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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">Início</div> </a> </li> <li id="toc-Equação_do_movimento" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equação_do_movimento"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Equação do movimento</span> </div> </a> <ul id="toc-Equação_do_movimento-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fórmula_do_período_para_pequenas_oscilações" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fórmula_do_período_para_pequenas_oscilações"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Fórmula do período para pequenas oscilações</span> </div> </a> <button aria-controls="toc-Fórmula_do_período_para_pequenas_oscilações-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>Alternar a subsecção Fórmula do período para pequenas oscilações</span> </button> <ul id="toc-Fórmula_do_período_para_pequenas_oscilações-sublist" class="vector-toc-list"> <li id="toc-Estimando_o_comprimento_do_pêndulo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estimando_o_comprimento_do_pêndulo"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Estimando o comprimento do pêndulo</span> </div> </a> <ul id="toc-Estimando_o_comprimento_do_pêndulo-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Pêndulo_físico" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pêndulo_físico"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Pêndulo físico</span> </div> </a> <ul id="toc-Pêndulo_físico-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conservação_de_energia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conservação_de_energia"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Conservação de energia</span> </div> </a> <ul id="toc-Conservação_de_energia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ver também</span> </div> </a> <ul id="toc-Ver_também-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notas"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notas</span> </div> </a> <ul id="toc-Notas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-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="Conteúdo" 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="Alternar o índice" > <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">Alternar o índice</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">Pêndulo</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="Ir para um artigo noutra língua. Disponível em 77 línguas" > <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-77" 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">77 línguas</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/%D9%86%D9%88%D8%A7%D8%B3" title="نواس — árabe" lang="ar" hreflang="ar" data-title="نواس" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/P%C3%A9ndulu" title="Péndulu — asturiano" lang="ast" hreflang="ast" data-title="Péndulu" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/K%C9%99fgir" title="Kəfgir — azerbaijano" lang="az" hreflang="az" data-title="Kəfgir" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B0%D1%8F%D1%82%D0%BD%D1%96%D0%BA" title="Маятнік — bielorrusso" lang="be" hreflang="be" data-title="Маятнік" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%85%D0%B0%D0%BB%D0%BE" title="Махало — búlgaro" lang="bg" hreflang="bg" data-title="Махало" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/Timbatu" title="Timbatu — Banjar" lang="bjn" hreflang="bjn" data-title="Timbatu" data-language-autonym="Banjar" data-language-local-name="Banjar" class="interlanguage-link-target"><span>Banjar</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%B0%E0%A6%B2_%E0%A6%A6%E0%A7%8B%E0%A6%B2%E0%A6%95" title="সরল দোলক — bengalês" lang="bn" hreflang="bn" data-title="সরল দোলক" data-language-autonym="বাংলা" data-language-local-name="bengalês" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Klatno" title="Klatno — bósnio" lang="bs" hreflang="bs" data-title="Klatno" data-language-autonym="Bosanski" data-language-local-name="bósnio" 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/P%C3%A8ndol" title="Pèndol — catalão" lang="ca" hreflang="ca" data-title="Pèndol" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kyvadlo" title="Kyvadlo — checo" lang="cs" hreflang="cs" data-title="Kyvadlo" data-language-autonym="Čeština" data-language-local-name="checo" 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/Pendul" title="Pendul — dinamarquês" lang="da" hreflang="da" data-title="Pendul" data-language-autonym="Dansk" data-language-local-name="dinamarquês" 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/Pendel" title="Pendel — alemão" lang="de" hreflang="de" data-title="Pendel" data-language-autonym="Deutsch" data-language-local-name="alemão" 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%95%CE%BA%CE%BA%CF%81%CE%B5%CE%BC%CE%AD%CF%82" title="Εκκρεμές — grego" lang="el" hreflang="el" data-title="Εκκρεμές" data-language-autonym="Ελληνικά" data-language-local-name="grego" 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/Pendulum" title="Pendulum — inglês" lang="en" hreflang="en" data-title="Pendulum" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Pendolo" title="Pendolo — esperanto" lang="eo" hreflang="eo" data-title="Pendolo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/P%C3%A9ndulo" title="Péndulo — espanhol" lang="es" hreflang="es" data-title="Péndulo" data-language-autonym="Español" data-language-local-name="espanhol" 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/Pendel" title="Pendel — estónio" lang="et" hreflang="et" data-title="Pendel" data-language-autonym="Eesti" data-language-local-name="estónio" 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/Pendulu" title="Pendulu — basco" lang="eu" hreflang="eu" data-title="Pendulu" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A2%D9%88%D9%86%DA%AF" title="آونگ — persa" lang="fa" hreflang="fa" data-title="آونگ" data-language-autonym="فارسی" data-language-local-name="persa" 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/Heiluri" title="Heiluri — finlandês" lang="fi" hreflang="fi" data-title="Heiluri" data-language-autonym="Suomi" data-language-local-name="finlandês" 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/Pendule_(physique)" title="Pendule (physique) — francês" lang="fr" hreflang="fr" data-title="Pendule (physique)" data-language-autonym="Français" data-language-local-name="francês" 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/Luascad%C3%A1n" title="Luascadán — irlandês" lang="ga" hreflang="ga" data-title="Luascadán" data-language-autonym="Gaeilge" data-language-local-name="irlandês" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E9%90%98%E6%93%BA" title="鐘擺 — gan" lang="gan" hreflang="gan" data-title="鐘擺" data-language-autonym="贛語" data-language-local-name="gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Pandoul_(fizik)" title="Pandoul (fizik) — Guianan Creole" lang="gcr" hreflang="gcr" data-title="Pandoul (fizik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/P%C3%A9ndulo" title="Péndulo — galego" lang="gl" hreflang="gl" data-title="Péndulo" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%98%D7%95%D7%98%D7%9C%D7%AA" title="מטוטלת — hebraico" lang="he" hreflang="he" data-title="מטוטלת" data-language-autonym="עברית" data-language-local-name="hebraico" 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%B2%E0%A5%8B%E0%A4%B2%E0%A4%95" title="लोलक — hindi" lang="hi" hreflang="hi" data-title="लोलक" data-language-autonym="हिन्दी" data-language-local-name="hindi" 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/Njihalo" title="Njihalo — croata" lang="hr" hreflang="hr" data-title="Njihalo" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Pandil" title="Pandil — haitiano" lang="ht" hreflang="ht" data-title="Pandil" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitiano" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Fizikai_inga" title="Fizikai inga — húngaro" lang="hu" hreflang="hu" data-title="Fizikai inga" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%83%D5%B8%D5%B3%D5%A1%D5%B6%D5%A1%D5%AF" title="Ճոճանակ — arménio" lang="hy" hreflang="hy" data-title="Ճոճանակ" data-language-autonym="Հայերեն" data-language-local-name="arménio" 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/Bandul" title="Bandul — indonésio" lang="id" hreflang="id" data-title="Bandul" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Pendulo" title="Pendulo — ido" lang="io" hreflang="io" data-title="Pendulo" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Pend%C3%BAll" title="Pendúll — islandês" lang="is" hreflang="is" data-title="Pendúll" data-language-autonym="Íslenska" data-language-local-name="islandês" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Pendolo" title="Pendolo — italiano" lang="it" hreflang="it" data-title="Pendolo" data-language-autonym="Italiano" data-language-local-name="italiano" 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%8C%AF%E3%82%8A%E5%AD%90" title="振り子 — japonês" lang="ja" hreflang="ja" data-title="振り子" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Penjulom" title="Penjulom — Jamaican Creole English" lang="jam" hreflang="jam" data-title="Penjulom" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A5%E1%83%90%E1%83%9C%E1%83%A5%E1%83%90%E1%83%A0%E1%83%90" title="ქანქარა — georgiano" lang="ka" hreflang="ka" data-title="ქანქარა" data-language-autonym="ქართული" data-language-local-name="georgiano" 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/%D0%9C%D0%B0%D1%8F%D1%82%D0%BD%D0%B8%D0%BA" title="Маятник — cazaque" lang="kk" hreflang="kk" data-title="Маятник" data-language-autonym="Қазақша" data-language-local-name="cazaque" 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%B2%E0%B3%8B%E0%B2%B2%E0%B2%95" title="ಲೋಲಕ — canarim" lang="kn" hreflang="kn" data-title="ಲೋಲಕ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="canarim" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%84%EC%9E%90" title="진자 — coreano" lang="ko" hreflang="ko" data-title="진자" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9C%D0%B0%D1%8F%D1%82%D0%BD%D0%B8%D0%BA" title="Маятник — quirguiz" lang="ky" hreflang="ky" data-title="Маятник" data-language-autonym="Кыргызча" data-language-local-name="quirguiz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Pendulum" title="Pendulum — latim" lang="la" hreflang="la" data-title="Pendulum" data-language-autonym="Latina" data-language-local-name="latim" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Fizin%C4%97_svyruokl%C4%97" title="Fizinė svyruoklė — lituano" lang="lt" hreflang="lt" data-title="Fizinė svyruoklė" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Sv%C4%81rsts" title="Svārsts — letão" lang="lv" hreflang="lv" data-title="Svārsts" data-language-autonym="Latviešu" data-language-local-name="letão" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%86%E0%B5%BB%E0%B4%A1%E0%B5%81%E0%B4%B2%E0%B4%82" title="പെൻഡുലം — malaiala" lang="ml" hreflang="ml" data-title="പെൻഡുലം" data-language-autonym="മലയാളം" data-language-local-name="malaiala" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Bandul" title="Bandul — malaio" lang="ms" hreflang="ms" data-title="Bandul" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/P%C3%A9ndulo_simples" title="Péndulo simples — mirandês" lang="mwl" hreflang="mwl" data-title="Péndulo simples" data-language-autonym="Mirandés" data-language-local-name="mirandês" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Slinger_(natuurkunde)" title="Slinger (natuurkunde) — neerlandês" lang="nl" hreflang="nl" data-title="Slinger (natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandês" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Pendel" title="Pendel — norueguês nynorsk" lang="nn" hreflang="nn" data-title="Pendel" data-language-autonym="Norsk nynorsk" data-language-local-name="norueguês nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Pendel" title="Pendel — norueguês bokmål" lang="nb" hreflang="nb" data-title="Pendel" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A9%88%E0%A8%82%E0%A8%A1%E0%A9%82%E0%A8%B2%E0%A8%AE" title="ਪੈਂਡੂਲਮ — panjabi" lang="pa" hreflang="pa" data-title="ਪੈਂਡੂਲਮ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Historia_i_zastosowania_wahad%C5%82a" title="Historia i zastosowania wahadła — polaco" lang="pl" hreflang="pl" data-title="Historia i zastosowania wahadła" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Pendul_gravita%C8%9Bional" title="Pendul gravitațional — romeno" lang="ro" hreflang="ro" data-title="Pendul gravitațional" data-language-autonym="Română" data-language-local-name="romeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%8F%D1%82%D0%BD%D0%B8%D0%BA" title="Маятник — russo" lang="ru" hreflang="ru" data-title="Маятник" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Pendle" title="Pendle — scots" lang="sco" hreflang="sco" data-title="Pendle" data-language-autonym="Scots" data-language-local-name="scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Klatno" title="Klatno — servo-croata" lang="sh" hreflang="sh" data-title="Klatno" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="servo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B7%90%E0%B6%AF%E0%B7%92%E0%B6%BD%E0%B6%B9%E0%B6%BA" title="පැදිලඹය — cingalês" lang="si" hreflang="si" data-title="පැදිලඹය" data-language-autonym="සිංහල" data-language-local-name="cingalês" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Pendulum" title="Pendulum — Simple English" lang="en-simple" hreflang="en-simple" data-title="Pendulum" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kyvadlo" title="Kyvadlo — eslovaco" lang="sk" hreflang="sk" data-title="Kyvadlo" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Nihalo" title="Nihalo — esloveno" lang="sl" hreflang="sl" data-title="Nihalo" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Klatno" title="Klatno — sérvio" lang="sr" hreflang="sr" data-title="Klatno" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Babandul" title="Babandul — sundanês" lang="su" hreflang="su" data-title="Babandul" data-language-autonym="Sunda" data-language-local-name="sundanês" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Pendel" title="Pendel — sueco" lang="sv" hreflang="sv" data-title="Pendel" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8A%E0%AE%9A%E0%AE%B2%E0%AF%8D_(%E0%AE%87%E0%AE%AF%E0%AE%B1%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="ஊசல் (இயற்பியல்) — tâmil" lang="ta" hreflang="ta" data-title="ஊசல் (இயற்பியல்)" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B2%E0%B1%8B%E0%B0%B2%E0%B0%95%E0%B0%AE%E0%B1%81" title="లోలకము — telugu" lang="te" hreflang="te" data-title="లోలకము" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%A0%D0%B0%D2%9B%D2%9B%D0%BE%D1%81%D0%B0%D0%BA" title="Раққосак — tajique" lang="tg" hreflang="tg" data-title="Раққосак" data-language-autonym="Тоҷикӣ" data-language-local-name="tajique" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Pendulo" title="Pendulo — tagalo" lang="tl" hreflang="tl" data-title="Pendulo" data-language-autonym="Tagalog" data-language-local-name="tagalo" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Sarka%C3%A7" title="Sarkaç — turco" lang="tr" hreflang="tr" data-title="Sarkaç" data-language-autonym="Türkçe" data-language-local-name="turco" 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%9C%D0%B0%D1%8F%D1%82%D0%BD%D0%B8%D0%BA" title="Маятник — tatar" lang="tt" hreflang="tt" data-title="Маятник" data-language-autonym="Татарча / tatarça" data-language-local-name="tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%8F%D1%82%D0%BD%D0%B8%D0%BA" title="Маятник — ucraniano" lang="uk" hreflang="uk" data-title="Маятник" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Mayatnik" title="Mayatnik — usbeque" lang="uz" hreflang="uz" data-title="Mayatnik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeque" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Con_l%E1%BA%AFc" title="Con lắc — vietnamita" lang="vi" hreflang="vi" data-title="Con lắc" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Pendulo" title="Pendulo — waray" lang="war" hreflang="war" data-title="Pendulo" data-language-autonym="Winaray" data-language-local-name="waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%91%86" title="摆 — wu" lang="wuu" hreflang="wuu" data-title="摆" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%93%BA" title="擺 — chinês" lang="zh" hreflang="zh" data-title="擺" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%93%BA" title="擺 — cantonês" lang="yue" hreflang="yue" data-title="擺" data-language-autonym="粵語" data-language-local-name="cantonês" class="interlanguage-link-target"><span>粵語</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/Q20702#sitelinks-wikipedia" title="Editar hiperligações interlínguas" class="wbc-editpage">Editar hiperligações</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="Espaços nominais"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/P%C3%AAndulo" title="Ver a página de conteúdo [c]" accesskey="c"><span>Artigo</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discuss%C3%A3o:P%C3%AAndulo" rel="discussion" 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.tmbox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-night .mw-parser-output .tmbox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-os .mw-parser-output .tmbox-speedy{background-color:#310402}}body.skin--responsive .mw-parser-output table.tmbox img{max-width:none!important}</style><table class="box-Mais_notas plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/Ficheiro:Question_book-new.svg" class="mw-file-description"><img alt="Esta página cita fontes, mas não cobrem todo o conteúdo" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">Esta página <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Cite_as_fontes" title="Wikipédia:Livro de estilo/Cite as fontes">cita fontes</a>, mas que <b><a href="/wiki/Wikip%C3%A9dia:V" class="mw-redirect" title="Wikipédia:V">não cobrem</a> todo o conteúdo</b>.<span class="hide-when-compact"> Ajude a <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Refer%C3%AAncias_e_notas_de_rodap%C3%A9" title="Wikipédia:Livro de estilo/Referências e notas de rodapé">inserir referências</a> (<small><i>Encontre fontes:</i> <span class="plainlinks"><a rel="nofollow" class="external text" href="https://wikipedialibrary.wmflabs.org/">ABW</a>  •  <a rel="nofollow" class="external text" href="https://www.periodicos.capes.gov.br">CAPES</a>  •  <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&as_epq=P%C3%AAndulo">Google</a> (<a rel="nofollow" class="external text" href="https://www.google.com/search?hl=pt&tbm=nws&q=P%C3%AAndulo&oq=P%C3%AAndulo">N</a> • <a rel="nofollow" class="external text" href="http://books.google.com/books?&as_brr=0&as_epq=P%C3%AAndulo">L</a> • <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?hl=pt&q=P%C3%AAndulo">A</a>)</span></small>).</span> <small class="date-container"><i>(<span class="date">Novembro de 2016</span>)</i></small></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Pendulo.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Pendulo.png/200px-Pendulo.png" decoding="async" width="200" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Pendulo.png/300px-Pendulo.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Pendulo.png/400px-Pendulo.png 2x" data-file-width="500" data-file-height="400" /></a><figcaption>Uma ilustração de um pêndulo simples.</figcaption></figure> <p>Em <a href="/wiki/Mec%C3%A2nica" title="Mecânica">mecânica</a>, um <b>pêndulo simples</b> é um dispositivo que consiste numa <a href="/wiki/Ponto_material" title="Ponto material">massa puntiforme</a> presa a um fio inextensível (a haste pendular) que oscila em torno de um ponto fixo.<sup id="cite_ref-Serway2_1-0" class="reference"><a href="#cite_note-Serway2-1"><span>[</span>1<span>]</span></a></sup> O braço executa movimentos alternados em torno da posição central. O pêndulo é muito utilizado em estudos da <a href="/wiki/For%C3%A7a_peso" class="mw-redirect" title="Força peso">força peso</a> e do <a href="/wiki/Movimento_oscilat%C3%B3rio" class="mw-redirect" title="Movimento oscilatório">movimento oscilatório</a>. </p><p>O pêndulo simples é um modelo teórico bastante simplificado e utilizado com frequência no <a href="/wiki/Ensino_de_F%C3%ADsica" class="mw-redirect" title="Ensino de Física">ensino de Física</a> como representação do <a href="/wiki/Movimento_harm%C3%B4nico_simples" title="Movimento harmônico simples">Movimento Harmônico Simples (MHS)</a>, um movimento periódico tomado como primeira aproximação na descrição de inúmeros <a href="/wiki/Fen%C3%B4meno_f%C3%ADsico" title="Fenômeno físico">fenômenos físicos</a> associados ao movimento de um objeto confinado em uma região de equilíbrio estável.