Dinàmica del sòlid rígid - Viquipèdia, l'enciclopèdia lliure

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="ca" dir="ltr"> <head> <meta charset="UTF-8"> <title>Dinàmica del sòlid rígid - Viquipèdia, l'enciclopèdia lliure</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )cawikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t.",".\t,"],"wgDigitTransformTable":["",""], "wgDefaultDateFormat":"dmy","wgMonthNames":["","gener","febrer","març","abril","maig","juny","juliol","agost","setembre","octubre","novembre","desembre"],"wgRequestId":"0f1b014a-da71-49a6-bed8-2f28a434b3bf","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Dinàmica_del_sòlid_rígid","wgTitle":"Dinàmica del sòlid rígid","wgCurRevisionId":34256409,"wgRevisionId":34256409,"wgArticleId":836859,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Mecànica"],"wgPageViewLanguage":"ca","wgPageContentLanguage":"ca","wgPageContentModel":"wikitext","wgRelevantPageName":"Dinàmica_del_sòlid_rígid","wgRelevantArticleId":836859,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags" :0,"wgVisualEditor":{"pageLanguageCode":"ca","pageLanguageDir":"ltr","pageVariantFallbacks":"ca"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q2037529","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options": "loading","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","wikibase.client.data-bridge.externalModifiers":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.UkensKonkurranse","ext.gadget.refToolbar","ext.gadget.charinsert","ext.gadget.AltresViccionari","ext.gadget.purgetab","ext.gadget.DocTabs","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns", "ext.cx.uls.quick.actions","wikibase.client.vector-2022","wikibase.client.data-bridge.init","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","oojs-ui.styles.icons-media","oojs-ui-core.icons","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=ca&modules=ext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.data-bridge.externalModifiers%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=ca&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=ca&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Dinàmica del sòlid rígid - Viquipèdia, l'enciclopèdia lliure"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//ca.m.wikipedia.org/wiki/Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid"> <link rel="alternate" type="application/x-wiki" title="Modifica" href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Viquipèdia (ca)"> <link rel="EditURI" type="application/rsd+xml" href="//ca.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://ca.wikipedia.org/wiki/Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.ca"> <link rel="alternate" type="application/atom+xml" title="Canal de sindicació Atom Viquipèdia" href="/w/index.php?title=Especial:Canvis_recents&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Dinàmica_del_sòlid_rígid rootpage-Dinàmica_del_sòlid_rígid skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Vés al contingut</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Lloc"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Menú principal" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Menú principal</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Menú principal</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.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-main-menu.unpin">amaga</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navegació </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Portada" title="Visiteu la pàgina principal [z]" accesskey="z"><span>Portada</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Especial:Article_aleatori" title="Carrega una pàgina a l’atzar [x]" accesskey="x"><span>Article a l'atzar</span></a></li><li id="n-Articles-de-qualitat" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Articles_de_qualitat"><span>Articles de qualitat</span></a></li> </ul> </div> </div> <div id="p-Comunitat" class="vector-menu mw-portlet mw-portlet-Comunitat" > <div class="vector-menu-heading"> Comunitat </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-portal" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Portal" title="Sobre el projecte, què podeu fer, on trobareu les coses"><span>Portal viquipedista</span></a></li><li id="n-Agenda-d'actes" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Trobades"><span>Agenda d'actes</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Especial:Canvis_recents" title="Una llista dels canvis recents al wiki [r]" accesskey="r"><span>Canvis recents</span></a></li><li id="n-La-taverna" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:La_taverna"><span>La taverna</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Contacte"><span>Contacte</span></a></li><li id="n-Xat" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Canals_IRC"><span>Xat</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Ajuda" title="El lloc per a saber més coses"><span>Ajuda</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Portada" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Viquipèdia" src="/static/images/mobile/copyright/wikipedia-wordmark-ca.svg" style="width: 7.5em; height: 1.4375em;"> <img class="mw-logo-tagline" alt="l'Enciclopèdia Lliure" src="/static/images/mobile/copyright/wikipedia-tagline-ca.svg" width="120" height="14" style="width: 7.5em; height: 0.875em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Especial:Cerca" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Cerca a la Viquipèdia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Cerca</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Cerca a Viquipèdia" aria-label="Cerca a Viquipèdia" autocapitalize="sentences" title="Cerca a la Viquipèdia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Especial:Cerca"> </div> <button class="cdx-button cdx-search-input__end-button">Cerca</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Eines personals"> <div class="vector-user-links-main"> <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=Din%C3%A0mica+del+s%C3%B2lid+r%C3%ADgid" 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=Din%C3%A0mica+del+s%C3%B2lid+r%C3%ADgid" 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 href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ca.wikipedia.org&uselang=ca"><span>Donatius</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Crea_compte&returnto=Din%C3%A0mica+del+s%C3%B2lid+r%C3%ADgid" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Crea un compte</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Registre_i_entrada&returnto=Din%C3%A0mica+del+s%C3%B2lid+r%C3%ADgid" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Inicia la sessió</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet 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-Cinemàtica_del_sòlid_rígid" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cinemàtica_del_sòlid_rígid"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Cinemàtica del sòlid rígid</span> </div> </a> <button aria-controls="toc-Cinemàtica_del_sòlid_rígid-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 Cinemàtica del sòlid rígid</span> </button> <ul id="toc-Cinemàtica_del_sòlid_rígid-sublist" class="vector-toc-list"> <li id="toc-Centre_de_gravetat" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Centre_de_gravetat"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Centre de gravetat</span> </div> </a> <ul id="toc-Centre_de_gravetat-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Velocitat_angular" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Velocitat_angular"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Velocitat angular</span> </div> </a> <ul id="toc-Velocitat_angular-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Moment_angular_o_cinètic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Moment_angular_o_cinètic"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Moment angular o cinètic</span> </div> </a> <ul id="toc-Moment_angular_o_cinètic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Espai_de_configuració_d'un_sòlid_rígid" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Espai_de_configuració_d'un_sòlid_rígid"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Espai de configuració d'un sòlid rígid</span> </div> </a> <ul id="toc-Espai_de_configuració_d'un_sòlid_rígid-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tensor_d'inèrcia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tensor_d'inèrcia"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Tensor d'inèrcia</span> </div> </a> <ul id="toc-Tensor_d'inèrcia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equacions_del_moviment" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equacions_del_moviment"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Equacions del moviment</span> </div> </a> <button aria-controls="toc-Equacions_del_moviment-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 Equacions del moviment</span> </button> <ul id="toc-Equacions_del_moviment-sublist" class="vector-toc-list"> <li id="toc-Angles_d'Euler" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Angles_d'Euler"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Angles