiacou v ťažisku. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definícia"><span id="Defin.C3.ADcia"></span>Definícia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%A4a%C5%BEisko_(fyzika)&veaction=edit&section=1" title="Upraviť sekciu: Definícia" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%A4a%C5%BEisko_(fyzika)&action=edit&section=1" title="Editovat zdrojový kód sekce Definícia"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ťažisko <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3357053e25be3943bc092c83ad09a76b8eb85a1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.74ex; height:2.843ex;" alt="{\displaystyle {\vec {T}}}"></span> sústavy hmotných bodov je rovnaké ako vážený priemer ich polôh <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aad177c5c2e4fd39ba2400511e620863b425552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.324ex; height:3.343ex;" alt="{\displaystyle {\vec {r_{i}}}}"></span>, kde váhou jednotlivých zložiek sú ich hmotnosti. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {T}}={\frac {\sum m_{i}{\vec {r_{i}}}}{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <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> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mi>M</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {T}}={\frac {\sum m_{i}{\vec {r_{i}}}}{M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc12facc6a57e4d3f9ec3364b95457f1b61e78f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.679ex; height:6.343ex;" alt="{\displaystyle {\vec {T}}={\frac {\sum m_{i}{\vec {r_{i}}}}{M}}}"></span></dd></dl> <p>Kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\sum m_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo>∑</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\sum m_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0672e6bdd7fa9711e14933c34f38ee983d4b5ad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.123ex; height:3.843ex;" alt="{\displaystyle M=\sum m_{i}}"></span> je celková hmotnosť sústavy. </p> <div class="mw-heading mw-heading2"><h2 id="Vlastnosti">Vlastnosti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%A4a%C5%BEisko_(fyzika)&veaction=edit&section=2" title="Upraviť sekciu: Vlastnosti" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%A4a%C5%BEisko_(fyzika)&action=edit&section=2" title="Editovat zdrojový kód sekce Vlastnosti"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/S%C3%BAbor:Center_gravity_2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Center_gravity_2.svg/220px-Center_gravity_2.svg.png" decoding="async" width="220" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Center_gravity_2.svg/330px-Center_gravity_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Center_gravity_2.svg/440px-Center_gravity_2.svg.png 2x" data-file-width="210" data-file-height="248" /></a><figcaption>Ťažisko telesa (čierny bod) sa nachádza na priesečníku polpriamok (modré polpriamky) pod bodom zavesenia (červený bod).</figcaption></figure> <p>Poloha ťažiska závisí od rozloženia látky v telese. Ak teleso podoprieme na hrote ihly, musí byť ťažisko nad hrotom (alebo pod hrotom), inak teleso spadne. Najjednoduchší spôsob zistenia ťažiska je narysovanie uhlopriečok pri homogénnych pravidelných rovinných útvaroch. Univerzálny spôsob je útvar zavesiť na bezmomentový záves, napríklad na niť. Potom sa útvar sa natočí tak, že ťažisko sa bude nachádzať na polpriamke pod závesom, smerom dole (v smere gravitačného zrýchlenia). Ak zavesíme teleso na dve rôzne miesta mimo ťažiska, tieto polpriamky sa pretnú v ťažisku. </p><p>Pre plošné homogénne útvary (napr. polkruh) tvoriace rotáciou okolo priamky známe rotačné telesá (napr. guľu), vieme vypočítať polohu ťažiska využitím <a href="/w/index.php?title=Guldinove_vety&action=edit&redlink=1" class="new" title="Guldinove vety (stránka neexistuje)">Guldinových viet</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Ťažisko_zjednotenia_niekoľkých_sústav"><span id=".C5.A4a.C5.BEisko_zjednotenia_nieko.C4.BEk.C3.BDch_s.C3.BAstav"></span>Ťažisko zjednotenia niekoľkých sústav</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%A4a%C5%BEisko_(fyzika)&veaction=edit&section=3" title="Upraviť sekciu: Ťažisko zjednotenia niekoľkých sústav" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%A4a%C5%BEisko_(fyzika)&action=edit&section=3" title="Editovat zdrojový kód sekce Ťažisko zjednotenia niekoľkých