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<div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Aparença" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Aparença</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ca.wikipedia.org&uselang=ca" class=""><span>Donatius</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Crea_compte&returnto=Graus+de+llibertat+%28f%C3%ADsica%29" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori" class=""><span>Crea un compte</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Registre_i_entrada&returnto=Graus+de+llibertat+%28f%C3%ADsica%29" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o" class=""><span>Inicia la sessió</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Més opcions" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Eines personals" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Eines personals</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menú d'usuari" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a 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mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pàgines per a editors no registrats <a href="/wiki/Ajuda:Introducci%C3%B3" aria-label="Vegeu més informació sobre l'edició"><span>més informació</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Especial:Contribucions_pr%C3%B2pies" title="Una llista de les modificacions fetes des d'aquesta adreça IP [y]" accesskey="y"><span>Contribucions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:Discussi%C3%B3_personal" title="Discussió sobre les edicions per aquesta adreça ip. [n]" accesskey="n"><span>Discussió per aquest IP</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Lloc"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contingut" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contingut</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">amaga</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inici</div> </a> </li> <li id="toc-Graus_de_llibertat_en_mecànica" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Graus_de_llibertat_en_mecànica"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Graus de llibertat en mecànica</span> </div> </a> <ul id="toc-Graus_de_llibertat_en_mecànica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graus_de_llibertat_en_mecànica_clàssica" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Graus_de_llibertat_en_mecànica_clàssica"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Graus de llibertat en mecànica clàssica</span> </div> </a> <button aria-controls="toc-Graus_de_llibertat_en_mecànica_clàssica-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>Commuta la subsecció Graus de llibertat en mecànica clàssica</span> </button> <ul id="toc-Graus_de_llibertat_en_mecànica_clàssica-sublist" class="vector-toc-list"> <li id="toc-Exemples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exemples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Exemples</span> </div> </a> <ul id="toc-Exemples-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Graus_de_llibertat_en_mecànica_estadística" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Graus_de_llibertat_en_mecànica_estadística"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Graus de llibertat en mecànica estadística</span> </div> </a> <ul id="toc-Graus_de_llibertat_en_mecànica_estadística-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referències" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referències"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Referències</span> </div> </a> <ul id="toc-Referències-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="Contingut" 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="Commuta la taula de continguts." > <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">Commuta la taula de continguts.</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">Graus de llibertat (física)</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="Vés a un article en una altra llengua. Disponible en 34 llengües" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-34" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">34 llengües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%B1%D8%AC%D8%A9_%D8%AD%D8%B1%D9%8A%D8%A9_(%D9%81%D9%8A%D8%B2%D9%8A%D8%A7%D8%A1)" title="درجة حرية (فيزياء) - àrab" lang="ar" hreflang="ar" data-title="درجة حرية (فيزياء)" data-language-autonym="العربية" data-language-local-name="àrab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D0%B8_%D0%BD%D0%B0_%D1%81%D0%B2%D0%BE%D0%B1%D0%BE%D0%B4%D0%B0" title="Степени на свобода - búlgar" lang="bg" hreflang="bg" data-title="Степени на