d'Euler</span> </div> </a> <ul id="toc-Angles_d'Euler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equacions_d'Euler" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equacions_d'Euler"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Equacions d'Euler</span> </div> </a> <ul id="toc-Equacions_d'Euler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Baldufa_simètrica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Baldufa_simètrica"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Baldufa simètrica</span> </div> </a> <ul id="toc-Baldufa_simètrica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Baldufa_asimètrica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Baldufa_asimètrica"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Baldufa asimètrica</span> </div> </a> <ul id="toc-Baldufa_asimètrica-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sòlid_rígid_en_mecànica_quàntica" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sòlid_rígid_en_mecànica_quàntica"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Sòlid rígid en mecànica quàntica</span> </div> </a> <ul id="toc-Sòlid_rígid_en_mecànica_quàntica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-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">Dinàmica del sòlid rígid</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 13 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-13" 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">13 llengües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/M%C3%BCtl%C9%99q_b%C9%99rk_cisim_dinamikas%C4%B1" title="Mütləq bərk cisim dinamikası - azerbaidjanès" lang="az" hreflang="az" data-title="Mütləq bərk cisim dinamikası" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaidjanès" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Starrk%C3%B6rpersimulation" title="Starrkörpersimulation - alemany" lang="de" hreflang="de" data-title="Starrkörpersimulation" data-language-autonym="Deutsch" data-language-local-name="alemany" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Rigid_body_dynamics" title="Rigid body dynamics - anglès" lang="en" hreflang="en" data-title="Rigid body dynamics" data-language-autonym="English" data-language-local-name="anglès" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Din%C3%A1mica_de_cuerpos_r%C3%ADgidos" title="Dinámica de cuerpos rígidos - espanyol" lang="es" hreflang="es" data-title="Dinámica de cuerpos rígidos" 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%DB%8C%D9%86%D8%A7%D9%85%DB%8C%DA%A9_%D8%A7%D8%AC%D8%B3%D8%A7%D9%85_%D8%B5%D9%84%D8%A8" title="دینامیک اجسام صلب - persa" lang="fa" hreflang="fa" data-title="دینامیک اجسام صلب" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%83%E0%A4%A2%E0%A4%BC_%E0%A4%AA%E0%A4%BF%E0%A4%A3%E0%A5%8D%E0%A4%A1_%E0%A4%97%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80" title="दृढ़ पिण्ड गतिकी - hindi" lang="hi" hreflang="hi" data-title="दृढ़ पिण्ड गतिकी" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Dinamika_benda_tegar" title="Dinamika benda tegar - indonesi" lang="id" hreflang="id" data-title="Dinamika benda tegar" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ruch_post%C4%99powy_bry%C5%82y_sztywnej" title="Ruch postępowy bryły sztywnej - polonès" lang="pl" hreflang="pl" data-title="Ruch postępowy bryły sztywnej" 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/Din%C3%A2mica_de_corpo_r%C3%ADgido" title="Dinâmica de corpo rígido - portuguès" lang="pt" hreflang="pt" data-title="Dinâmica de corpo rígido" data-language-autonym="Português" data-language-local-name="portuguès" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0_%D1%82%D0%B2%D1%91%D1%80%D0%B4%D0%BE%D0%B3%D0%BE_%D1%82%D0%B5%D0%BB%D0%B0" title="Механика твёрдого тела - rus" lang="ru" hreflang="ru" data-title="Механика твёрдого тела" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Kat%C4%B1_cisim_dinami%C4%9Fi" title="Katı cisim dinamiği - turc" lang="tr" hreflang="tr" data-title="Katı cisim dinamiği" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%88%9A%E4%BD%93%E5%8A%A8%E5%8A%9B%E5%AD%A6" title="刚体动力学 - xinès" lang="zh" hreflang="zh" data-title="刚体动力学" data-language-autonym="中文" data-language-local-name="xinès" 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/%E5%89%9B%E9%AB%94%E5%8B%95%E5%8A%9B%E5%AD%B8" title="剛體動力學 - cantonès" lang="yue" hreflang="yue" data-title="剛體動力學" data-language-autonym="粵語" data-language-local-name="cantonès" 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/Q2037529#sitelinks-wikipedia" title="Modifica enllaços interlingües" class="wbc-editpage">Modifica els enllaços</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="Espais de noms"> <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/Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid" title="Vegeu el contingut de la pàgina [c]" accesskey="c"><span>Pàgina</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discussi%C3%B3:Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid" rel="discussion" title="Discussió sobre el contingut d'aquesta pàgina [t]" accesskey="t"><span>Discussió</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Canvia la variant de llengua" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">català</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistes"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid"><span>Mostra</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit" title="Modifica el codi font d'aquesta pàgina [e]" accesskey="e"><span>Modifica</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=history" title="Versions antigues d'aquesta pàgina [h]" accesskey="h"><span>Mostra l'historial</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Eines" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Eines</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Eines</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.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-page-tools.unpin">amaga</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Més opcions" > <div class="vector-menu-heading"> Accions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid"><span>Mostra</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit" title="Modifica el codi font d'aquesta pàgina [e]" accesskey="e"><span>Modifica</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=history"><span>Mostra l'historial</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:Enlla%C3%A7os/Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid" title="Una llista de totes les pàgines wiki que enllacen amb aquesta [j]" accesskey="j"><span>Què hi enllaça</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Seguiment/Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid" rel="nofollow" title="Canvis recents a pàgines enllaçades des d'aquesta pàgina [k]" accesskey="k"><span>Canvis relacionats</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A0gines_especials" title="Llista totes les pàgines especials [q]" accesskey="q"><span>Pàgines especials</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&oldid=34256409" title="Enllaç permanent a aquesta revisió de la pàgina"><span>Enllaç permanent</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=info" title="Més informació sobre aquesta pàgina"><span>Informació de la pàgina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citau&page=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&id=34256409&wpFormIdentifier=titleform" title="Informació sobre com citar aquesta pàgina"><span>Citau aquest article</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:UrlShortener&url=https%3A%2F%2Fca.wikipedia.org%2Fwiki%2FDin%25C3%25A0mica_del_s%25C3%25B2lid_r%25C3%25ADgid"><span>Obtén una URL abreujada</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fca.wikipedia.org%2Fwiki%2FDin%25C3%25A0mica_del_s%25C3%25B2lid_r%25C3%25ADgid"><span>Descarrega el codi QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimeix/exporta </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Llibre&bookcmd=book_creator&referer=Din%C3%A0mica+del+s%C3%B2lid+r%C3%ADgid"><span>Crea un llibre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=show-download-screen"><span>Baixa com a PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&printable=yes" title="Versió per a impressió d'aquesta pàgina [p]" accesskey="p"><span>Versió per a impressora</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> En altres projectes </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2037529" title="Enllaç a l'element del repositori de dades connectat [g]" accesskey="g"><span>Element a Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aparença</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.