sústav"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ak máme <i>n</i> sústav hmotných bodov s celkovými hmotnosťami <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1},M_{2},\dots ,M_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1},M_{2},\dots ,M_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0a0512f081073b354d87166dd0f96a4e98decf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.301ex; height:2.509ex;" alt="{\displaystyle M_{1},M_{2},\dots ,M_{n}}"></span> a s ťažiskami <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {T_{1}}},{\vec {T_{2}}},\dots ,{\vec {T_{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {T_{1}}},{\vec {T_{2}}},\dots ,{\vec {T_{n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1213c5ae67909d8c19f4feae93dfd6bfba008d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.612ex; height:4.009ex;" alt="{\displaystyle {\vec {T_{1}}},{\vec {T_{2}}},\dots ,{\vec {T_{n}}}}"></span>, výsledné ťažisko zjednotenia týchto sústav má rovnakú polohu, ako poloha ťažiska sústavy <i>n</i> hmotných bodov nachádzajúcich sa v ťažiskách jednotlivých sústav majúcich hmotnosti zodpovedajúce celkovým hmotnostiam sústav. Táto dôležitá vlastnosť ťažiska plynie z vlastností váženého aritmetického priemeru a pomáha nám určiť polohy ťažísk zložitejších telies. </p> <div class="mw-heading mw-heading3"><h3 id="Pohyb_ťažiska"><span id="Pohyb_.C5.A5a.C5.BEiska"></span>Pohyb ťažiska</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%A4a%C5%BEisko_(fyzika)&veaction=edit&section=4" title="Upraviť sekciu: Pohyb ťažiska" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%A4a%C5%BEisko_(fyzika)&action=edit&section=4" title="Editovat zdrojový kód sekce Pohyb ťažiska"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Poloha ťažiska spĺňa rovnicu: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {T}}={\frac {\sum m_{i}{\vec {r_{i}}}}{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <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> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mi>M</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {T}}={\frac {\sum m_{i}{\vec {r_{i}}}}{M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc12facc6a57e4d3f9ec3364b95457f1b61e78f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.679ex; height:6.343ex;" alt="{\displaystyle {\vec {T}}={\frac {\sum m_{i}{\vec {r_{i}}}}{M}}}"></span></dd></dl> <p>Derivovaním oboch strán rovnice podľa času dostávame: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}={\frac {\sum m_{i}{\vec {v_{i}}}}{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <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> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mi>M</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}={\frac {\sum m_{i}{\vec {v_{i}}}}{M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dccc21399bc129a9975ce4b55575a0aca1abcf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.115ex; height:6.343ex;" alt="{\displaystyle {\vec {v}}={\frac {\sum m_{i}{\vec {v_{i}}}}{M}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}={\frac {\sum m_{i}{\vec {a_{i}}}}{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mi>M</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}={\frac {\sum m_{i}{\vec {a_{i}}}}{M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6f36de6e8a2b698fbd3840d76ee831bdd37689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.169ex; height:6.343ex;" alt="{\displaystyle {\vec {a}}={\frac {\sum m_{i}{\vec {a_{i}}}}{M}}}"></span></dd></dl> <p>S využitím odvodených rovníc je možné odvodiť, že ťažisko sústavy sa pohybuje, ako keby v ňom bola sústredená celá <a href="/wiki/Hmotnos%C5%A5" title="Hmotnosť">hmotnosť</a> sústavy a pôsobili v ňom všetky <a href="/wiki/Sila" title="Sila">sily</a> pôsobiace na sústavu. Odvodenie nájdete v článku o <a href="/wiki/Sila" title="Sila">sile</a>. </p><p>Kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v_{i}}},{\vec {a_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v_{i}}},{\vec {a_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e48ad46f60c6283baf5f6fd1d004c0ac5cc1c344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.681ex; height:3.343ex;" alt="{\displaystyle {\vec {v_{i}}},{\vec {a_{i}}}}"></span> sú rýchlosti a zrýchlenia jednotlivých bodov