свобода" data-language-autonym="Български" data-language-local-name="búlgar" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Stupe%C5%88_volnosti" title="Stupeň volnosti - txec" lang="cs" hreflang="cs" data-title="Stupeň volnosti" data-language-autonym="Čeština" data-language-local-name="txec" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D1%80%C4%95%D0%BA%D0%BB%C4%95%D1%85_%D0%BA%D0%B0%D0%BF%D0%B0%D1%88%C4%95_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Ирĕклĕх капашĕ (физика) - txuvaix" lang="cv" hreflang="cv" data-title="Ирĕклĕх капашĕ (физика)" data-language-autonym="Чӑвашла" data-language-local-name="txuvaix" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Frihedsgrad" title="Frihedsgrad - danès" lang="da" hreflang="da" data-title="Frihedsgrad" data-language-autonym="Dansk" data-language-local-name="danès" 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/Freiheitsgrad" title="Freiheitsgrad - alemany" lang="de" hreflang="de" data-title="Freiheitsgrad" data-language-autonym="Deutsch" data-language-local-name="alemany" 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%92%CE%B1%CE%B8%CE%BC%CF%8C%CF%82_%CE%B5%CE%BB%CE%B5%CF%85%CE%B8%CE%B5%CF%81%CE%AF%CE%B1%CF%82_%CF%83%CF%85%CF%83%CF%84%CE%AE%CE%BC%CE%B1%CF%84%CE%BF%CF%82" title="Βαθμός ελευθερίας συστήματος - grec" lang="el" hreflang="el" data-title="Βαθμός ελευθερίας συστήματος" data-language-autonym="Ελληνικά" data-language-local-name="grec" 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/Degrees_of_freedom_(physics_and_chemistry)" title="Degrees of freedom (physics and chemistry) - anglès" lang="en" hreflang="en" data-title="Degrees of freedom (physics and chemistry)" data-language-autonym="English" data-language-local-name="anglès" 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/Grado_de_libereco" title="Grado de libereco - esperanto" lang="eo" hreflang="eo" data-title="Grado de libereco" 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/Grado_de_libertad_(f%C3%ADsica)" title="Grado de libertad (física) - espanyol" lang="es" hreflang="es" data-title="Grado de libertad (física)" data-language-autonym="Español" data-language-local-name="espanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B1%D8%AC%D9%87_%D8%A2%D8%B2%D8%A7%D8%AF%DB%8C_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9_%D9%88_%D8%B4%DB%8C%D9%85%DB%8C)" title="درجه آزادی (فیزیک و شیمی) - persa" lang="fa" hreflang="fa" data-title="درجه آزادی (فیزیک و شیمی)" 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/Vapausaste_(termodynamiikka)" title="Vapausaste (termodynamiikka) - finès" lang="fi" hreflang="fi" data-title="Vapausaste (termodynamiikka)" data-language-autonym="Suomi" data-language-local-name="finès" 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/Degr%C3%A9_de_libert%C3%A9_(physique)" title="Degré de liberté (physique) - francès" lang="fr" hreflang="fr" data-title="Degré de liberté (physique)" data-language-autonym="Français" data-language-local-name="francès" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Graos_de_liberdade_(f%C3%ADsica)" title="Graos de liberdade (física) - gallec" lang="gl" hreflang="gl" data-title="Graos de liberdade (física)" data-language-autonym="Galego" data-language-local-name="gallec" 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%93%D7%A8%D7%92%D7%95%D7%AA_%D7%97%D7%95%D7%A4%D7%A9_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94_%D7%95%D7%9B%D7%99%D7%9E%D7%99%D7%94)" title="דרגות חופש (פיזיקה וכימיה) - hebreu" lang="he" hreflang="he" data-title="דרגות חופש (פיזיקה וכימיה)" data-language-autonym="עברית" data-language-local-name="hebreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Szabads%C3%A1gi_fok" title="Szabadsági fok - hongarès" lang="hu" hreflang="hu" data-title="Szabadsági fok" data-language-autonym="Magyar" data-language-local-name="hongarès" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Derajat_kebebasan_(fisika_dan_kimia)" title="Derajat kebebasan (fisika dan kimia) - indonesi" lang="id" hreflang="id" data-title="Derajat kebebasan (fisika dan kimia)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesi" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it badge-Q70893996 mw-list-item" title=""><a href="https://it.wikipedia.org/wiki/Grado_di_libert%C3%A0_(chimica)" title="Grado di libertà (chimica) - italià" lang="it" hreflang="it" data-title="Grado di libertà (chimica)" data-language-autonym="Italiano" data-language-local-name="italià" 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/%E8%87%AA%E7%94%B1%E5%BA%A6" title="自由度 - japonès" lang="ja" hreflang="ja" data-title="自由度" data-language-autonym="日本語" data-language-local-name="japonès" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%95%D1%80%D0%BA%D1%96%D0%BD%D0%B4%D1%96%D0%BA_%D0%B4%D3%99%D1%80%D0%B5%D0%B6%D0%B5%D0%BB%D0%B5%D1%80%D1%96" title="Еркіндік