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-appearance.unpin">amaga</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">De la 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"><table class="vertical-navbox nowraplinks" style="float:right;clear:right;width:22.0em;margin:0 0 1.0em 1.0em;color:#202122;background-color:#f8f9fa;border:1px solid #a2a9b1;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%;width:220px"><tbody><tr><th style="padding:0.2em 0.4em 0.2em;font-size:145%;line-height:1.2em"><a href="/wiki/Mec%C3%A0nica_cl%C3%A0ssica" title="Mecànica clàssica">Mecànica clàssica</a></th></tr><tr><td style="padding:0 0.1em 0.4em"> <a href="/w/index.php?title=Hist%C3%B2ria_de_la_mec%C3%A0nica_cl%C3%A0ssica&action=edit&redlink=1" class="new" title="Història de la mecànica clàssica (encara no existeix)">Història</a><br /><a href="/wiki/Cronologia_de_la_mec%C3%A0nica_cl%C3%A0ssica" title="Cronologia de la mecànica clàssica">Cronologia</a></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;text-align:left">Branques</div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><a href="/wiki/Est%C3%A0tica" title="Estàtica">Estàtica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Din%C3%A0mica" title="Dinàmica">Dinàmica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Cinem%C3%A0tica" title="Cinemàtica">Cinemàtica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Mec%C3%A0nica_aplicada" title="Mecànica aplicada">Mecànica aplicada</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Mec%C3%A0nica_celeste" title="Mecànica celeste">Mecànica celeste</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Mec%C3%A0nica_dels_medis_continus" title="Mecànica dels medis continus">Mecànica dels medis continus</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Mec%C3%A0nica_estad%C3%ADstica" title="Mecànica estadística">Mecànica estadística</a></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;text-align:left">Formulacions</div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><div style="text-align:left;"> <ul><li><a href="/wiki/Lleis_de_Newton" title="Lleis de Newton">Mecànica newtoniana</a></li> <li><a href="/wiki/Mec%C3%A0nica_anal%C3%ADtica" title="Mecànica analítica">Mecànica analítica</a>: <ul><li><a href="/wiki/Formulaci%C3%B3_lagrangiana" title="Formulació lagrangiana">Mecànica lagrangiana</a></li> <li><a href="/wiki/Formulaci%C3%B3_hamiltoniana" title="Formulació hamiltoniana">Mecànica hamiltoniana</a></li></ul></li></ul></div></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;text-align:left">Conceptes fonamentals</div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><a href="/wiki/Espai" title="Espai">Espai</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Temps" title="Temps">Temps</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Velocitat" title="Velocitat">Velocitat</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Celeritat" title="Celeritat">Celeritat</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Massa" title="Massa">Massa</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Acceleraci%C3%B3" title="Acceleració">Acceleració</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Gravetat" title="Gravetat">Gravetat</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/For%C3%A7a" title="Força">Força</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Impuls_(f%C3%ADsica)" title="Impuls (física)">Impuls</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Parell_de_forces" title="Parell de forces">Parell</a> / <a href="/wiki/Moment" title="Moment">Moment</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Quantitat_de_moviment" title="Quantitat de moviment">Quantitat de moviment</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Moment_angular" title="Moment angular">Moment angular</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/In%C3%A8rcia" title="Inèrcia">Inèrcia</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Moment_d%27in%C3%A8rcia" title="Moment d'inèrcia">Moment d'inèrcia</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Sistema_de_refer%C3%A8ncia" title="Sistema de referència">Sistema de referència</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Energia" title="Energia">Energia</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Energia_cin%C3%A8tica" title="Energia cinètica">Energia cinètica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Energia_potencial" title="Energia potencial">Energia potencial</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Treball_mec%C3%A0nic" class="mw-redirect" title="Treball mecànic">Treball mecànic</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Treball_virtual" title="Treball virtual">Treball virtual</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Principi_de_d%27Alembert" title="Principi de d'Alembert">Principi de d'Alembert</a></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;text-align:left">Temes principals</div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><a href="/wiki/S%C3%B2lid_r%C3%ADgid" title="Sòlid rígid">Sòlid rígid</a><span style="font-weight:bold;"> ·</span> <a class="mw-selflink selflink">Dinàmica del sòlid rígid</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Equacions_d%27Euler" class="mw-redirect" title="Equacions d'Euler">Equacions d'Euler</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Moviment_(f%C3%ADsica)" class="mw-redirect" title="Moviment (física)">Moviment</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Lleis_de_Newton" title="Lleis de Newton">Lleis de Newton</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Llei_de_la_gravitaci%C3%B3_universal" title="Llei de la gravitació universal">Llei de la gravitació universal</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Equacions_del_moviment" class="mw-redirect" title="Equacions del moviment">Equacions del moviment</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Sistema_de_refer%C3%A8ncia_inercial" title="Sistema de referència inercial">Sistema de referència inercial</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Sistema_de_refer%C3%A8ncia_no_inercial" title="Sistema de referència no inercial">Sistema de referència no inercial</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Sistema_de_refer%C3%A8ncia_amb_rotaci%C3%B3" class="mw-redirect" title="Sistema de referència amb rotació">Sistema de referència amb rotació</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/For%C3%A7a_fict%C3%ADcia" class="mw-redirect" title="Força fictícia">Força fictícia</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Moviment_rectilini" title="Moviment rectilini">Moviment rectilini</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Despla%C3%A7ament_(cinem%C3%A0tica)" title="Desplaçament (cinemàtica)">Desplaçament</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Velocitat_relativa" title="Velocitat relativa">Velocitat relativa</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Fricci%C3%B3" title="Fricció">Fricció</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Moviment_harm%C3%B2nic_simple" title="Moviment harmònic simple">Moviment harmònic simple</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Oscil%C2%B7lador_harm%C3%B2nic" title="Oscil·lador harmònic">Oscil·lador harmònic</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Vibraci%C3%B3" title="Vibració">Vibració</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Esmorte%C3%AFment" title="Esmorteïment">Esmorteïment</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Moviment_de_rotaci%C3%B3" class="mw-redirect" title="Moviment de rotació">Moviment de rotació</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Moviment_circular" title="Moviment circular">Moviment circular</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/For%C3%A7a_centr%C3%ADpeta" title="Força centrípeta">Força centrípeta</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/For%C3%A7a_centr%C3%ADfuga" title="Força centrífuga">Força centrífuga</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Efecte_de_Coriolis" title="Efecte de Coriolis">Efecte de Coriolis</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/P%C3%A8ndol_(matem%C3%A0tiques)" title="Pèndol (matemàtiques)">Pèndol</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Acceleraci%C3%B3_angular" title="Acceleració angular">Acceleració angular</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Velocitat_angular" title="Velocitat angular">Velocitat angular</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Freq%C3%BC%C3%A8ncia_angular" class="mw-redirect" title="Freqüència angular">Freqüència angular</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Despla%C3%A7ament_angular" title="Desplaçament angular">Desplaçament angular</a></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;text-align:left">Científics</div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Jeremiah_Horrocks" title="Jeremiah Horrocks">Jeremiah Horrocks</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">Jean le Rond d'Alembert</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Alexis_Clairaut" class="mw-redirect" title="Alexis Clairaut">Alexis Clairaut</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Joseph_Louis_Lagrange" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Sim%C3%A9on-Denis_Poisson" class="mw-redirect" title="Siméon-Denis Poisson">Siméon-Denis Poisson</a></div></div></td> </tr><tr><td style="text-align:right;font-size:115%;padding-top: 0.6em;"><div class="plainlinks hlist navbar mini"><ul><li class="nv-view"><span typeof="mw:File"><a href="/wiki/Plantilla:Mec%C3%A0nica_cl%C3%A0ssica" title="Plantilla:Mecànica clàssica"><img alt="Vegeu aquesta plantilla" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/18px-Commons-emblem-notice.