a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}},{\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}},{\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97df7ddcedf973b6d2791c76bc3913f4b5f1816e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.439ex; height:2.676ex;" alt="{\displaystyle {\vec {v}},{\vec {a}}}"></span> je rýchlosť a zrýchlenie ťažiska. </p> <div class="mw-heading mw-heading3"><h3 id="Pôsobisko_gravitačnej_a_zotrvačnej_sily_v_homogénnom_poli"><span id="P.C3.B4sobisko_gravita.C4.8Dnej_a_zotrva.C4.8Dnej_sily_v_homog.C3.A9nnom_poli"></span>Pôsobisko gravitačnej a zotrvačnej sily v homogénnom poli</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%A4a%C5%BEisko_(fyzika)&veaction=edit&section=5" title="Upraviť sekciu: Pôsobisko gravitačnej a zotrvačnej sily v homogénnom poli" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%A4a%C5%BEisko_(fyzika)&action=edit&section=5" title="Editovat zdrojový kód sekce Pôsobisko gravitačnej a zotrvačnej sily v homogénnom poli"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Na všetky body telesa pôsobia gravitačná aj zotrvačná sila v rovnakom smere a ich veľkosť je úmerná hmotnosti daného bodu. Ide teda o <a href="/wiki/Sila" title="Sila">skladanie rovnobežných síl</a> a pôsobisko výslednice dostaneme v bode: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {R}}={\frac {\sum {\vec {F_{i}}}{\vec {r_{i}}}}{\vec {F}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {R}}={\frac {\sum {\vec {F_{i}}}{\vec {r_{i}}}}{\vec {F}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85abc457cce0ff0be2f814bb6a8b901ee4a41e77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.187ex; height:7.843ex;" alt="{\displaystyle {\vec {R}}={\frac {\sum {\vec {F_{i}}}{\vec {r_{i}}}}{\vec {F}}}}"></span></dd></dl> <p>Kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}=\sum {\vec {F_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}=\sum {\vec {F_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46a635ee2567f84848e207371e792cc03ad551b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.935ex; height:4.676ex;" alt="{\displaystyle {\vec {F}}=\sum {\vec {F_{i}}}}"></span>. Keďže sú však sily pôsobiace na jednotlivé elementy <a href="/wiki/Tuh%C3%A9_teleso" title="Tuhé teleso">tuhého telesa</a> úmerné ich hmotnostiam, prejde rovnica do tvaru: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {R}}={\frac {\sum m_{i}{\vec {r_{i}}}}{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∑</mo> <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> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mi>M</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {R}}={\frac {\sum m_{i}{\vec {r_{i}}}}{M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c92a635a418dd649a364b41d4554acad18e493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.703ex; height:6.343ex;" alt="{\displaystyle {\vec {R}}={\frac {\sum m_{i}{\vec {r_{i}}}}{M}}}"></span></dd></dl> <p>Čo je rovnica ťažiska. V homogénnom poli môžeme gravitačné a zotrvačné sily pôsobiace na jednotlivé elementy tuhého telesa nahradiť jednou silou, ktorej veľkosť bude úmerná celkovej hmotnosti telesa a pôsobisko bude mať v ťažisku. </p> <div class="mw-heading mw-heading2"><h2 id="Referencie">Referencie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C5%A4a%C5%BEisko_(fyzika)&veaction=edit&section=6" title="Upraviť sekciu: Referencie" class="mw-editsection-visualeditor"><span>upraviť</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C5%A4a%C5%BEisko_(fyzika)&action=edit&section=6" title="Editovat zdrojový kód sekce Referencie"><span>upraviť zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://www.math.sk/skripta2/node68.html">https://www.math.sk/skripta2/node68.html</a></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Zdroj: „<a dir="ltr" href="https://sk.wikipedia.org/w/index.php?title=Ťažisko_(fyzika)&oldid=7708005">https://sk.wikipedia.org/w/index.php?title=Ťažisko_(fyzika)&oldid=7708005</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%C5%A0peci%C3%A1lne:Kateg%C3%B3rie" title="Špeciálne:Kategórie">Kategória</a>: <ul><li><a href="/wiki/Kateg%C3%B3ria:Dynamika" title="Kategória:Dynamika">Dynamika</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Dátum a čas poslednej úpravy tejto stránky: 15. október 2023, 16:41.</li> <li id="footer-info-copyright">Text je dostupný za podmienok <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution/Share-Alike License 4.0 Unported</a>; prípadne za ďalších podmienok. 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