дәрежелері - kazakh" lang="kk" hreflang="kk" data-title="Еркіндік дәрежелері" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Fr%C3%A4iheetsgrad" title="Fräiheetsgrad - luxemburguès" lang="lb" hreflang="lb" data-title="Fräiheetsgrad" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxemburguès" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fridomsgrad_i_fysikk" title="Fridomsgrad i fysikk - noruec nynorsk" lang="nn" hreflang="nn" data-title="Fridomsgrad i fysikk" data-language-autonym="Norsk nynorsk" data-language-local-name="noruec 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/Frihetsgrad_(fysikk)" title="Frihetsgrad (fysikk) - noruec bokmål" lang="nb" hreflang="nb" data-title="Frihetsgrad (fysikk)" data-language-autonym="Norsk bokmål" data-language-local-name="noruec bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Stopie%C5%84_swobody_(fizyka)" title="Stopień swobody (fizyka) - polonès" lang="pl" hreflang="pl" data-title="Stopień swobody (fizyka)" data-language-autonym="Polski" data-language-local-name="polonès" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Graus_de_liberdade_(f%C3%ADsica)" title="Graus de liberdade (física) - portuguès" lang="pt" hreflang="pt" data-title="Graus de liberdade (física)" data-language-autonym="Português" data-language-local-name="portuguès" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Grad_de_libertate" title="Grad de libertate - romanès" lang="ro" hreflang="ro" data-title="Grad de libertate" data-language-autonym="Română" data-language-local-name="romanès" 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%A1%D1%82%D0%B5%D0%BF%D0%B5%D0%BD%D0%B8_%D1%81%D0%B2%D0%BE%D0%B1%D0%BE%D0%B4%D1%8B_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Степени свободы (физика) - rus" lang="ru" hreflang="ru" data-title="Степени свободы (физика)" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Degree_of_freedom" title="Degree of freedom - Simple English" lang="en-simple" hreflang="en-simple" data-title="Degree of freedom" 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-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Tingkat_kab%C3%A9basan" title="Tingkat kabébasan - sondanès" lang="su" hreflang="su" data-title="Tingkat kabébasan" data-language-autonym="Sunda" data-language-local-name="sondanès" 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/Frihetsgrad" title="Frihetsgrad - suec" lang="sv" hreflang="sv" data-title="Frihetsgrad" data-language-autonym="Svenska" data-language-local-name="suec" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Serbestlik_derecesi_(fizik_ve_kimya)" title="Serbestlik derecesi (fizik ve kimya) - turc" lang="tr" hreflang="tr" data-title="Serbestlik derecesi (fizik ve kimya)" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D1%82%D1%83%D0%BF%D0%B5%D0%BD%D1%96_%D0%B2%D1%96%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%96" title="Ступені вільності - ucraïnès" lang="uk" hreflang="uk" data-title="Ступені вільності" data-language-autonym="Українська" data-language-local-name="ucraïnès" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/B%E1%BA%ADc_t%E1%BB%B1_do_(l%C3%BD_h%C3%B3a)" title="Bậc tự do (lý hóa) - vietnamita" lang="vi" hreflang="vi" data-title="Bậc tự do (lý hóa)" 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-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%87%AA%E7%94%B1%E5%BA%A6_(%E7%89%A9%E7%90%86%E5%AD%A6)" title="自由度 (物理学) - xinès" lang="zh" hreflang="zh" data-title="自由度 (物理学)" data-language-autonym="中文" data-language-local-name="xinès" class="interlanguage-link-target"><span>中文</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/Q274502#sitelinks-wikipedia" title="Modifica 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">De la Viquipèdia, l'enciclopèdia lliure</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ca" dir="ltr"><p>El nombre de <b>graus de llibertat</b> en un sistema físic es refereix al nombre mínim de variables que cal especificar per determinar completament l'<a href="/wiki/Estat_f%C3%ADsic" title="Estat físic">estat físic</a>. El concepte apareix en <a href="/wiki/Mec%C3%A0nica_cl%C3%A0ssica" title="Mecànica clàssica">mecànica clàssica</a> i en <a href="/wiki/Termodin%C3%A0mica" title="Termodinàmica">termodinàmica</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Graus_de_llibertat_en_mecànica"><span id="Graus_de_llibertat_en_mec.C3.A0nica"></span>Graus de llibertat en mecànica</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graus_de_llibertat_(f%C3%ADsica)&action=edit&section=1" title="Modifica