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/27px-Commons-emblem-notice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/36px-Commons-emblem-notice.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></li></ul></div></td></tr></tbody></table> <p>La <b>dinàmica d'un sòlid rígid</b> estudia el moviment i equilibri de <a href="/wiki/S%C3%B2lid" title="Sòlid">sòlids</a> materials ignorant-ne les <a href="/wiki/Deformaci%C3%B3" title="Deformació">deformacions</a>. Es tracta, per tant, d'un model matemàtic útil per estudiar una part de la mecànica de sòlids, ja que tots els sòlids reals són deformables. S'entén per <i>sòlid rígid</i> un conjunt de punts de l'espai que es mouen de tal manera que no se n'alteren les <a href="/wiki/Dist%C3%A0ncia" title="Distància">distàncies</a>, sigui quina sigui la força que hi actua (matemàticament, el moviment d'un sòlid rígid ve donat per un <a href="/w/index.php?title=Grup_uniparam%C3%A8tric&action=edit&redlink=1" class="new" title="Grup uniparamètric (encara no existeix)">grup uniparamètric</a> d'<a href="/wiki/Isometria" title="Isometria">isometries</a>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Cinemàtica_del_sòlid_rígid"><span id="Cinem.C3.A0tica_del_s.C3.B2lid_r.C3.ADgid"></span>Cinemàtica del sòlid rígid</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit&section=1" title="Modifica la secció: Cinemàtica del sòlid rígid"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Centre_de_gravetat">Centre de gravetat</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit&section=2" title="Modifica la secció: Centre de gravetat"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El <a href="/wiki/Centre_de_massa" title="Centre de massa">centre de massa</a> d'un sistema continu és el <a href="/wiki/Punt_(geometria)" title="Punt (geometria)">punt geomètric</a> definit com: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_1" style="font-style: normal;"><a href="#Eqnref_1">1</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{\text{CM}}={\frac {\int \mathbf {r} dm}{\int dm}}={\frac {\int \mathbf {r} dm}{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>CM</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mi>d</mi> <mi>m</mi> </mrow> <mrow> <mo>∫</mo> <mi>d</mi> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mi>d</mi> <mi>m</mi> </mrow> <mi>M</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{\text{CM}}={\frac {\int \mathbf {r} dm}{\int dm}}={\frac {\int \mathbf {r} dm}{M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9f25edba440fa97e05bd044e446957c0a9baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.224ex; height:6.843ex;" alt="{\displaystyle \mathbf {r} _{\text{CM}}={\frac {\int \mathbf {r} dm}{\int dm}}={\frac {\int \mathbf {r} dm}{M}}}"></span> </p> </blockquote> <p>En mecànica del sòlid rígid, el centre de massa s'usa perquè prenent un sistema de coordenades que s'hi centra, l'<a href="/wiki/Energia_cin%C3%A8tica" title="Energia cinètica">energia cinètica</a> total <b><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/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span></b> pot expressar-se com: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_2" style="font-style: normal;"><a href="#Eqnref_2">2</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {K={1 \over 2}Mv^{2}+K_{rot}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>M</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {K={1 \over 2}Mv^{2}+K_{rot}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80ed5d8e968919312f6743e1d6ec74dc88561695" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.965ex; height:5.176ex;" alt="{\displaystyle {K={1 \over 2}Mv^{2}+K_{rot}}}"></span> </p> </blockquote> <p>sent <b><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> </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/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span></b> la <a href="/wiki/Massa" title="Massa">massa</a> total del cos, <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span></b> la <a href="/wiki/Velocitat" title="Velocitat">velocitat</a> de translació del centre de masses i <b><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_{rot}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{rot}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2a057a1d2b1da5d962565ff389452823f52efcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.338ex; height:2.509ex;" alt="{\displaystyle K_{rot}}"></span></b> l'energia cinètica de <a href="/wiki/Moviment_circular" title="Moviment circular">rotació</a> del cos, expressable en termes de <a href="/wiki/Velocitat_angular" title="Velocitat angular">velocitat angular</a> i el <a href="/wiki/Tensor" title="Tensor">tensor</a> d'<a href="/wiki/In%C3%A8rcia" title="Inèrcia">inèrcia</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Velocitat_angular">Velocitat angular</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit&section=3" title="Modifica la secció: Velocitat angular"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sigui una partícula qualsevol d'un sòlid rígid el qual es desplaça girant. Atès que tots els punts estan rígidament connectats podem fer la següent descomposició de posició i velocitats, prenent un punt de referència arbitrari <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 \mathbf {r} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe39f0fedae3334af5c4ffaedf25c9778363400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.156ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{0}}"></span>: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_3a" style="font-style: normal;"><a href="#Eqnref_3a">3a</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} (t,\mathbf {r} _{0})=\mathbf {r} _{c}(t)+\mathbf {r} (t,\mathbf {r} _{0})=\mathbf {r} _{c}(t)+A(t)\mathbf {r} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t,\mathbf {r} _{0})=\mathbf {r} _{c}(t)+\mathbf {r} (t,\mathbf {r} _{0})=\mathbf {r} _{c}(t)+A(t)\mathbf {r} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1dd11a01c6fc8896f3a9bd3819a28d269009ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.699ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} (t,\mathbf {r} _{0})=\mathbf {r} _{c}(t)+\mathbf {r} (t,\mathbf {r} _{0})=\mathbf {r} _{c}(t)+A(t)\mathbf {r} _{0}}"></span> </p> </blockquote> <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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_3b" style="font-style: normal;"><a href="#Eqnref_3b">3b</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} (t,\mathbf {r} _{0})=\mathbf {v} _{c}(t)+{\boldsymbol {\omega }}(t)\times \mathbf {R} (t,\mathbf {r} _{0})=\mathbf {v} _{c}(t)+{\boldsymbol {\omega }}(t)\times (\mathbf {r} (t,\mathbf {r} _{0})-\mathbf {y} _{c}(t))=\mathbf {v} _{c}(t)+{\boldsymbol {\omega }}(t)\times A(t)\mathbf {y} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} (t,\mathbf {r} _{0})=\mathbf {v} _{c}(t)+{\boldsymbol {\omega }}(t)\times \mathbf {R} (t,\mathbf {r} _{0})=\mathbf {v} _{c}(t)+{\boldsymbol {\omega }}(t)\times (\mathbf {r} (t,\mathbf {r} _{0})-\mathbf {y} _{c}(t))=\mathbf {v} _{c}(t)+{\boldsymbol {\omega }}(t)\times A(t)\mathbf {y} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56aa52c123f30905d6ca1deaf02df5cb4b0437f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:92.848ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} (t,\mathbf {r} _{0})=\mathbf {v} _{c}(t)+{\boldsymbol {\omega }}(t)\times \mathbf {R} (t,\mathbf {r} _{0})=\mathbf {v} _{c}(t)+{\boldsymbol {\omega }}(t)\times (\mathbf {r} (t,\mathbf {r} _{0})-\mathbf {y} _{c}(t))=\mathbf {v} _{c}(t)+{\boldsymbol {\omega }}(t)\times A(t)\mathbf {y} _{0}}"></span> </p> </blockquote> <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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_3c" style="font-style: normal;"><a href="#Eqnref_3c">3c</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'(t)\mathbf {r} _{0}={\boldsymbol {\omega }}(t)\times A(t)\mathbf {y} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'(t)\mathbf {r} _{0}={\boldsymbol {\omega }}(t)\times A(t)\mathbf {y} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a276b0cda40eb18d63f365945b1b8ce4d8b34a1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.347ex; height:3.009ex;" alt="{\displaystyle A'(t)\mathbf {r} _{0}={\boldsymbol {\omega }}(t)\times A(t)\mathbf {y} _{0}}"></span> </p> </blockquote> <p>On </p> <dl><dd><ul><li><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 \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span> és el vector posició del punt o partícula</li> <li><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 \mathbf {r} _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae202b0f8d86faafc2341193fd9cd4c46159526a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.046ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{c}}"></span> és la posició d'un punt de referència del sòlid</li> <li><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(t)\in SO(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(t)\in SO(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ce683f4bef9e395dcf42cdd63447102dbc5e62c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.477ex; height:2.843ex;" alt="{\displaystyle A(t)\in SO(3)}"></span> és l'orientació, que ve donada per una matriu ortogonal</li> <li><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 \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"></span> és la posició de la partícula respecte al punt de referència del cos al llarg del temps amb una orientació variable.