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span>[</span>2<span>]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>nota 1<span>]</span></a></sup> </p><p>A descoberta da periodicidade do movimento pendular foi feita por <a href="/wiki/Galileu_Galilei" title="Galileu Galilei">Galileu Galilei</a>. O movimento de um pêndulo simples envolve basicamente uma grandeza chamada <a href="/wiki/Per%C3%ADodo_(f%C3%ADsica)" title="Período (física)">período</a> (simbolizada por <i>T</i>): é o intervalo de tempo que o objecto leva para percorrer toda a trajectória (ou seja, retornar a sua posição original de lançamento, uma vez que o movimento pendular é periódico). Derivada dessa grandeza, existe a <a href="/wiki/Frequ%C3%AAncia" title="Frequência">frequência</a> (<i>f</i>), numericamente igual ao inverso do período (f = 1 / T), e que portanto se caracteriza pelo número de vezes (ciclos) que o objecto percorre a trajectória pendular num intervalo de tempo específico. A unidade da frequência no <a href="/wiki/Sistema_Internacional_de_Unidades" title="Sistema Internacional de Unidades">SI</a> é o <a href="/wiki/Hertz" title="Hertz">hertz</a>, equivalente a um ciclo por segundo(1/s). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Equação_do_movimento"><span id="Equa.C3.A7.C3.A3o_do_movimento"></span>Equação do movimento</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=P%C3%AAndulo&veaction=edit&section=1" title="Editar secção: Equação do movimento" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=P%C3%AAndulo&action=edit&section=1" title="Editar código-fonte da secção: Equação do movimento"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Equa%C3%A7%C3%A3o_do_p%C3%AAndulo" title="Equação do pêndulo">Equação do pêndulo</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Pendulosimple.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Pendulosimple.jpg/267px-Pendulosimple.jpg" decoding="async" width="267" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Pendulosimple.jpg/401px-Pendulosimple.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/f/f9/Pendulosimple.jpg 2x" data-file-width="532" data-file-height="411" /></a><figcaption>Descrição do movimento de um pêndulo</figcaption></figure> <p>Denota-se o ângulo formado entre a vertical e o braço de pêndulo. Faz-se as seguintes hipóteses: </p> <ol><li>O braço é formado por um fio não flexível que se mantém sempre com o mesmo formato e comprimento.</li> <li>Toda a massa, <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>, do pêndulo está concentrada na ponta do braço a uma distância constante <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 L\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d330bc0cd693cc87e3943137dc591038a89f77e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.97ex; height:2.176ex;" alt="{\displaystyle L\,}"></span> do eixo.</li> <li>Não existem outras forças a actuar no sistema senão a gravidade e a força que mantém o eixo do pêndulo fixo. (O movimento é portanto conservativo).</li> <li>O pêndulo realiza um movimento bidimensional no plano <b>xy</b>.</li></ol> <p>É fácil ver que a <a href="/wiki/Segunda_lei_de_Newton" class="mw-redirect" title="Segunda lei de Newton">segunda lei de Newton</a> fornece a seguinte <a href="/wiki/Equa%C3%A7%C3%A3o_diferencial_ordin%C3%A1ria" title="Equação diferencial ordinária">equação diferencial ordinária</a> não-linear conhecida como <a href="/wiki/Equa%C3%A7%C3%A3o_do_p%C3%AAndulo" title="Equação do pêndulo">equação do pêndulo</a>: </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 {d^{2}\theta \over dt^{2}}+{g \over L}\sin \theta =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mi>L</mi> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d^{2}\theta \over dt^{2}}+{g \over L}\sin \theta =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aea0f293d59a10bf4e366339714a37c656735038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.086ex; height:6.009ex;" alt="{\displaystyle {d^{2}\theta \over dt^{2}}+{g \over L}\sin \theta =0.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Fórmula_do_período_para_pequenas_oscilações"><span id="F.C3.B3rmula_do_per.C3.ADodo_para_pequenas_oscila.C3.A7.C3.B5es"></span>Fórmula do período para pequenas oscilações</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=P%C3%AAndulo&veaction=edit&section=2" title="Editar secção: Fórmula do período para pequenas oscilações" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=P%C3%AAndulo&action=edit&section=2" title="Editar código-fonte da secção: Fórmula do período para pequenas oscilações"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Para pequenas oscilações, Huygens mostrou no seu livro <i>Horologium oscillatorium</i> (1673) que a aproximação <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 \sin \theta \simeq \theta \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>≃</mo> <mi>θ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \theta \simeq \theta \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c8e5c328257cbac07c2f8c01270809bcf8ca3dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.909ex; height:2.176ex;" alt="{\displaystyle \sin \theta \simeq \theta \,}"></span> fornece a seguinte expressão para o período do pêndulo: </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 T_{0}=2\pi {\sqrt {\frac {L}{g}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>L</mi> <mi>g</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}=2\pi {\sqrt {\frac {L}{g}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42171c34f42e118ea0adf5ab79c42342596ba7e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.747ex; height:7.509ex;" alt="{\displaystyle T_{0}=2\pi {\sqrt {\frac {L}{g}}}}"></span></dd></dl> <p><a href="/wiki/T" title="T">T</a>: <a href="/wiki/Per%C3%ADodo_(f%C3%ADsica)" title="Período (física)">período</a> </p> <p><a href="/wiki/L" title="L">L</a>: comprimento do fio</p><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Pendulo_Simples.