la secció: Graus de llibertat en mecànica"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En mecànica, per cada partícula lliure del sistema i per cada direcció en què aquesta és capaç de moure's hi ha dos graus de llibertat, un relacionat amb la <a href="/wiki/Posici%C3%B3" title="Posició">posició</a> i l'altre amb la <a href="/wiki/Velocitat" title="Velocitat">velocitat</a>. El nombre de graus de llibertat d'un sistema quan hi ha lligams entre les partícules, serà el nombre total de variables, menys el nombre de lligams que les relacionen. Observeu que aquesta definició no coincideix ni amb la definició de <a href="/wiki/Graus_de_llibertat_(enginyeria)" title="Graus de llibertat (enginyeria)">graus de llibertat</a> que s'usa en enginyeria de màquines, ni amb la que s'usa en enginyeria estructural. </p> <div class="mw-heading mw-heading2"><h2 id="Graus_de_llibertat_en_mecànica_clàssica"><span id="Graus_de_llibertat_en_mec.C3.A0nica_cl.C3.A0ssica"></span>Graus de llibertat en mecànica clàssica</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graus_de_llibertat_(f%C3%ADsica)&action=edit&section=2" title="Modifica la secció: Graus de llibertat en mecànica clàssica"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Mec%C3%A0nica_hamiltoniana" class="mw-redirect" title="Mecànica hamiltoniana">mecànica hamiltoniana</a> el nombre de graus de llibertat d'un sistema coincideix amb la dimensió topològica de l'<a href="/wiki/Espai_de_fases" title="Espai de fases">espai de fases</a> del sistema. En <a href="/wiki/Mec%C3%A0nica_lagrangiana" class="mw-redirect" title="Mecànica lagrangiana">mecànica lagrangiana</a> el nombre de graus de llibertat coincideix la dimensió del <a href="/wiki/Espai_de_configuraci%C3%B3" title="Espai de configuració">fibrat tangent de l'espai de configuració</a> del sistema. </p><p>Un conjunt de N partícules intereactuants però movent-se sense restriccions en l'espai tridimensional té 6 N graus de llibertat (tres coordenades de <a href="/wiki/Posici%C3%B3" title="Posició">posició</a> i tres <a href="/w/index.php?title=Velocitats&action=edit&redlink=1" class="new" title="Velocitats (encara no existeix)">velocitatss</a>). Si el conjunt de partícules es mou sobre un estat d -dimensional el nombre de graus de llibertat és 2 d · N . </p><p>Si hi ha <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> lligams entre les partícules, el nombre de graus de llibertat serà </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 GL=6N-k\leq 6N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>L</mi> <mo>=</mo> <mn>6</mn> <mi>N</mi> <mo>−</mo> <mi>k</mi> <mo>≤</mo> <mn>6</mn> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GL=6N-k\leq 6N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a17d1b488e5f7e1bae5588a09104df2ae21b9db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.11ex; height:2.343ex;" alt="{\displaystyle GL=6N-k\leq 6N}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Exemples">Exemples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graus_de_llibertat_(f%C3%ADsica)&action=edit&section=3" title="Modifica la secció: Exemples"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Partícula lliure: Una sola partícula lliure té 6 graus de llibertat</li> <li>Partícula obligada a moure's sobre una superfície</li></ul> <p>La superfície suposa un lligam per a les posicions, ja que s'ha de complir </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 F(x,y,z)=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x,y,z)=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38ab7a5063743741c91058a8aebed9498071a335" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.839ex; height:2.843ex;" alt="{\displaystyle F(x,y,z)=0\,}"></span></dd></dl> <p>i una altra per les velocitats, ja que la velocitat ha de ser en tot moment tangent a la superfície, per la qual cosa </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 0=\mathbf {v} \cdot \mathbf {n} =\mathbf {v} \cdot \nabla F=v_{x}{\frac {\partial F}{\partial x}}+v_{y}{\frac {\partial F}{\partial y}}+v_{z}{\frac {\partial F}{\partial z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>F</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>F</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>F</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>F</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\mathbf {v} \cdot \mathbf {n} =\mathbf {v} \cdot \nabla F=v_{x}{\frac {\partial F}{\partial x}}+v_{y}{\frac {\partial F}{\partial y}}+v_{z}{\frac {\partial F}{\partial z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/116146d09b29038ce82d24c6e186fba633f55d4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.772ex; height:6.009ex;" alt="{\displaystyle 