</li> <li><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 \mathbf {r} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe39f0fedae3334af5c4ffaedf25c9778363400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.156ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{0}}"></span> és la posició de la partícula respecte al punt de referència del cos en l'orientació de referència inicial.</li> <li><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 {\boldsymbol {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb8af7a2f64af348e559652b6b1f0d2415ba444" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\omega }}}"></span> és la <a href="/wiki/Velocitat_angular" title="Velocitat angular">velocitat angular</a></li> <li><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 \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> és la velocitat total de la partícula</li> <li><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 \mathbf {v} _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dfdda3c94d8d32e13871189ca9dd313d9f4bc6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.355ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{c}}"></span> és la velocitat de translació o velocitat del punt de referència.</li></ul></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Moment_angular_o_cinètic"><span id="Moment_angular_o_cin.C3.A8tic"></span>Moment angular o cinètic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit&section=4" title="Modifica la secció: Moment angular o cinètic"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Article principal: <a href="/wiki/Moment_angular" title="Moment angular">Moment angular</a> </p><p>El moment angular és una <a href="/wiki/Magnitud_f%C3%ADsica" title="Magnitud física">magnitud física</a> important perquè en molts sistemes físics constitueix una magnitud conservada, que sota certes condicions de les forces que s'apliquen al cos és possible associar-li una <a href="/wiki/Llei_de_conservaci%C3%B3" title="Llei de conservació">llei de conservació</a>. El fet que el moment angular, sota certes circumstàncies, sigui una magnitud constant, pot ser aprofitat en la resolució de les <a href="/wiki/Equacions_de_moviment" class="mw-redirect" title="Equacions de moviment">equacions de moviment</a>. En un instant donat, i fixat un punt de l'espai en el punt origen O, es defineix el moment angular <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 \mathbf {L} _{O}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} _{O}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79893bc4a0fcbf432098de00cd65c7dbc9cbb23f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.509ex;" alt="{\displaystyle \mathbf {L} _{O}}"></span> d'un sistema de partícules respecte a aquest punt com la integral següent: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_4" style="font-style: normal;"><a href="#Eqnref_4">4</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} _{O}=\int _{V}\rho (\mathbf {r} _{O}\times \mathbf {v} _{O})\quad dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>ρ</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="1em" /> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} _{O}=\int _{V}\rho (\mathbf {r} _{O}\times \mathbf {v} _{O})\quad dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8006deec43d610463157d79d7eabd6fc058d8ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.031ex; height:5.676ex;" alt="{\displaystyle \mathbf {L} _{O}=\int _{V}\rho (\mathbf {r} _{O}\times \mathbf {v} _{O})\quad dV}"></span> </p> </blockquote> <p>On <b><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 \mathbf {V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0048514530d0c0fb8a7beb795110815a818784d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {V} }"></span></b> és el <a href="/wiki/Volum" title="Volum">volum</a> del sòlid, <b><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 \rho (\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f477411625125978c0a18946bdfae2c1f13bcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.113ex; height:2.843ex;" alt="{\displaystyle \rho (\mathbf {r} )}"></span></b> és la <a href="/wiki/Densitat" title="Densitat">densitat màssica</a> en cada punt, <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 \mathbf {v} _{O},\mathbf {r} _{O}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{O},\mathbf {r} _{O}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4573976cbc69d88b655c72454812cb2717f7951" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.519ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{O},\mathbf {r} _{O}}"></span> són la velocitat d'una partícula del cos i el vector de posició respecte a O. Convé recordar que el valor de <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 \mathbf {r} _{O}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{O}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b843abddb32761bcc7498fb3b778bbe0d4c56a25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.588ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{O}}"></span> depèn del punt O que es triï. Per a l'estudi de sòlids rígids en moviment convé escollir un "punt mòbil" (és a dir, per a cada instant del temps considerarem un punt diferent de l'espai). Per exemple podem avaluar el moment angular respecte al centre de masses <b><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span></b> del sòlid: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_5" style="font-style: normal;"><a href="#Eqnref_5">5</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{G}={\boldsymbol {\omega }}\times \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{G}={\boldsymbol {\omega }}\times \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6229ee463700c04bbaebafeea4259b944a2385be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.645ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{G}={\boldsymbol {\omega }}\times \mathbf {r} }"></span> </p> </blockquote> <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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_6" style="font-style: normal;"><a href="#Eqnref_6">6</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} _{G}=\int _{V}\rho \left[\mathbf {r} \times ({\boldsymbol {\omega }}\times \mathbf {r} )\right]dV=\int _{V}\rho \left[(\mathbf {r} \cdot \mathbf {r} ){\boldsymbol {\omega }}-\mathbf {r} (\mathbf {r} \cdot {\boldsymbol {\omega }})\right]dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>ρ</mi> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mi>d</mi> <mi>V</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>ρ</mi> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} _{G}=\int _{V}\rho \left[\mathbf {r} \times ({\boldsymbol {\omega }}\times \mathbf {r} )\right]dV=\int _{V}\rho \left[(\mathbf {r} \cdot \mathbf {r} ){\boldsymbol {\omega }}-\mathbf {r} (\mathbf {r} \cdot {\boldsymbol {\omega }})\right]dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8438a21f8d45baa046fb87440ae03ceb3c66872f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:57.152ex; height:5.676ex;" alt="{\displaystyle \mathbf {L} _{G}=\int _{V}\rho \left[\mathbf {r} \times ({\boldsymbol {\omega }}\times \mathbf {r} )\right]dV=\int _{V}\rho \left[(\mathbf {r} \cdot \mathbf {r} ){\boldsymbol {\omega }}-\mathbf {r} (\mathbf {r} \cdot {\boldsymbol {\omega }})\right]dV}"></span> </p> </blockquote> <p><br /> On s'ha introduït l'abreujament <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 \mathbf {r} =\mathbf {r} _{G}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =\mathbf {r} _{G}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ae5d88563f9c090133f50b43eddf16886ab6f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.827ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} =\mathbf {r} _{G}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Espai_de_configuració_d'un_sòlid_rígid"><span id="Espai_de_configuraci.C3.B3_d.27un_s.C3.B2lid_r.C3.ADgid"></span>Espai de configuració d'un sòlid rígid</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit&section=5" title="Modifica la secció: Espai de configuració d'un sòlid rígid"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <a href="/wiki/Formulaci%C3%B3_lagrangiana" title="Formulació lagrangiana">mecànica lagrangiana</a> per descriure un sistema mecànic amb un nombre finit de <a href="/wiki/Graus_de_llibertat_(f%C3%ADsica)" title="Graus de llibertat (física)">graus de llibertat</a> es defineix com una <a href="/wiki/Varietat_diferenciable" title="Varietat diferenciable">varietat diferenciable</a> anomenada <a href="/wiki/Espai_de_configuraci%C3%B3" title="Espai de configuració">espai de configuració</a>. El moviment del sistema es descriu com un conjunt de