Analiză dimensională

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<button aria-controls="toc-Teorema_invarianței-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>Toggle Teorema invarianței subsection</span> </button> <ul id="toc-Teorema_invarianței-sublist" class="vector-toc-list"> <li id="toc-Exemplu" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exemplu"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Exemplu</span> </div> </a> <ul id="toc-Exemplu-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Teorema_Produselor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Teorema_Produselor"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Teorema Produselor</span> </div> </a> <button aria-controls="toc-Teorema_Produselor-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>Toggle Teorema Produselor subsection</span> </button> <ul id="toc-Teorema_Produselor-sublist" class="vector-toc-list"> <li id="toc-Exemplu_(Similitudine)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exemplu_(Similitudine)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Exemplu (Similitudine)</span> </div> </a> <ul id="toc-Exemplu_(Similitudine)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bibliografie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografie"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Bibliografie</span> </div> </a> <ul id="toc-Bibliografie-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="Cuprins" 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="Comută cuprinsul" > <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">Comută cuprinsul</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">Analiză dimensională</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="Mergeți la un articol în altă limbă. Disponibil în 34 limbi" > <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-34" 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">34 limbi</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D8%A8%D8%B9%D8%AF%D9%8A" title="تحليل بعدي – arabă" lang="ar" hreflang="ar" data-title="تحليل بعدي" data-language-autonym="العربية" data-language-local-name="arabă" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D1%96%D0%B7_%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BD%D0%B0%D1%81%D1%86%D0%B5%D0%B9" title="Аналіз размернасцей – belarusă" lang="be" hreflang="be" data-title="Аналіз размернасцей" data-language-autonym="Беларуская" data-language-local-name="belarusă" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AE%E0%A6%BE%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BE_%E0%A6%B8%E0%A6%AE%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="মাত্রা সমীকরণ – bengaleză" lang="bn" hreflang="bn" data-title="মাত্রা সমীকরণ" data-language-autonym="বাংলা" data-language-local-name="bengaleză" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/An%C3%A0lisi_dimensional" title="Anàlisi dimensional – catalană" lang="ca" hreflang="ca" data-title="Anàlisi dimensional" data-language-autonym="Català" data-language-local-name="catalană" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dimensionsanalyse" title="Dimensionsanalyse – germană" lang="de" hreflang="de" data-title="Dimensionsanalyse" data-language-autonym="Deutsch" data-language-local-name="germană" 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%94%CE%B9%CE%B1%CF%83%CF%84%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CE%B1%CE%BD%CE%AC%CE%BB%CF%85%CF%83%CE%B7" title="Διαστατική ανάλυση – greacă" lang="el" hreflang="el" data-title="Διαστατική ανάλυση" data-language-autonym="Ελληνικά" data-language-local-name="greacă" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Dimensional_analysis" title="Dimensional analysis – engleză" lang="en" hreflang="en" data-title="Dimensional analysis" data-language-autonym="English" data-language-local-name="engleză" 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/An%C3%A1lisis_dimensional" title="Análisis dimensional – spaniolă" lang="es" hreflang="es" data-title="Análisis dimensional" data-language-autonym="Español" data-language-local-name="spaniolă" 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/Dimensionaalanal%C3%BC%C3%BCs" title="Dimensionaalanalüüs – estonă" lang="et" hreflang="et" data-title="Dimensionaalanalüüs" data-language-autonym="Eesti" data-language-local-name="estonă" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%DB%8C%D9%84_%D8%A7%D8%A8%D8%B9%D8%A7%D8%AF%DB%8C" title="تحلیل ابعادی – persană" lang="fa" hreflang="fa" data-title="تحلیل ابعادی" data-language-autonym="فارسی" data-language-local-name="persană" 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/Dimensioanalyysi" title="Dimensioanalyysi – finlandeză" lang="fi" hreflang="fi" data-title="Dimensioanalyysi" data-language-autonym="Suomi" data-language-local-name="finlandeză" 