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/c8/Pendulo_Simples.gif" decoding="async" width="170" height="280" class="mw-file-element" data-file-width="170" data-file-height="280" /></a><figcaption>Descrição do movimento de um pêndulo</figcaption></figure> <p>Uma aproximação para o período <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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> válida para amplitudes tão grandes quanto <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 \theta _{0}=30^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msup> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{0}=30^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b52e3ffb45a324d95074bf9b99d5ef00c6f8e5ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.622ex; height:2.676ex;" alt="{\displaystyle \theta _{0}=30^{\circ }}"></span> foi obtida por Bernoulli em 1749: </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 T\approx T_{0}\left(1+{\theta _{0}^{2} \over 16}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>≈</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>16</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\approx T_{0}\left(1+{\theta _{0}^{2} \over 16}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e71e4ea484e6723a546c5fc194ebfaf41082f15f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.379ex; height:7.509ex;" alt="{\displaystyle T\approx T_{0}\left(1+{\theta _{0}^{2} \over 16}\right)}"></span>.</dd></dl> <p>A obtenção de fórmulas aproximadas para o período do pêndulo no regime não-linear (i.e., para grandes amplitudes) tem recebido muita atenção nos últimos 20 anos. Por exemplo, Kidd e Fogg propuseram, em 2002, a seguinte fórmula prática, válida para <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 \theta _{0}\leq 60^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{0}\leq 60^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398b70c46426e31d7eed20bf4c130d63e67fcc14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.622ex; height:2.676ex;" alt="{\displaystyle \theta _{0}\leq 60^{\circ }}"></span>: </p><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 T\approx T_{0}\,{\frac {1}{\sqrt {\cos(\theta _{0}/2)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>≈</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\approx T_{0}\,{\frac {1}{\sqrt {\cos(\theta _{0}/2)}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c00d3140be6c248d47075838a2a67a754a20a4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.084ex; height:6.509ex;" alt="{\displaystyle T\approx T_{0}\,{\frac {1}{\sqrt {\cos(\theta _{0}/2)}}}}"></span>. </p><p>Em seguida, o brasileiro Fábio Lima, físico da Universidade de Brasília, propôs em 2006 uma fórmula simples ainda mais precisa, válida para qualquer amplitude <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 \theta _{0}\leq 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{0}\leq 90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/033b5c6fe5ea6efe6068598a8394dd2190fd8f66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.622ex; height:2.676ex;" alt="{\displaystyle \theta _{0}\leq 90^{\circ }}"></span>:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span>[</span>3<span>]</span></a></sup> </p><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 T\approx -\,T_{0}\,{\frac {\ln {a}}{1-a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>≈</mo> <mo>−</mo> <mspace width="thinmathspace" /> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\approx -\,T_{0}\,{\frac {\ln {a}}{1-a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8149bbc2f86c0deeec3258745fb448a6a78caddc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.798ex; height:5.509ex;" alt="{\displaystyle T\approx -\,T_{0}\,{\frac {\ln {a}}{1-a}}}"></span>, </p><p>onde <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 a\equiv \cos(\theta _{0}/2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≡</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\equiv \cos(\theta _{0}/2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61e44b9dcb1b7fcc2b7c70818a2174c7789cb505" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.718ex; height:2.843ex;" alt="{\displaystyle a\equiv \cos(\theta _{0}/2)}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Estimando_o_comprimento_do_pêndulo"><span id="Estimando_o_comprimento_do_p.C3.AAndulo"></span>Estimando o comprimento do pêndulo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=P%C3%AAndulo&veaction=edit&section=3" title="Editar secção: Estimando o comprimento do pêndulo" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=P%C3%AAndulo&action=edit&section=3" title="Editar código-fonte da secção: Estimando o comprimento do pêndulo"><span>editar código-fonte</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 T_{0}=2\pi {\sqrt {\frac {\ell }{g}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>ℓ</mi> <mi>g</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}=2\pi {\sqrt {\frac {\ell }{g}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45513d0af67aac50e49d0d27c23441a819f68d67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.281ex; height:7.509ex;" alt="{\displaystyle T_{0}=2\pi {\sqrt {\frac {\ell }{g}}}}"></span> pode ser expresso como <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 \ell ={\frac {g}{\pi ^{2}}}\times {\frac {T_{0}^{2}}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ={\frac {g}{\pi ^{2}}}\times {\frac {T_{0}^{2}}{4}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f68e4939e0ffe0b7633c38a233b776e257b4e39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.39ex; height:6.343ex;" alt="{\displaystyle \ell ={\frac {g}{\pi ^{2}}}\times {\frac {T_{0}^{2}}{4}}.}"></span></dd></dl> <p>Se usarmos o <a href="/wiki/Sistema_internacional_de_unidades" class="mw-redirect" title="Sistema internacional de unidades">Sistema internacional de unidades</a> (isto é, comprimento em <a href="/wiki/Metro" title="Metro">metros</a> e tempo em segundos), então, na superfície da <a href="/wiki/Terra" title="Terra">Terra</a> (<i>g</i> = 9.80665 m/s²), o comprimento do pêndulo pode ser estimado de forma simples a partir do seu período: </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 \ell \approx {\frac {T_{0}^{2}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell \approx {\frac {T_{0}^{2}}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a102a41e9749fe846311e1dd9cc3535688dd32a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.679ex; height:6.009ex;" alt="{\displaystyle \ell \approx {\frac {T_{0}^{2}}{4}}}"></span></dd></dl> <p>Em outras palavras: </p><p>Na superfície da Terra, o número que representa o comprimento de um pêndulo em metros é igual ao obtido calculando-se um quarto do quadrado do seu período em segundos. </p> <div class="mw-heading mw-heading2"><h2 id="Pêndulo_físico"><span id="P.C3.AAndulo_f.C3.ADsico"></span>Pêndulo físico</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=P%C3%AAndulo&veaction=edit&section=4" title="Editar secção: Pêndulo físico" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=P%C3%AAndulo&action=edit&section=4" title="Editar código-fonte da secção: Pêndulo físico"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:P%C3%AAndulo_simples_com_a_base_em_rota%C3%A7%C3%A3o_no_plano_horizontal_e_diagrama_de_for%C3%A7as_externas.