trajectòries al llarg de l'espai de configuració. Per a un sòlid rígid amb un punt immòbil (només existeix rotació) l'espai de configuració ve donat per la varietat diferenciable del grup de rotació <b>SO(3)</b>. Quan el sòlid té translació i rotació de tots els seus punts l'espai de configuració és <b><i>E</i><sup>+</sup>(<i>n</i>)</b>, el subgrup d'<a href="/wiki/Isometria" title="Isometria">isometria</a> del <a href="/wiki/Geometria_euclidiana" title="Geometria euclidiana">grup euclidià</a> (combinacions de <a href="/wiki/Translaci%C3%B3_(geometria)" title="Translació (geometria)">translacions</a> i <a href="/wiki/Rotaci%C3%B3_(matem%C3%A0tiques)" title="Rotació (matemàtiques)">rotacions</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Tensor_d'inèrcia"><span id="Tensor_d.27in.C3.A8rcia"></span>Tensor d'inèrcia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit&section=6" title="Modifica la secció: Tensor d'inèrcia"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Article principal:</i> <a href="/wiki/Tensor_d%27in%C3%A8rcia" title="Tensor d'inèrcia">Tensor d'inèrcia</a> </p><p>Quan s'estudia el moviment d'un sòlid rígid resulta convenient descompondre'l en un moviment de translació més un moviment de rotació: </p> <ol><li>Per descriure la <b>translació</b> només necessitem calcular les <a href="/wiki/For%C3%A7a" title="Força">forces</a> resultants i aplicar les <a href="/wiki/Lleis_de_Newton" title="Lleis de Newton">lleis de Newton</a> com si es tractés de punts materials.</li> <li>En canvi la descripció de la <b>rotació</b> és més complexa, ja que necessitem alguna magnitud que defineixi com està distribuïda la massa al voltant de cert punt o de l'eix de rotació (per exemple un eix que passi pel centre de massa). Aquesta magnitud és el <b>tensor d'inèrcia</b> que caracteritza la inèrcia rotacional del sòlid.</li></ol> <p>Aquest tensor d'inèrcia del sòlid rígid es defineix com un tensor simètric de segon ordre tal que la forma quadràtica construïda a partir del <a href="/wiki/Tensor" title="Tensor">tensor</a> i la <a href="/wiki/Velocitat_angular" title="Velocitat angular">velocitat angular</a> <b><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span></b> dona l'<a href="/wiki/Energia_cin%C3%A8tica" title="Energia cinètica">energia cinètica de rotació</a>, és a dir: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_7" style="font-style: normal;"><a href="#Eqnref_7">7</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{rot}={\frac {1}{2}}\left({\begin{matrix}\omega _{x}&\omega _{y}&\omega _{z}\\\end{matrix}}\right)\left({\begin{matrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{matrix}}\right)\left({\begin{matrix}\omega _{x}\\\omega _{y}\\\omega _{z}\\\end{matrix}}\right)={\frac {1}{2}}\sum _{j}\sum _{k}I_{jk}\omega _{j}\omega _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{rot}={\frac {1}{2}}\left({\begin{matrix}\omega _{x}&\omega _{y}&\omega _{z}\\\end{matrix}}\right)\left({\begin{matrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{matrix}}\right)\left({\begin{matrix}\omega _{x}\\\omega _{y}\\\omega _{z}\\\end{matrix}}\right)={\frac {1}{2}}\sum _{j}\sum _{k}I_{jk}\omega _{j}\omega _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0052b88ad6732bded100c028057e978d837c9cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:71.766ex; height:9.843ex;" alt="{\displaystyle E_{rot}={\frac {1}{2}}\left({\begin{matrix}\omega _{x}&\omega _{y}&\omega _{z}\\\end{matrix}}\right)\left({\begin{matrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{matrix}}\right)\left({\begin{matrix}\omega _{x}\\\omega _{y}\\\omega _{z}\\\end{matrix}}\right)={\frac {1}{2}}\sum _{j}\sum _{k}I_{jk}\omega _{j}\omega _{k}}"></span> </p> </blockquote> <p>No només l'energia cinètica es pot expressar senzillament en termes del tensor d'inèrcia, si reescrivim l'expressió (<span id="Eqnref_6" class="plainlinksneverexpand"><a href="#Equation_6">6</a></span>) per al moment angular introduint-hi la definició del tensor d'inèrcia, veiem que aquest tensor és l'aplicació lineal que relaciona la velocitat angular i el moment angular: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_8" style="font-style: normal;"><a href="#Eqnref_8">8</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} _{G}=\mathbf {I} {\boldsymbol {\omega }}=\left({\begin{matrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{matrix}}\right)\left({\begin{matrix}\omega _{x}\\\omega _{y}\\\omega _{z}\\\end{matrix}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} _{G}=\mathbf {I} {\boldsymbol {\omega }}=\left({\begin{matrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{matrix}}\right)\left({\begin{matrix}\omega _{x}\\\omega _{y}\\\omega _{z}\\\end{matrix}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af562ca33de24fde85b4d2028e26657129558e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:38.413ex; height:9.843ex;" alt="{\displaystyle \mathbf {L} _{G}=\mathbf {I} {\boldsymbol {\omega }}=\left({\begin{matrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{matrix}}\right)\left({\begin{matrix}\omega _{x}\\\omega _{y}\\\omega _{z}\\\end{matrix}}\right)}"></span> </p> </blockquote> <div class="mw-heading mw-heading2"><h2 id="Equacions_del_moviment">Equacions del moviment</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit&section=7" title="Modifica la secció: Equacions del moviment"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Angles_d'Euler"><span id="Angles_d.27Euler"></span>Angles d'Euler</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit&section=8" title="Modifica la secció: Angles d'Euler"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Article principal:</i> <a href="/wiki/Angles_d%27Euler" title="Angles d'Euler">angles d'Euler</a> </p><p>Els <b>angles d'Euler</b> són tres coordenades angulars que permeten relacionar l'orientació d'un sistema d'eixos respecte a un altre. En mecànica del sòlid rígid es consideren normalment dos sistemes de referència: un sistema d'eixos fix o associat a un observador inercial i un altre mòbil respecte al primer però solidari amb el sòlid rígid. Encara que tècnicament és possible plantejar les equacions de Newton per al sistema inercial relacionant les magnituds del sistema associat al sòlid rígid mitjançant la <a href="/wiki/Matriu_de_rotaci%C3%B3" title="Matriu de rotació">matriu de rotació</a> associada als angles d'Euler, resulta un sistema d'equacions poc pràctic a causa que en aquest sistema el tensor d'inèrcia varia amb el temps. D'altra banda, els àngles d'Euler proporcionen tres coordenades generalitzades adequades per descriure el moviment de sòlids rígids mitjançant els mètodes de la mecànica lagrangiana. </p> <div class="mw-heading mw-heading3"><h3 id="Equacions_d'Euler"><span id="Equacions_d.27Euler"></span>Equacions d'Euler</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit&section=9" title="Modifica la secció: Equacions d'Euler"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Article principal:</i> <a href="/wiki/Equacions_d%27Euler" class="mw-redirect" title="Equacions d'Euler">Equacions d'Euler</a> </p><p>Quan les equacions del moviment d'un sòlid rígid s'expressen en un sistema de referència no inercial solidari amb els eixos principals d'inèrcia del sòlid rígid prenen una fórmula particularment simple coneguda com a equacions d'Euler. En general, en aquest sistema de referència és molt més senzill integrar les equacions de moviments que en un sistema de referència inercial i no solidari amb el cos. Les equacions d'Euler per al moviment d'un sòlid rígid tenen la forma: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_9" style="font-style: normal;"><a href="#Eqnref_9">9</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}I_{1}{\dot {\omega }}_{1}+(I_{3}-I_{2})\omega _{2}\omega _{3}&=&M_{1}\\I_{2}{\dot {\omega }}_{2}+(I_{1}-I_{3})\omega _{3}\omega _{1}&=&M_{2}\\I_{3}{\dot {\omega }}_{3}+(I_{2}-I_{1})\omega _{1}\omega _{2}&=&M_{3}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}I_{1}{\dot {\omega }}_{1}+(I_{3}-I_{2})\omega _{2}\omega _{3}&=&M_{1}\\I_{2}{\dot {\omega }}_{2}+(I_{1}-I_{3})\omega _{3}\omega _{1}&=&M_{2}\\I_{3}{\dot {\omega }}_{3}+(I_{2}-I_{1})\omega _{1}\omega _{2}&=&M_{3}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966a0abdde76d94543dd18978b8908d53d740bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:31.735ex; height:9.843ex;" alt="{\displaystyle {\begin{matrix}I_{1}{\dot {\omega }}_{1}+(I_{3}-I_{2})\omega _{2}\omega _{3}&=&M_{1}\\I_{2}{\dot {\omega }}_{2}+(I_{1}-I_{3})\omega _{3}\omega _{1}&=&M_{2}\\I_{3}{\dot {\omega }}_{3}+(I_{2}-I_{1})\omega _{1}\omega _{2}&=&M_{3}\end{matrix}}}"></span> </p> </blockquote> <p>on <b><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_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43175c381ee6ba2694520af1f1a26c676a2726ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.343ex; height:2.509ex;" alt="{\displaystyle M_{k}}"></span></b>, són les components vectorials del moment o torque total aplicat, <b><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 I_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d658e7f6b34dd1d3025a7c9a72efba5b9f46475d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.112ex; height:2.509ex;" alt="{\displaystyle I_{k}}"></span></b>, són els moments principals d'inèrcia i <b><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 _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e636bf7531cecd91206a36f038cf869e7934932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.535ex; height:2.009ex;" alt="{\displaystyle \omega _{k}}"></span></b> són les components del vector velocitat angular <b><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span></b> segons els eixos principals d'inèrcia. </p> <div class="mw-heading mw-heading3"><h3 id="Baldufa_simètrica"><span id="Baldufa_sim.C3.A8trica"></span>Baldufa simètrica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit&section=10" title="Modifica la secció: Baldufa simètrica"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Frame"><a href="/wiki/Fitxer:Precessing-top.