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/Analyse_dimensionnelle" title="Analyse dimensionnelle – franceză" lang="fr" hreflang="fr" data-title="Analyse dimensionnelle" data-language-autonym="Français" data-language-local-name="franceză" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Anail%C3%ADs_thoiseach" title="Anailís thoiseach – irlandeză" lang="ga" hreflang="ga" data-title="Anailís thoiseach" data-language-autonym="Gaeilge" data-language-local-name="irlandeză" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%9E%D7%93%D7%99%D7%AA" title="אנליזה ממדית – ebraică" lang="he" hreflang="he" data-title="אנליזה ממדית" data-language-autonym="עברית" data-language-local-name="ebraică" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%AE%E0%A5%80%E0%A4%AF_%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%A3" title="विमीय विश्लेषण – hindi" lang="hi" hreflang="hi" data-title="विमीय विश्लेषण" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Analiz_dimansyon%C3%A8l" title="Analiz dimansyonèl – haitiană" lang="ht" hreflang="ht" data-title="Analiz dimansyonèl" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitiană" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%89%D5%A1%D6%83%D5%A1%D5%B5%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Չափայնություն – armeană" lang="hy" hreflang="hy" data-title="Չափայնություն" data-language-autonym="Հայերեն" data-language-local-name="armeană" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Analisis_dimensi" title="Analisis dimensi – indoneziană" lang="id" hreflang="id" data-title="Analisis dimensi" data-language-autonym="Bahasa Indonesia" data-language-local-name="indoneziană" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Analisi_dimensionale" title="Analisi dimensionale – italiană" lang="it" hreflang="it" data-title="Analisi dimensionale" data-language-autonym="Italiano" data-language-local-name="italiană" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AC%A1%E5%85%83%E8%A7%A3%E6%9E%90" title="次元解析 – japoneză" lang="ja" hreflang="ja" data-title="次元解析" data-language-autonym="日本語" data-language-local-name="japoneză" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%90%E1%83%9C%E1%83%96%E1%83%9D%E1%83%9B%E1%83%98%E1%83%9A%E1%83%94%E1%83%91%E1%83%90_(%E1%83%A4%E1%83%98%E1%83%96%E1%83%98%E1%83%99%E1%83%90)" title="განზომილება (ფიზიკა) – georgiană" lang="ka" hreflang="ka" data-title="განზომილება (ფიზიკა)" data-language-autonym="ქართული" data-language-local-name="georgiană" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D3%A8%D0%BB%D1%88%D0%B5%D0%BC%D0%B4%D1%96%D0%BB%D1%96%D0%BA%D1%82%D0%B5%D1%80%D0%B4%D1%96_%D1%82%D0%B0%D0%BB%D0%B4%D0%B0%D1%83" title="Өлшемділіктерді талдау – kazahă" lang="kk" hreflang="kk" data-title="Өлшемділіктерді талдау" data-language-autonym="Қазақша" data-language-local-name="kazahă" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B0%A8%EC%9B%90_%ED%95%B4%EC%84%9D" title="차원 해석 – coreeană" lang="ko" hreflang="ko" data-title="차원 해석" data-language-autonym="한국어" data-language-local-name="coreeană" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Dimensjonsanalyse" title="Dimensjonsanalyse – norvegiană nynorsk" lang="nn" hreflang="nn" data-title="Dimensjonsanalyse" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegiană nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Analiza_wymiarowa" title="Analiza wymiarowa – poloneză" lang="pl" hreflang="pl" data-title="Analiza wymiarowa" data-language-autonym="Polski" data-language-local-name="poloneză" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/An%C3%A1lise_dimensional" title="Análise dimensional – portugheză" lang="pt" hreflang="pt" data-title="Análise dimensional" data-language-autonym="Português" data-language-local-name="portugheză" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7_%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Анализ размерности – rusă" lang="ru" hreflang="ru" data-title="Анализ размерности" data-language-autonym="Русский" data-language-local-name="rusă" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Dimensional_analysis" title="Dimensional analysis – Simple English" lang="en-simple" hreflang="en-simple" data-title="Dimensional analysis" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Razse%C5%BEnostna_analiza" title="Razsežnostna analiza – slovenă" lang="sl" hreflang="sl" data-title="Razsežnostna analiza" data-language-autonym="Slovenščina" data-language-local-name="slovenă" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Dimensionsanalys" title="Dimensionsanalys – suedeză" lang="sv" hreflang="sv" data-title="Dimensionsanalys" data-language-autonym="Svenska" data-language-local-name="suedeză" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Boyut_analizi" title="Boyut analizi – turcă" lang="tr" hreflang="tr" data-title="Boyut analizi" data-language-autonym="Türkçe" data-language-local-name="turcă" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7%D1%83_%D1%80%D0%BE%D0%B7%D0%BC%D1%96%D1%80%D0%BD%D0%BE%D1%81%D1%82%D0%B5%D0%B9" title="Метод аналізу розмірностей – ucraineană" lang="uk" hreflang="uk" data-title="Метод аналізу розмірностей" data-language-autonym="Українська" data-language-local-name="ucraineană" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_t%C3%ADch_th%E1%BB%A9_nguy%C3%AAn" title="Phân tích thứ nguyên – vietnameză" lang="vi" hreflang="vi" data-title="Phân tích thứ nguyên" data-language-autonym="Tiếng Việt" 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data-event-name="pinnable-header.vector-appearance.unpin">ascunde</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">De la Wikipedia, enciclopedia liberă</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="ro" dir="ltr"><p><b>Analiza dimensională</b> este un instrument de principiu folosit în <a href="/wiki/Fizic%C4%83" title="Fizică">fizică</a>, <a href="/wiki/Chimie" title="Chimie">chimie</a> și <a href="/wiki/Tehnic%C4%83" title="Tehnică">tehnică</a> la înțelegerea situațiilor care implică utilizarea combinată a mai multor <a href="/wiki/M%C4%83rime_fizic%C4%83" title="Mărime fizică">mărimi fizice</a>. Este un instrument uzual al <a href="/wiki/Om_de_%C8%99tiin%C8%9B%C4%83" title="Om de știință">oamenilor de știință</a> și <a href="/wiki/Inginer" title="Inginer">inginerilor</a> pentru a verifica plauzibilitatea diferitelor tipuri de <a href="/wiki/Unitate_de_m%C4%83sur%C4%83" title="Unitate de măsură">unități de măsură</a> derivate, a consistenței <a href="/wiki/Ecua%C8%9Bie" title="Ecuație">ecuațiilor</a> și a metodelor de calcul. Este folosită de asemenea pentru a face ipoteze pertinente asupra <a href="/wiki/Fenomen" title="Fenomen">fenomenelor</a> fizice care să fie verificate <a href="/wiki/Experiment" title="Experiment">experimental</a> sau prin teorii mai evoluate. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Metoda_de_lucru_algebric_cu_dimensiuni">Metoda de lucru algebric cu dimensiuni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analiz%C4%83_dimensional%C4%83&veaction=edit&section=1" title="Modifică secțiunea: Metoda de lucru algebric cu dimensiuni" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analiz%C4%83_dimensional%C4%83&action=edit&section=1" title="Edit section's source code: Metoda de lucru algebric cu dimensiuni"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>verificarea corectitudinii scrierii relațiilor fizice;</li> <li>obține rezultate noi din considerente pur dimensionale;</li></ul> <div class="mw-heading mw-heading2"><h2 id="Principiul_omogenității"><span id="Principiul_omogenit.C4.83.C8.9Bii"></span>Principiul omogenității</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analiz%C4%83_dimensional%C4%83&veaction=edit&section=2" title="Modifică secțiunea: Principiul omogenității" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analiz%C4%83_dimensional%C4%83&action=edit&section=2" title="Edit section's source code: Principiul omogenității"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Orice relație fizică (între mărimi) trebuie să treacă într-o relație matematică între numere. Pentru acestea termenii unei relații trebuie să fie omogeni = să aibă aceaṣi dimensiune = <a href="/w/index.php?title=Echidimensionali&action=edit&redlink=1" class="new" title="Echidimensionali — pagină inexistentă">echidimensionali</a> </p> <div class="mw-heading mw-heading2"><h2 id="Teorema_invarianței"><span id="Teorema_invarian.C8.9Bei"></span>Teorema invarianței</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analiz%C4%83_dimensional%C4%83&veaction=edit&section=3" title="Modifică secțiunea: Teorema invarianței" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analiz%C4%83_dimensional%C4%83&action=edit&section=3" title="Edit section's source code: Teorema invarianței"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pentru ca o relație fizică să fie invariantă la schimbarea unității de măsură este necesar ca mărimile derivate să se exprime în funcție de mărimile fundamentale ca un produs de puteri. </p> <div class="mw-heading mw-heading3"><h3 id="Exemplu">Exemplu</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analiz%C4%83_dimensional%C4%83&veaction=edit&section=4" title="Modifică secțiunea: Exemplu" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analiz%C4%83_dimensional%C4%83&action=edit&section=4" title="Edit section's source code: Exemplu"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fie <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 