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/P%C3%AAndulo_simples_com_a_base_em_rota%C3%A7%C3%A3o_no_plano_horizontal_e_diagrama_de_for%C3%A7as_externas.png/220px-P%C3%AAndulo_simples_com_a_base_em_rota%C3%A7%C3%A3o_no_plano_horizontal_e_diagrama_de_for%C3%A7as_externas.png" decoding="async" width="220" height="114" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/P%C3%AAndulo_simples_com_a_base_em_rota%C3%A7%C3%A3o_no_plano_horizontal_e_diagrama_de_for%C3%A7as_externas.png/330px-P%C3%AAndulo_simples_com_a_base_em_rota%C3%A7%C3%A3o_no_plano_horizontal_e_diagrama_de_for%C3%A7as_externas.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/P%C3%AAndulo_simples_com_a_base_em_rota%C3%A7%C3%A3o_no_plano_horizontal_e_diagrama_de_for%C3%A7as_externas.png/440px-P%C3%AAndulo_simples_com_a_base_em_rota%C3%A7%C3%A3o_no_plano_horizontal_e_diagrama_de_for%C3%A7as_externas.png 2x" data-file-width="490" data-file-height="253" /></a><figcaption>Pêndulo Físico</figcaption></figure> <p>O <b>pêndulo físico</b> pode ser chamado de pêndulo real, pois não tem uma distribuição uniforme de massa. Ele consiste em um objeto que oscila em torno de um eixo de rotação perpendicular ao plano em que se movimenta. Pela Segunda Lei de Newton para corpos extensos, temos: </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 {\tau }}=I{\vec {\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>α</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\tau }}=I{\vec {\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80be770c1e5cd1f929f2072af5143125c7b8f34f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.091ex; height:2.343ex;" alt="{\displaystyle {\vec {\tau }}=I{\vec {\alpha }}}"></span></dd></dl> <p>em que </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 {\tau }}={\vec {r}}\times {\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>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\tau }}={\vec {r}}\times {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac803a111e6f9d07dbb0e251e3764d5bdd1ac09c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.266ex; height:2.843ex;" alt="{\displaystyle {\vec {\tau }}={\vec {r}}\times {\vec {F}}}"></span></dd></dl> <p>Em que τ é o <a href="/wiki/Torque" title="Torque">torque</a> atuante no corpo, α é a <a href="/wiki/Acelera%C3%A7%C3%A3o_angular" title="Aceleração angular">aceleração angular</a>, <b>I</b> é o <a href="/wiki/Momento_de_in%C3%A9rcia" title="Momento de inércia">momento de inércia</a> . Temos também que a força <i>F</i> que realiza torque no corpo é a <a href="/wiki/For%C3%A7a_peso" class="mw-redirect" title="Força peso">força peso</a>, e o braço <b>r</b> é a distância <b>d</b> entre o <a href="/wiki/Centro_de_massa" title="Centro de massa">centro de massa</a> do objeto e o eixo de rotação. Assim temos: </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 \tau =-mgdsen\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mo>−</mo> <mi>m</mi> <mi>g</mi> <mi>d</mi> <mi>s</mi> <mi>e</mi> <mi>n</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =-mgdsen\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24191b580a41270b6d1092d038248518eb029b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.14ex; height:2.509ex;" alt="{\displaystyle \tau =-mgdsen\theta }"></span></dd></dl> <p>e, igualando as equações, tem-se a equação diferencial </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 {\frac {d^{2}\theta }{dt^{2}}}+{\frac {mgd}{I}}sen\theta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>g</mi> <mi>d</mi> </mrow> <mi>I</mi> </mfrac> </mrow> <mi>s</mi> <mi>e</mi> <mi>n</mi> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {mgd}{I}}sen\theta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b2909356bcca6c98a4cb6b93f15864bb91273c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.168ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {mgd}{I}}sen\theta =0}"></span>.</dd></dl> <p>Para pequenos ângulos, podemos usar a aproximação <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 sen\theta =\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mi>e</mi> <mi>n</mi> <mi>θ</mi> <mo>=</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle sen\theta =\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b01534c338c8e640030676b092bc3a542293521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.848ex; height:2.176ex;" alt="{\displaystyle sen\theta =\theta }"></span> sendo possível obter uma solução para a equação diferencial </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 {\frac {d^{2}\theta }{dt^{2}}}+{\frac {mgd}{I}}\theta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>g</mi> <mi>d</mi> </mrow> <mi>I</mi> </mfrac> </mrow> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {mgd}{I}}\theta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf6881c9c17ae9ad7a87fee7e22d25181cb4e56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.599ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {mgd}{I}}\theta =0}"></span></dd></dl> <p>que é homogênea e linear. Assim, temos </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 \omega ={\sqrt {\frac {mgd}{I}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>m</mi> <mi>g</mi> <mi>d</mi> </mrow> <mi>I</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {\frac {mgd}{I}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dac52acd5c54e04bb8693ce94d93660f156606c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.076ex; height:6.343ex;" alt="{\displaystyle \omega ={\sqrt {\frac {mgd}{I}}}}"></span></dd></dl> <p>e </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 T=2\pi {\sqrt {\frac {I}{mgd}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>I</mi> <mrow> <mi>m</mi> <mi>g</mi> <mi>d</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=2\pi {\sqrt {\frac {I}{mgd}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b18b92f608cbe173cf91d7e27bf3ca51a5fc2bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.761ex; height:7.676ex;" alt="{\displaystyle T=2\pi {\sqrt {\frac {I}{mgd}}}}"></span>,</dd></dl> <p>onde <b>T</b> é o <a href="/wiki/Per%C3%ADodo_(f%C3%ADsica)" title="Período (física)">período</a> de oscilação, <b>I</b> é o <a href="/wiki/Momento_de_in%C3%A9rcia" title="Momento de inércia">momento de inércia</a>, <b>m</b> é a <a href="/wiki/Massa" title="Massa">massa</a> do pêndulo, <b>g</b> é o valor da <a href="/wiki/Gravidade" title="Gravidade">aceleração da gravidade</a> e <b>d</b> é a distância do ponto de pivô onde está preso até seu <a href="/wiki/Centro_de_massa" title="Centro de massa">centro de massa</a>. </p><p>Se o ponto de pivô estiver em seu centro de massa, não haverá oscilação. </p> <div class="mw-heading mw-heading2"><h2 id="Conservação_de_energia"><span id="Conserva.C3.A7.C3.A3o_de_energia"></span>Conservação de energia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=P%C3%AAndulo&veaction=edit&section=5" title="Editar secção: Conservação de energia" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=P%C3%AAndulo&action=edit&section=5" title="Editar código-fonte da secção: Conservação de energia"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Pendulo_imagem_CAD.