0=\mathbf {v} \cdot \mathbf {n} =\mathbf {v} \cdot \nabla F=v_{x}{\frac {\partial F}{\partial x}}+v_{y}{\frac {\partial F}{\partial y}}+v_{z}{\frac {\partial F}{\partial z}}}"></span></dd></dl> <p>per tant el nombre de graus de llibertat és </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 GL=6-2=4\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>L</mi> <mo>=</mo> <mn>6</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>4</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GL=6-2=4\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ef169b97e75e20830941c942be523e7d3e8019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.321ex; height:2.343ex;" alt="{\displaystyle GL=6-2=4\,}"></span></dd></dl> <p>valor que coincideix amb el que s'espera per a un moviment en una varietat bidimensional. </p> <ul><li>Dues partícules en els extrems d'una vareta</li></ul> <p>Per tenir dues partícules tenim 12 graus de llibertat, però la condició que la distància entre les partícules sigui fixada suposa un lligam per a les seves posicions i una altra per a les seves velocitats, el que ens dona </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 GL=12-2=10\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>L</mi> <mo>=</mo> <mn>12</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>10</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GL=12-2=10\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/220aea2080640560b9190dd18081ae99eb4f0714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.646ex; height:2.343ex;" alt="{\displaystyle GL=12-2=10\,}"></span></dd></dl> <p>Aquests graus de llibertat es poden representar per variables diferents (les tres coordenades del centre de la vareta i els dos ángulos que donen l'orientació d'aquesta, amb les seves corresponents velocitats). </p> <ul><li>Un sòlid rígid</li></ul> <p>Un sòlid format per <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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> partícules té en principi <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 6N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37b183457644de3b21f0eb69d056ce6f33d3580c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.226ex; height:2.176ex;" alt="{\displaystyle 6N}"></span> variables. Però el nombre de lligadures és: </p> <ul><li><ul><li>Per a la primera partícula, cap</li> <li>Per a la segona partícula, 2 (la distància a la primera i la seva velocitat, com en el cas de dues partícules unides per una vareta)</li> <li>Per a la tercera partícula, 4 (les distàncies a les dues primeres partícules i les seves corresponents velocitats)</li> <li>Per a la quarta i següents, 6, ja que un cop donada la distància a tres partícules, la distància a totes les altres està també fixada).</li></ul></li></ul> <p>Per tant el nombre de graus de llibertat és </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 GL=6N-2-4-6(N-3)=12\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>L</mi> <mo>=</mo> <mn>6</mn> <mi>N</mi> <mo>−</mo> <mn>2</mn> <mo>−</mo> <mn>4</mn> <mo>−</mo> <mn>6</mn> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>12</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GL=6N-2-4-6(N-3)=12\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c66a353dc46d5f6519a0be5c0a55344254725f5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.429ex; height:2.843ex;" alt="{\displaystyle GL=6N-2-4-6(N-3)=12\,}"></span></dd></dl> <p>que es poden representar per sis variables (la posició del <a href="/wiki/Centre_de_massa" title="Centre de massa">centre de massa</a> i els <a href="/wiki/Angles_d%27Euler" title="Angles d'Euler">angles d'Euler</a>) i les seves corresponents velocitats. </p><p><br /> En general, no tots els lligams poden representar-se mitjançant una reducció en el nombre de variables (encara que sí en el nombre de variables independents). Quan tenim un sistema en el qual els lligams no són integrables, es diu que el sistema és no holònom. </p><p>És important assenyalar que la convenció per comptabilitzar els <a href="/wiki/Graus_de_llibertat_(enginyeria)" title="Graus de llibertat (enginyeria)">graus de llibertat en enginyeria</a> mecànica és diferent, i justament la meitat que en els casos(<span id="Eqnref_1" class="plainlinksneverexpand"><a href="#Equation_1">1</a></span>)i(<span id="Eqnref_2" class="plainlinksneverexpand"><a href="#Equation_2">2</a></span>). </p> <div class="mw-heading mw-heading2"><h2 id="Graus_de_llibertat_en_mecànica_estadística"><span id="Graus_de_llibertat_en_mec.C3.A0nica_estad.C3.ADstica"></span>Graus de llibertat en mecànica