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d9/Precessing-top.gif" decoding="async" width="200" height="150" class="mw-file-element" data-file-width="200" data-file-height="150" /></a><figcaption>Moviment complex d'un sòlid rígid que presenta <a href="/wiki/Precessi%C3%B3" title="Precessió">precessió</a> al voltant de la direcció del <a href="/wiki/Moment_angular" title="Moment angular">moment angular</a> a més <a href="/wiki/Moviment_circular" title="Moviment circular">rotació</a> respecte al seu <a href="/wiki/Simetria" title="Simetria">eix de simetria</a></figcaption></figure> <p>Es diu <b>baldufa simètrica</b> a un sòlid rígid de revolució, amb dos dels seus moments d'inèrcia principals iguals <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 I_{1}=I_{2}\neq I_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≠</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{1}=I_{2}\neq I_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e298edfaa4377954f15d3e42072f757129f5c16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.429ex; height:2.676ex;" alt="{\displaystyle I_{1}=I_{2}\neq I_{3}}"></span>. Com en una baldufa simètrica es poden escollir arbitràriament els eixos 1 i 2, convé aprofitar aquest fet per simplificar les expressions prenent l'eix 1 paral·lel a la línia nodal dels <a href="/wiki/Angles_d%27Euler" title="Angles d'Euler">angles d'Euler</a> la qual cosa equival a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c96cdab103e6c884877c86d6e5db6e471a167d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.646ex; height:2.509ex;" alt="{\displaystyle \phi =0}"></span>. La qual cosa fa que les velocitats angulars en el sistema de referència no inercial vinguin donades per: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_10" style="font-style: normal;"><a href="#Eqnref_10">10</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}={\begin{Bmatrix}\omega _{1}\\\omega _{2}\\\omega _{3}\end{Bmatrix}}={\begin{Bmatrix}{\dot {\theta }}\\{\dot {\phi }}\sin \theta \\{\dot {\phi }}\cos \theta +{\dot {\psi }}\end{Bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}={\begin{Bmatrix}\omega _{1}\\\omega _{2}\\\omega _{3}\end{Bmatrix}}={\begin{Bmatrix}{\dot {\theta }}\\{\dot {\phi }}\sin \theta \\{\dot {\phi }}\cos \theta +{\dot {\psi }}\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1556421752e8935b3832828ad43217cac8272c80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:31.012ex; height:10.176ex;" alt="{\displaystyle {\boldsymbol {\omega }}={\begin{Bmatrix}\omega _{1}\\\omega _{2}\\\omega _{3}\end{Bmatrix}}={\begin{Bmatrix}{\dot {\theta }}\\{\dot {\phi }}\sin \theta \\{\dot {\phi }}\cos \theta +{\dot {\psi }}\end{Bmatrix}}}"></span> </p> </blockquote> <p>L'energia cinètica de rotació una baldufa simètrica pot expressar-se en termes dels angles d'Euler senzillament: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_11" style="font-style: normal;"><a href="#Eqnref_11">11</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{c}={\frac {1}{2}}\left(I_{1}\omega _{1}^{2}+I_{1}\omega _{2}^{2}+I_{3}\omega _{3}^{2}\right)={\frac {I_{1}}{2}}\left({\dot {\phi }}^{2}\sin ^{2}\theta +{\dot {\theta }}^{2}\right)+{\frac {I_{3}}{2}}\left({\dot {\phi }}\cos \theta +{\dot {\psi }}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{c}={\frac {1}{2}}\left(I_{1}\omega _{1}^{2}+I_{1}\omega _{2}^{2}+I_{3}\omega _{3}^{2}\right)={\frac {I_{1}}{2}}\left({\dot {\phi }}^{2}\sin ^{2}\theta +{\dot {\theta }}^{2}\right)+{\frac {I_{3}}{2}}\left({\dot {\phi }}\cos \theta +{\dot {\psi }}\right)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e4fcfae437cc9e25d8538a2f2170cafc597d121" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:72.87ex; height:5.343ex;" alt="{\displaystyle E_{c}={\frac {1}{2}}\left(I_{1}\omega _{1}^{2}+I_{1}\omega _{2}^{2}+I_{3}\omega _{3}^{2}\right)={\frac {I_{1}}{2}}\left({\dot {\phi }}^{2}\sin ^{2}\theta +{\dot {\theta }}^{2}\right)+{\frac {I_{3}}{2}}\left({\dot {\phi }}\cos \theta +{\dot {\psi }}\right)^{2}}"></span> </p> </blockquote> <p>D'altra banda si es pren l'eix Z del sistema de referència alineat amb el moment angular del sòlid rígid es té de les components del moment angular i la relació amb la velocitat angular són: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_12" style="font-style: normal;"><a href="#Eqnref_12">12</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {M} ={\begin{Bmatrix}0\\M\sin \theta \\M\cos \theta \end{Bmatrix}}={\begin{bmatrix}I_{1}&0&0\\0&I_{1}&0\\0&0&I_{3}\end{bmatrix}}{\begin{Bmatrix}\omega _{1}\\\omega _{2}\\\omega _{3}\end{Bmatrix}}={\begin{Bmatrix}I_{1}{\dot {\theta }}\\I_{1}{\dot {\phi }}\sin \theta \\I_{3}{\dot {\phi }}\cos \theta +{\dot {\psi }}\end{Bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>M</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>M</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {M} ={\begin{Bmatrix}0\\M\sin \theta \\M\cos \theta \end{Bmatrix}}={\begin{bmatrix}I_{1}&0&0\\0&I_{1}&0\\0&0&I_{3}\end{bmatrix}}{\begin{Bmatrix}\omega _{1}\\\omega _{2}\\\omega _{3}\end{Bmatrix}}={\begin{Bmatrix}I_{1}{\dot {\theta }}\\I_{1}{\dot {\phi }}\sin \theta \\I_{3}{\dot {\phi }}\cos \theta +{\dot {\psi }}\end{Bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a532e9192b3ec95573dcca69464d6bbd953735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:64.087ex; height:10.176ex;" alt="{\displaystyle \mathbf {M} ={\begin{Bmatrix}0\\M\sin \theta \\M\cos \theta \end{Bmatrix}}={\begin{bmatrix}I_{1}&0&0\\0&I_{1}&0\\0&0&I_{3}\end{bmatrix}}{\begin{Bmatrix}\omega _{1}\\\omega _{2}\\\omega _{3}\end{Bmatrix}}={\begin{Bmatrix}I_{1}{\dot {\theta }}\\I_{1}{\dot {\phi }}\sin \theta \\I_{3}{\dot {\phi }}\cos \theta +{\dot {\psi }}\end{Bmatrix}}}"></span> </p> </blockquote> <p>Escrivint component a component aquestes equacions es té que: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_13" style="font-style: normal;"><a href="#Eqnref_13">13</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\theta }}=0\qquad I_{1}{\dot {\phi }}=M\qquad I_{3}\omega _{3}=I_{3}({\dot {\phi }}\cos \theta +{\dot {\psi }})=M\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>M</mi> <mspace width="2em" /> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ψ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>M</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\theta }}=0\qquad I_{1}{\dot {\phi }}=M\qquad I_{3}\omega _{3}=I_{3}({\dot {\phi }}\cos \theta +{\dot {\psi }})=M\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3029af376ee2a031a1b76d35ffc71aa2a0b206a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.951ex; height:3.343ex;" alt="{\displaystyle {\dot {\theta }}=0\qquad I_{1}{\dot {\phi }}=M\qquad I_{3}\omega _{3}=I_{3}({\dot {\phi }}\cos \theta +{\dot {\psi }})=M\cos \theta }"></span> </p> </blockquote> <p>La primera equació ens diu que en el moviment lliure d'una baldufa simètrica aquesta no remata, és a dir, no hi ha moviment de <a href="/wiki/Nutaci%C3%B3" title="Nutació">nutació</a>, ja que l'angle format per eix de rotació i el moment angular es manté constant en el moviment. La segona descriu el moviment de precessió d'acord amb el qual l'eix de rotació (que coincideix amb la direcció de la velocitat angular) gira al voltant de la direcció del moment angular (eix Z). La tercera equació dona la velocitat de rotació del sòlid al voltant del seu tercer eix d'inèrcia. </p> <div class="mw-heading mw-heading3"><h3 id="Baldufa_asimètrica"><span id="Baldufa_asim.C3.A8trica"></span>Baldufa asimètrica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit&section=11" title="Modifica la secció: Baldufa asimètrica"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una baldufa asimètrica és un sòlid rígid tal que cap dels seus tres moments principals d'inèrcia té el mateix valor, és comú anomenar en ordre ascendent com: <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 I1<I2<I3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mn>1</mn> <mo><</mo> <mi>I</mi> <mn>2</mn> <mo><</mo> <mi>I</mi> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I1<I2<I3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9b400a3d46dc4e965d02684ce371bd383be790f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.2ex; height:2.176ex;" alt="{\displaystyle I1<I2<I3}"></span>. En el moviment de gir lliure d'una baldufa té dues