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Pendulo_imagem_CAD.png/220px-Pendulo_imagem_CAD.png" decoding="async" width="220" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Pendulo_imagem_CAD.png/330px-Pendulo_imagem_CAD.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Pendulo_imagem_CAD.png/440px-Pendulo_imagem_CAD.png 2x" data-file-width="880" data-file-height="645" /></a><figcaption></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Pendulo_energia.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/10/Pendulo_energia.gif" decoding="async" width="200" height="205" class="mw-file-element" data-file-width="200" data-file-height="205" /></a><figcaption>Comparação entre a energia cinética e energia potencial gravitacional em um pendulo</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:1806-9126-RBEF-43-e20200360.pdf" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/1806-9126-RBEF-43-e20200360.pdf/page1-220px-1806-9126-RBEF-43-e20200360.pdf.jpg" decoding="async" width="220" height="311" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/1806-9126-RBEF-43-e20200360.pdf/page1-330px-1806-9126-RBEF-43-e20200360.pdf.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/1806-9126-RBEF-43-e20200360.pdf/page1-440px-1806-9126-RBEF-43-e20200360.pdf.jpg 2x" data-file-width="1239" data-file-height="1752" /></a><figcaption>Artigo acadêmico mostrando um experimento utilizando a <a href="/wiki/Arduino" title="Arduino">plataforma arduino</a> para medição da dissipação da energia em um pêndulo devido a resistência do ar.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span>[</span>2<span>]</span></a></sup></figcaption></figure> <p>Na ausência de forças dissipativas, a <a href="/wiki/Energia_mec%C3%A2nica" title="Energia mecânica">energia mecânica</a> de um Pêndulo <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_{m}ec)}"> <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>m</mi> </mrow> </msub> <mi>e</mi> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E_{m}ec)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b58bec0254c5323e4422db4a8848cec67e9c5792" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.29ex; height:2.843ex;" alt="{\displaystyle (E_{m}ec)}"></span> , tem valor constante dado por: </p><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 E_{m}ec=U(x)+K(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>e</mi> <mi>c</mi> <mo>=</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{m}ec=U(x)+K(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1b413929dbf942cd37f65a319151aa4f33a323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.546ex; height:2.843ex;" alt="{\displaystyle E_{m}ec=U(x)+K(x)}"></span>  </p><p>onde <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 U(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d626d3a1e65c94535c811c73fa83389cfb76683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.922ex; height:2.843ex;" alt="{\displaystyle U(x)}"></span> é a energia potencial e <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 K(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a621a6fbcc971649157a42ea42a599c22eee94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.205ex; height:2.843ex;" alt="{\displaystyle K(x)}"></span> é a energia cinética, e ambas são funções da posição <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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1f6d437f1742bfd5ffbdccd2477c07e9909e04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.139ex; height:2.843ex;" alt="{\displaystyle (x)}"></span> da partícula. </p><p>Considerando o sistema apresentado na figura ao lado, se uma partícula é liberada da posição (X2,h2), nesse ponto a velocidade desenvolvida por ela é igual a zero (partindo do repouso), logo sua <a href="/wiki/Energia_cin%C3%A9tica" title="Energia cinética">energia cinética</a> é nula, e a <a href="/wiki/Energia_potencial_gravitacional" title="Energia potencial gravitacional">energia potencial gravitacional</a> é máxima. Ao passar pelo ponto (X1,h1), a energia potencial e a energia cinética, contém o mesmo valor, isso devido ao fato de que a partícula se encontra em movimento, e não está localizada no ponto mais baixo da sua trajetória. Já no ponto (0,0), a energia cinética assume valor máximo e a energia potencial é nula, visto que nesse ponto a velocidade desenvolvida pela partícula é máxima, consequentemente pelo fato de (h) ter o seu valor igual a zero. </p><p>Como o sistema apresentado é conservativo, ou seja, não há perda de energia, podemos considerar que a mesma energia aplicada no início do movimento, é a mesma no final do movimento. Dessa forma, se a partícula é perturbada na posição (0,0) com uma velocidade (V) conhecida, é possível determinar as coordenadas da sua posição final, já que a gravidade atuará sobre a partícula, fazendo com que sua velocidade seja nula em um ponto acima da sua origem. Nessa situação temos: </p><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 E_{m}eci=E_{m}ecf}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>e</mi> <mi>c</mi> <mi>i</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>e</mi> <mi>c</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{m}eci=E_{m}ecf}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6f4a53dbf4d0631d97f1c0c3de65c7c9554475b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.141ex; height:2.509ex;" alt="{\displaystyle E_{m}eci=E_{m}ecf}"></span> </p><p>Como no ponto mais baixo temos somente energia cinética, e no ponto mais alto temos somente energia potencial, podemos reescrever a equação como sendo. </p><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 U(x)=K(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(x)=K(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1934c5ccaee86b99cf211ceb94bb94f6f5034ed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.225ex; height:2.843ex;" alt="{\displaystyle U(x)=K(x)}"></span> </p><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 mg\Delta h=\left({\frac {mv^{2}}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>g</mi> <mi mathvariant="normal">Δ</mi> <mi>h</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mg\Delta h=\left({\frac {mv^{2}}{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95b5cd0f819d68bcb41603d6d0bca57ed7293cfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.009ex; height:6.343ex;" alt="{\displaystyle mg\Delta h=\left({\frac {mv^{2}}{2}}\right)}"></span> </p><p>Visto que tanto a energia potencial, como a energia cinética são dependentes da