f(m,v,E_{c})=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(m,v,E_{c})=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d90c8518055a0e386418ad25e78515cc79cbad4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.631ex; height:2.843ex;" alt="{\displaystyle f(m,v,E_{c})=0\,}"></span>, o relație funcțională pentru <a href="/wiki/Energie_cinetic%C4%83" title="Energie cinetică">energia cinetică</a> a <a href="/wiki/Punct_material" title="Punct material">punctului material</a>, unde: <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\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37129e832c2c81b9f146dd22228d409bd099b295" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.427ex; height:1.676ex;" alt="{\displaystyle m\,}"></span> este masa, <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 v\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b67d1fd725a759a151374b793113d7a78a65da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.515ex; height:1.676ex;" alt="{\displaystyle v\,}"></span> este viteza și <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 E_{c}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{c}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d9227418822da7ccc84fac1363975549971348f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.047ex; height:2.509ex;" alt="{\displaystyle E_{c}\,}"></span> este energia cinetică. </p><p>Mărimi fundamentale pentru <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 E_{c}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{c}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d9227418822da7ccc84fac1363975549971348f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.047ex; height:2.509ex;" alt="{\displaystyle E_{c}\,}"></span>: <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\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37129e832c2c81b9f146dd22228d409bd099b295" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.427ex; height:1.676ex;" alt="{\displaystyle m\,}"></span>, <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 v\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b67d1fd725a759a151374b793113d7a78a65da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.515ex; height:1.676ex;" alt="{\displaystyle v\,}"></span> </p><p>Mărimea derivată: <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 E_{c}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{c}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d9227418822da7ccc84fac1363975549971348f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.047ex; height:2.509ex;" alt="{\displaystyle E_{c}\,}"></span>. </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 E_{c}=m^{r_{1}}v^{r_{2}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{c}=m^{r_{1}}v^{r_{2}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41b1880789ff62c96eee73f9b79814e00ec2a1c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.924ex; height:2.676ex;" alt="{\displaystyle E_{c}=m^{r_{1}}v^{r_{2}}\,}"></span></dd> <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 [E_{c}]=ML^{2}T^{-2};\quad [m]=M;\quad [v]=LT^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mi>M</mi> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>;</mo> <mspace width="1em" /> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>M</mi> <mo>;</mo> <mspace width="1em" /> <mo stretchy="false">[</mo> <mi>v</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>L</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [E_{c}]=ML^{2}T^{-2};\quad [m]=M;\quad [v]=LT^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e440821b27553fa9cd211fb5cb4a9e45d8dde5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.927ex; height:3.176ex;" alt="{\displaystyle [E_{c}]=ML^{2}T^{-2};\quad [m]=M;\quad [v]=LT^{-1}}"></span></dd> <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 ML^{2}T^{-2}={(M)}^{r_{1}}{(LT^{-1})}^{r_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>L</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ML^{2}T^{-2}={(M)}^{r_{1}}{(LT^{-1})}^{r_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2479b35559bea08fc68ae9ada6b9d77c7ba9e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.538ex; height:3.176ex;" alt="{\displaystyle ML^{2}T^{-2}={(M)}^{r_{1}}{(LT^{-1})}^{r_{2}}}"></span></dd></dl> <p>deci <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 r_{1}=1;r_{2}=2\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>;</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}=1;r_{2}=2\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b041897fe26edbc48ae51def6a7e7674a25d942" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.149ex; height:2.509ex;" alt="{\displaystyle r_{1}=1;r_{2}=2\,}"></span> </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 \Rightarrow E_{c}=Ct\cdot mv^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi>C</mi> <mi>t</mi> <mo>⋅</mo> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow E_{c}=Ct\cdot mv^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7446f20124e118c2887e7bd878173dcd578de12e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.234ex; height:3.009ex;" alt="{\displaystyle \Rightarrow E_{c}=Ct\cdot mv^{2}}"></span></dd></dl> <p>unde <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 Ct}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ct}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa48bb3e826405c1af8566df7fb374a114b363fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.606ex; height:2.176ex;" alt="{\displaystyle Ct}"></span> este o constantă. </p> <div class="mw-heading mw-heading2"><h2 id="Teorema_Produselor">Teorema Produselor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analiz%C4%83_dimensional%C4%83&veaction=edit&section=5" title="Modifică secțiunea: Teorema Produselor" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analiz%C4%83_dimensional%C4%83&action=edit&section=5" title="Edit section's source code: Teorema Produselor"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 f(x_{1},\dots ,x_{i},y_{k+1},\dots ,y_{k+j},\dots ,y_{n})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{1},\dots ,x_{i},y_{k+1},\dots ,y_{k+j},\dots ,y_{n})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a9e3fcfe6de8cd56937db74748fea68ae5a55a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.301ex; height:3.009ex;" alt="{\displaystyle f(x_{1},\dots ,x_{i},y_{k+1},\dots ,y_{k+j},\dots ,y_{n})=0}"></span></dd> <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 f(\prod _{i=1}^{N}\dots \prod _{i=N-k}^{N})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</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>…</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>N</mi> <mo>−</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\prod _{i=1}^{N}\dots \prod _{i=N-k}^{N})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/490bfa50c9f5241520c118c12b7bdf53b97ea7dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.256ex; height:7.509ex;" alt="{\displaystyle f(\prod _{i=1}^{N}\dots \prod _{i=N-k}^{N})=0}"></span></dd></dl> <p>unde <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 \prod _{i=N-k}^{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mi>N</mi> <mo>−</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{i=N-k}^{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cfd55d6754875cf1d330093fcdd1c713a6c4bff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:5.44ex; height:7.509ex;" alt="{\displaystyle \prod _{i=N-k}^{N}}"></span> - complexe adimensionale; k-rangul matricii dimensionale </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 x_{1}\dots x_{i}\dots x_{k}\dots x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>…</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>…</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}\dots x_{i}\dots x_{k}\dots x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51b1383b56bdc85a70941d1038deefdcf65a74a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.972ex; height:2.009ex;" alt="{\displaystyle x_{1}\dots x_{i}\dots x_{k}\dots x_{n}}"></span></dd> <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 [x_{i}]=L^{\alpha i}M^{\beta i}\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>i</mi> </mrow> </msup> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>i</mi> </mrow> </msup> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{i}]=L^{\alpha i}M^{\beta i}\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8b2085d789985bbbe4febd47c861203a91a911" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.307ex; height:3.176ex;" alt="{\displaystyle [x_{i}]=L^{\alpha i}M^{\beta i}\dots }"></span></dd> <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 {\begin{bmatrix}\alpha _{1}&\cdots &\alpha _{n}\\\vdots &\ddots &\vdots \\\dots &\cdots &\dots \end{bmatrix}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mo>…</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mo>…</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\alpha _{1}&\cdots &\alpha _{n}\\\vdots &\ddots &\vdots \\\dots &\cdots &\dots \end{bmatrix}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7115305b37eaec2ddc21b85c8f90ef1c7dfbcf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:16.535ex; height:11.009ex;" alt="{\displaystyle {\begin{bmatrix}\alpha _{1}&\cdots &\alpha _{n}\\\vdots &\ddots &\vdots \\\dots &\cdots &\dots \end{bmatrix}}\!