estadística</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graus_de_llibertat_(f%C3%ADsica)&action=edit&section=4" title="Modifica la secció: Graus de llibertat en mecànica estadística"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r30997230">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Teorema_d%27equipartici%C3%B3" title="Teorema d'equipartició">Teorema d'equipartició</a></div> <p>Al <a href="/w/index.php?title=L%C3%ADmit_cl%C3%A0ssic&action=edit&redlink=1" class="new" title="Límit clàssic (encara no existeix)">límit clàssic</a> de la <a href="/wiki/Mec%C3%A0nica_estad%C3%ADstica" title="Mecànica estadística">mecànica estadística</a> l'energia d'un sistema en equilibri tèrmic amb n graus de llibertat quadràtics i independents és:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:Left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <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=\langle I\rangle =n{\frac {k_{B}T}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>I</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\langle I\rangle =n{\frac {k_{B}T}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1393c8ff127ce905189bf526514b59cad64170be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.518ex; height:5.343ex;" alt="{\displaystyle U=\langle I\rangle =n{\frac {k_{B}T}{2}}}"></span> </p> </blockquote> <p>On: </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 K_{B}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{B}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e2bcba13ac511d40055b43cb0ee63e9ea737af8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.84ex; height:2.509ex;" alt="{\displaystyle K_{B}\,}"></span> és la <a href="/wiki/Constant_de_Boltzmann" title="Constant de Boltzmann">constant de Boltzmann</a></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mspace width="thinmathspace" /> </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/476a8389064c06ab89963a2467aef525838da0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.023ex; height:2.176ex;" alt="{\displaystyle T\,}"></span> és la <a href="/wiki/Temperatura" title="Temperatura">temperatura</a></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4f97d800eaacd982789661ac3896e1181d6137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.451ex; height:2.176ex;" alt="{\displaystyle N\,}"></span> és el nombre de graus de llibertat del sistema</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Referències"><span id="Refer.C3.A8ncies"></span>Referències</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Graus_de_llibertat_(f%C3%ADsica)&action=edit&section=5" title="Modifica la secció: Referències"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist {{#if: | references-column-count references-column-count-{{{col}}}" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="https://www.termcat.cat/ca/diccionaris-en-linia/212/fitxa/NDM1NjQ4NQ%3D%3D">graus de llibertat</a>». <i>Diccionari de química</i>.  Termcat. [Consulta: 27 gener 2022].</span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFReif2009"><span style="font-variant: small-caps;">Reif</span>, F. <i>Fundamentals of Statistical and Thermal Physics</i> (en anglès).  Long Grove, IL: Waveland Press, Inc., 2009, p. 51. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-1-57766-612-7" title="Especial:Fonts bibliogràfiques/978-1-57766-612-7">ISBN 978-1-57766-612-7</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Statistical+and+Thermal+Physics&rft.aulast=Reif&rft.aufirst=F.&rft.date=2009&rft.pub=Waveland+Press%2C+Inc.&rft.place=Long+Grove%2C+IL&rft.pages=51&rft.isbn=978-1-57766-612-7"><span style="display: none;"> </span></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtingut de «<a dir="ltr" href="https://ca.wikipedia.org/w/index.php?title=Graus_de_llibertat_(física)&oldid=33691875">https://ca.wikipedia.org/w/index.php?title=Graus_de_llibertat_(física)&oldid=33691875</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categories</a>: <ul><li><a href="/wiki/Categoria:F%C3%ADsica" title="Categoria:Física">Física</a></li><li><a href="/wiki/Categoria:Dimensi%C3%B3" title="Categoria:Dimensió">Dimensió</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> La pàgina va ser modificada per darrera vegada el 14 jul 2024 a les 19:48.</li> <li id="footer-info-copyright">El text està disponible sota la <a href="/wiki/Viquip%C3%A8dia:Text_de_la_llic%C3%A8ncia_de_Creative_Commons_Reconeixement-Compartir_Igual_4.0_No_adaptada" title="Viquipèdia:Text de la llicència de Creative Commons Reconeixement-Compartir Igual 4.0 No adaptada"> Llicència de Creative Commons Reconeixement i Compartir-Igual</a>; es poden aplicar termes addicionals. 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