massa da partícula, no pêndulo, devido a existência da conservação da energia aplicada, a massa pode ser desprezada, resultando na equação a seguir. </p><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 g\Delta h=\left({\frac {v^{2}}{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi mathvariant="normal">Δ</mi> <mi>h</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\Delta h=\left({\frac {v^{2}}{2}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ba6fdd72cb41b0760cdea980473cd05cff3e41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.928ex; height:6.343ex;" alt="{\displaystyle g\Delta h=\left({\frac {v^{2}}{2}}\right)}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Ver_também"><span id="Ver_tamb.C3.A9m"></span>Ver também</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=P%C3%AAndulo&veaction=edit&section=6" title="Editar secção: Ver também" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=P%C3%AAndulo&action=edit&section=6" title="Editar código-fonte da secção: Ver também"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/P%C3%AAndulo_de_Kater" title="Pêndulo de Kater">Pêndulo de Kater</a></li> <li><a href="/wiki/P%C3%AAndulo_de_Foucault" title="Pêndulo de Foucault">Pêndulo de Foucault</a></li> <li><a href="/wiki/P%C3%AAndulo_de_Newton" title="Pêndulo de Newton">Pêndulo de Newton</a></li> <li><a href="/wiki/Rel%C3%B3gio_de_p%C3%AAndulo" title="Relógio de pêndulo">Relógio de pêndulo</a></li></ul> <h2 id="Notas" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso.">Notas</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Este trecho incorpora texto em licença CC-BY-4.0 da obra citada.</span> </li> </ol></div></div> <div class="reflist" style="list-style-type: lower-alpha;"></div> <h2 id="Referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Refer.C3.AAncias"></span>Referências</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Serway2-1"><span class="mw-cite-backlink"><a href="#cite_ref-Serway2_1-0">↑</a></span> <span class="reference-text"><cite class="citation book">Serway, Raymond A; Jewett Jr, John W (2007). <i>Princípios de Física</i>. <b>2</b>. São Paulo: Thomson. p. 423-425. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Especial:Fontes_de_livros/85-221-0413-1" title="Especial:Fontes de livros/85-221-0413-1">85-221-0413-1</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AP%C3%AAndulo&rft.au=Jewett+Jr%2C+John+W&rft.aufirst=Raymond+A&rft.aulast=Serway&rft.btitle=Princ%C3pios+de+F%C3sica&rft.date=2007&rft.genre=book&rft.isbn=85-221-0413-1&rft.pages=423-425&rft.place=S%C3%A3o+Paulo&rft.pub=Thomson&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">↑ <sup><i><b><a href="#cite_ref-:0_2-0">a</a></b></i></sup> <sup><i><b><a href="#cite_ref-:0_2-1">b</a></b></i></sup></span> <span class="reference-text"><cite id="CITEREFMarcos_M._de_Almeida" class="citation">Marcos M. de Almeida (2021), <a rel="nofollow" class="external text" href="http://www.scielo.br/scielo.php?script=sci_abstract&pid=S1806-11172021000100704&lng=en&nrm=iso&tlng=pt">«Experimento de baixo custo para medição da dissipação da energia em um pêndulo»</a>, <i><a href="/wiki/Revista_Brasileira_de_Ensino_de_F%C3%ADsica" title="Revista Brasileira de Ensino de Física">Revista Brasileira de Ensino de Física</a></i>, <a href="/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/1806-1117">1806-1117</a>, <b>43</b>, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1590%2F1806-9126-RBEF-2020-0360">10.1590/1806-9126-RBEF-2020-0360</a>, <a href="/wiki/Wikidata" title="Wikidata">Wikidata</a> <a href="https://www.wikidata.org/wiki/Q106728917" class="extiw" title="d:Q106728917">Q106728917</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AP%C3%AAndulo&rft.atitle=Experimento+de+baixo+custo+para+medi%C3%A7%C3%A3o+da+dissipa%C3%A7%C3%A3o+da+energia+em+um+p%C3%AAndulo&rft.au=Marcos+M.+de+Almeida&rft.chron=2021&rft.genre=article&rft.issn=1806-1117&rft.jtitle=Revista+Brasileira+de+Ensino+de+F%C3sica&rft.volume=43&rft_id=http%3A%2F%2Fwww.scielo.br%2Fscielo.php%3Fscript%3Dsci_abstract%26pid%3DS1806-11172021000100704%26lng%3Den%26nrm%3Diso%26tlng%3Dpt&rft_id=info%3Adoi%2F10.1590%2F1806-9126-RBEF-2020-0360&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><cite class="citation journal">Lima, Fábio M. S. (2006). <a rel="nofollow" class="external text" href="https://aapt.scitation.org/doi/full/10.1119/1.2215616">«An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime»</a>. <i>American Journal of Physics</i><span class="reference-accessdate">. Consultado em 24 de junho de 2019</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AP%C3%AAndulo&rft.atitle=An+accurate+formula+for+the+period+of+a+simple+pendulum+oscillating+beyond+the+small+angle+regime&rft.aufirst=F%C3%A1bio+M.+S.&rft.aulast=Lima&rft.date=2006&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft_id=https%3A%2F%2Faapt.scitation.org%2Fdoi%2Ffull%2F10.1119%2F1.2215616&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="noprint" style="clear: right; border: solid #aaa 1px; margin: 0 0 1em 1em; font-size: 90%; background: var(--background-color-neutral-subtle, #f9f9f9); color: inherit; width: 238px; padding: 5px; spacing: 0px; text-align: center; float: right;"> <div style="float: left; vertical-align:middle;"><figure class="mw-halign-none" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/22px-Commons-logo.svg.png" decoding="async" width="22" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/33px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/44px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span><figcaption></figcaption></figure></div> <div style="margin-left: 30px; line-height:normal; vertical-align:middle;">O <a href="https://commons.wikimedia.org/wiki/P%C3%A1gina_principal" class="extiw" title="commons:Página principal">Commons</a> possui uma <i>categoria</i> com imagens e outros ficheiros sobre <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Pendulums?uselang=pt">Pêndulo</a></span></b></div> </div> <table class="noprint" style="border-top:none; border-bottom:none; background:transparent; color: inherit"><tbody><tr> <td style="text-align: center;"><span typeof="mw:File"><a href="/wiki/Ficheiro:U%2B269B.svg" class="mw-file-description"><img alt="Ícone de esboço" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/U%2B269B.svg/35px-U%2B269B.svg.png" decoding="async" width="35" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/U%2B269B.svg/53px-U%2B269B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/U%2B269B.svg/70px-U%2B269B.svg.png 2x" data-file-width="180" data-file-height="180" /></a></span></td> <td><i>Este artigo sobre <a href="/wiki/F%C3%ADsica" title="Física">física</a> é um <a href="/wiki/Wikip%C3%A9dia:Esbo%C3%A7o" title="Wikipédia:Esboço">esboço</a>. 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