}"></span></dd></dl> <p><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 {\begin{matrix}\prod _{a}\end{matrix}}={\frac {a}{\prod fundamentale}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </munder> </mtd> </mtr> </mtable> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mo>∏</mo> <mi>f</mi> <mi>u</mi> <mi>n</mi> <mi>d</mi> <mi>a</mi> <mi>m</mi> <mi>e</mi> <mi>n</mi> <mi>t</mi> <mi>a</mi> <mi>l</mi> <mi>e</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\prod _{a}\end{matrix}}={\frac {a}{\prod fundamentale}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8909f564d36289fbd78b9db7598f2df4a042fa2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.376ex; height:5.509ex;" alt="{\displaystyle {\begin{matrix}\prod _{a}\end{matrix}}={\frac {a}{\prod fundamentale}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Exemplu_(Similitudine)"><span id="Exemplu_.28Similitudine.29"></span>Exemplu (Similitudine)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Analiz%C4%83_dimensional%C4%83&veaction=edit&section=6" title="Modifică secțiunea: Exemplu (Similitudine)" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Analiz%C4%83_dimensional%C4%83&action=edit&section=6" title="Edit section's source code: Exemplu (Similitudine)"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Acceleraṭia căderii libere a unui <a href="/wiki/Corp_(fizic%C4%83)" title="Corp (fizică)">corp</a> la suprafaṭa unui astru sferic omogen de rază R și masă m depinde de: m, R, k unde k este constanta atracției universale. Dacă pentru un astru cu raza R ṣi masa m corpurile cad liber cu accelerația g=10 m/s la pătrat, cu ce accelerație vor cădea corpurile la suprafața unui astru cu raza R'=R/2 și de masă m'=m/10? (Planeta Marte). </p> <ul><li>Rezolvare:</li></ul> <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 g=f(m,R,K)\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>R</mi> <mo>,</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=f(m,R,K)\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd329c5aaec628c639e9025403d1ffdb3578f464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.166ex; width:15.02ex; height:2.843ex;" alt="{\displaystyle g=f(m,R,K)\!}"></span></dd> <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 [g]=LT^{-2};[m]=M;[k]=L^{3}T^{-2}M^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>g</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>L</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>;</mo> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>M</mi> <mo>;</mo> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [g]=LT^{-2};[m]=M;[k]=L^{3}T^{-2}M^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eaa47eea50071faaf7245d36316ce0f53ad682b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.211ex; height:3.176ex;" alt="{\displaystyle [g]=LT^{-2};[m]=M;[k]=L^{3}T^{-2}M^{-1}}"></span></dd> <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 f(m,R,K,g)=0\rightharpoondown n=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>R</mi> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇁</mo> <mi>n</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(m,R,K,g)=0\rightharpoondown n=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22636db575e606f48e0074f11f4e0ae2d8c69e62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.707ex; height:2.843ex;" alt="{\displaystyle f(m,R,K,g)=0\rightharpoondown n=4}"></span></dd> <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 rang=3;k=3;n-k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>a</mi> <mi>n</mi> <mi>g</mi> <mo>=</mo> <mn>3</mn> <mo>;</mo> <mi>k</mi> <mo>=</mo> <mn>3</mn> <mo>;</mo> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rang=3;k=3;n-k=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/617e3779e2f0859175f5467d25784a4bf0b6cebd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.297ex; height:2.509ex;" alt="{\displaystyle rang=3;k=3;n-k=1}"></span></dd> <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 {\begin{matrix}\prod _{g}\end{matrix}}={\frac {g}{m^{r_{1}}R^{r_{2}}k^{r_{3}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </munder> </mtd> </mtr> </mtable> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mrow> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\prod _{g}\end{matrix}}={\frac {g}{m^{r_{1}}R^{r_{2}}k^{r_{3}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47f0b4a4a7ec221cfe878d2e52445e53deca54c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.333ex; height:5.009ex;" alt="{\displaystyle {\begin{matrix}\prod _{g}\end{matrix}}={\frac {g}{m^{r_{1}}R^{r_{2}}k^{r_{3}}}}}"></span></dd> <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 r_{2}+3r_{3}=1;r_{1}-r_{3}=0;-2r_{3}=-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>;</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>;</mo> <mo>−</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}+3r_{3}=1;r_{1}-r_{3}=0;-2r_{3}=-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b739ee4ba2bbfcf54694d1665a3d384f731bfb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.987ex; height:2.509ex;" alt="{\displaystyle r_{2}+3r_{3}=1;r_{1}-r_{3}=0;-2r_{3}=-2}"></span></dd> <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 r_{3}=1;r_{1}=1;r_{2}=-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>;</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>;</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}=1;r_{1}=1;r_{2}=-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3910821c193eff25e84dfb21bbf591f3ae6e21ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.967ex; height:2.509ex;" alt="{\displaystyle r_{3}=1;r_{1}=1;r_{2}=-2}"></span></dd> <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 {\begin{matrix}\prod _{g}\end{matrix}}={\frac {gR^{2}}{mk}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </munder> </mtd> </mtr> </mtable> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <mi>k</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\prod _{g}\end{matrix}}={\frac {gR^{2}}{mk}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f891849601a1f04dc8bc3627fb31da77e475fdea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.835ex; height:5.843ex;" alt="{\displaystyle {\begin{matrix}\prod _{g}\end{matrix}}={\frac {gR^{2}}{mk}}}"></span></dd> <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 {\begin{matrix}\prod _{g}'\end{matrix}}={\frac {g'R'^{2}}{m'k'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </munder> <mo>′</mo> </msup> </mtd> </mtr> </mtable> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <msup> <mi>R</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mrow> <mrow> <msup> <mi>m</mi> <mo>′</mo> </msup> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\prod _{g}'\end{matrix}}={\frac {g'R'^{2}}{m'k'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/736469666e8fca817fe01f2a5a9f1bc7e097a0aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.659ex; height:5.843ex;" alt="{\displaystyle {\begin{matrix}\prod _{g}'\end{matrix}}={\frac {g'R'^{2}}{m'k'}}}"></span></dd></dl> <ul><li>Din Teorema lui Newton:</li></ul> <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 {\begin{matrix}\prod _{g}'\end{matrix}}={\begin{matrix}\prod _{g}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </munder> <mo>′</mo> </msup> </mtd> </mtr> </mtable> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </munder> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\prod _{g}'\end{matrix}}={\begin{matrix}\prod _{g}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dec503264a081ab1cf8a041518aa0531939e1d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.937ex; margin-bottom: -0.235ex; width:11.716ex; height:3.509ex;" alt="{\displaystyle {\begin{matrix}\prod _{g}'\end{matrix}}={\begin{matrix}\prod _{g}\end{matrix}}}"></span></dd> <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 {\frac {gR^{2}}{mk}}={\frac {g'R'^{2}}{m'k'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <msup> <mi>R</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mrow> <mrow> <msup> <mi>m</mi> <mo>′</mo> </msup> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {gR^{2}}{mk}}={\frac {g'R'^{2}}{m'k'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca2fdfcd6de9d1fb9017db563859801979d9ef3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.779ex; height:5.843ex;" alt="{\displaystyle {\frac {gR^{2}}{mk}}={\frac {g'R'^{2}}{m'k'}}}"></span></dd> <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 {\frac {gR^{2}}{mk}}={\frac {g'R'^{2}}{m'k'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <msup> <mi>R</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mrow> <mrow> <msup> <mi>m</mi> <mo>′</mo> </msup> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {gR^{2}}{mk}}={\frac {g'R'^{2}}{m'k'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca2fdfcd6de9d1fb9017db563859801979d9ef3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.779ex; height:5.843ex;" alt="{\displaystyle {\frac {gR^{2}}{mk}}={\frac {g'R'^{2}}{m'k'}}}"></span></dd> <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 {\frac {gR^{2}}{mk}}={\frac {g{\frac {R^{2}}{4}}}{{\frac {m}{10}}k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>m</mi> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mn>10</mn> </mfrac> </mrow> <mi>k</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {gR^{2}}{mk}}={\frac {g{\frac {R^{2}}{4}}}{{\frac {m}{10}}k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9675e055403fdfe0634ef4f09c15d27e46b608d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.736ex; height:8.009ex;" alt="{\displaystyle {\frac {gR^{2}}{mk}}={\frac {g{\frac {R^{2}}{4}}}{{\frac {m}{10}}k}}}"></span></dd> <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 \Rightarrow g'=4{\frac {m}{s^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow g'=4{\frac {m}{s^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff15dc62a033d4337cfc6aafeecfb226dced67ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; 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