integrals de moviment: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_14a" style="font-style: normal;"><a href="#Eqnref_14a">14a</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {L_{1}^{2}}{I_{1}}}+{\frac {L_{2}^{2}}{I_{2}}}+{\frac {L_{3}^{2}}{I_{3}}}=2E\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>E</mi> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L_{1}^{2}}{I_{1}}}+{\frac {L_{2}^{2}}{I_{2}}}+{\frac {L_{3}^{2}}{I_{3}}}=2E\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/576163f634563425fd2392be2948c1afbb19fcd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.782ex; height:6.509ex;" alt="{\displaystyle {\frac {L_{1}^{2}}{I_{1}}}+{\frac {L_{2}^{2}}{I_{2}}}+{\frac {L_{3}^{2}}{I_{3}}}=2E\;}"></span> </p> </blockquote> <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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_14b" style="font-style: normal;"><a href="#Eqnref_14b">14b</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{1}^{2}+L_{2}^{2}+L_{3}^{2}=L^{2}\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{1}^{2}+L_{2}^{2}+L_{3}^{2}=L^{2}\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6453114cff5bdf0e545d474c5fe3d1aacc855382" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.973ex; height:3.343ex;" alt="{\displaystyle L_{1}^{2}+L_{2}^{2}+L_{3}^{2}=L^{2}\;}"></span> </p> </blockquote> <p>Com que només hi ha tres coordenades angulars i existeixen aquestes dues restriccions, les components del moment angular només poden variar al llarg d'una corba donada per la intersecció de l'el·lipsoide (<span id="Eqnref_14a" class="plainlinksneverexpand"><a href="#Equation_14a">14a</a></span>) i l'esfera (<span id="Eqnref_14b" class="plainlinksneverexpand"><a href="#Equation_14b">14b</a></span>). Així mateix, es pot veure que el gir al voltant dels eixos d'inèrcia associat als moments <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 I_{1},I_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{1},I_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50e3d706b63d63c5f780e2122af16cebf3a1f067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.189ex; height:2.509ex;" alt="{\displaystyle I_{1},I_{3}}"></span> és estable mentre que l'associat a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3506ae39df854f347365bae6f326ef4f565be5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.077ex; height:2.509ex;" alt="{\displaystyle I_{2}}"></span> és inestable, és a dir, qualsevol petita pertorbació canvia dràsticament les trajectòries del moviment. 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 L^{2}>2EI_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>2</mn> <mi>E</mi> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}>2EI_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a844ff9a9357676bab32629dfdcecaf16309739c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.751ex; height:3.009ex;" alt="{\displaystyle L^{2}>2EI_{2}}"></span> les equacions paramètriques de variació de les velocitats angulars venen donades per les <a href="/wiki/Funci%C3%B3_el%C2%B7l%C3%ADptica_de_Jacobi" class="mw-redirect" title="Funció el·líptica de Jacobi">funcions el·líptiques de Jacobi</a>: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_15" style="font-style: normal;"><a href="#Eqnref_15">15</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\omega _{1}={\sqrt {\frac {2EI_{3}-L^{2}}{I_{1}(I_{3}-I_{1})}}}{\mbox{cn}}\tau \\\omega _{2}={\sqrt {\frac {2EI_{3}-L^{2}}{I_{1}(I_{3}-I_{2})}}}{\mbox{sn}}\tau \\\omega _{3}={\sqrt {\frac {L^{2}-2EI_{1}}{I_{1}(I_{3}-I_{2})}}}{\mbox{dn}}\tau \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>2</mn> <mi>E</mi> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>cn</mtext> </mstyle> </mrow> <mi>τ</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>2</mn> <mi>E</mi> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>sn</mtext> </mstyle> </mrow> <mi>τ</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>E</mi> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>dn</mtext> </mstyle> </mrow> <mi>τ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\omega _{1}={\sqrt {\frac {2EI_{3}-L^{2}}{I_{1}(I_{3}-I_{1})}}}{\mbox{cn}}\tau \\\omega _{2}={\sqrt {\frac {2EI_{3}-L^{2}}{I_{1}(I_{3}-I_{2})}}}{\mbox{sn}}\tau \\\omega _{3}={\sqrt {\frac {L^{2}-2EI_{1}}{I_{1}(I_{3}-I_{2})}}}{\mbox{dn}}\tau \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b50fcc28a25d74a771f687a85e3fe526bb803e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.005ex; width:24.179ex; height:23.176ex;" alt="{\displaystyle {\begin{aligned}\omega _{1}={\sqrt {\frac {2EI_{3}-L^{2}}{I_{1}(I_{3}-I_{1})}}}{\mbox{cn}}\tau \\\omega _{2}={\sqrt {\frac {2EI_{3}-L^{2}}{I_{1}(I_{3}-I_{2})}}}{\mbox{sn}}\tau \\\omega _{3}={\sqrt {\frac {L^{2}-2EI_{1}}{I_{1}(I_{3}-I_{2})}}}{\mbox{dn}}\tau \end{aligned}}}"></span> </p> </blockquote> <p>amb: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_16" style="font-style: normal;"><a href="#Eqnref_16">16</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =t{\sqrt {\frac {(I_{3}-I_{2})(L^{2}-2EI_{1})}{I_{1}I_{2}I_{3}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>E</mi> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =t{\sqrt {\frac {(I_{3}-I_{2})(L^{2}-2EI_{1})}{I_{1}I_{2}I_{3}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d3bb4558de39381eec4746939f164bba337ced7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.406ex; height:7.676ex;" alt="{\displaystyle \tau =t{\sqrt {\frac {(I_{3}-I_{2})(L^{2}-2EI_{1})}{I_{1}I_{2}I_{3}}}}}"></span> </p> </blockquote> <p>Si <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^{2}<2EI_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>2</mn> <mi>E</mi> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}<2EI_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71dad8a90a8d4c624ea42ea0be6b0799acf48732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.751ex; height:3.009ex;" alt="{\displaystyle L^{2}<2EI_{2}}"></span> n'hi ha prou intercanviar els subíndexs 1 i 3 en les anteriors expressions. </p><p>Finalment convé observar que quan <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 I_{1}\to I_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{1}\to I_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8196f6f48ab2cabab3cac841def201fa14741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.769ex; height:2.509ex;" alt="{\displaystyle I_{1}\to I_{2}}"></span> les funcions el·líptiques de Jacobi es redueixen a funcions trigonomètriques ordinàries, i les equacions del moviment es redueixen a les d'una baldufa simètrica: </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 style="float:right; width:10%; text-align:right;">(<cite id="Equation_17" style="font-style: normal;"><a href="#Eqnref_17">17</a></cite>)</span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{sn}}\tau \to \sin \tau ,\quad {\mbox{cn}}\tau \to \cos \tau ,\quad {\mbox{dn}}\tau \to 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>sn</mtext> </mstyle> </mrow> <mi>τ</mi> <mo stretchy="false">→</mo> <mi>sin</mi> <mo>⁡</mo> <mi>τ</mi> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>cn</mtext> </mstyle> </mrow> <mi>τ</mi> <mo stretchy="false">→</mo> <mi>cos</mi> <mo>⁡</mo> <mi>τ</mi> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>dn</mtext> </mstyle> </mrow> <mi>τ</mi> <mo stretchy="false">→</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mbox{sn}}\tau \to \sin \tau ,\quad {\mbox{cn}}\tau \to \cos \tau ,\quad {\mbox{dn}}\tau \to 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56a0c2edfc47a4c104f541ae8a433f110807928c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.587ex; height:2.509ex;" alt="{\displaystyle {\mbox{sn}}\tau \to \sin \tau ,\quad {\mbox{cn}}\tau \to \cos \tau ,\quad {\mbox{dn}}\tau \to 1}"></span> </p> </blockquote> <div class="mw-heading mw-heading2"><h2 id="Sòlid_rígid_en_mecànica_quàntica"><span id="S.C3.B2lid_r.C3.ADgid_en_mec.C3.A0nica_qu.C3.A0ntica"></span>Sòlid rígid en mecànica quàntica</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit&section=12" title="Modifica la secció: Sòlid rígid en mecànica quàntica"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La formulació <a href="/wiki/Mec%C3%A0nica_qu%C3%A0ntica" title="Mecànica quàntica">quàntica</a> es realitza mitjançant la quantificació de la <a href="/wiki/Varietat_simpl%C3%A8ctica" title="Varietat simplèctica">varietat simplèctica</a> de 12 dimensions associada a un sòlid rígid. L'<a href="/wiki/Espai_de_configuraci%C3%B3" title="Espai de configuració">espai de configuració</a> d'un sòlid rígid 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 SO(3)\times \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SO(3)\times \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea68561bdba0f3f06e2f4d6f83a60f1d6ae1851b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.817ex; height:3.176ex;" alt="{\displaystyle SO(3)\times \mathbb {R} ^{3}}"></span></dd></dl> <p>i per tant un <a href="/wiki/Espai_de_Hilbert" title="Espai de Hilbert">espai de Hilbert</a> adequat per al sòlid rígid és isomorf al producte tensorial d'espais de funcions de quadrat integrable </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 L^{2}(SO(3),\mu _{H})\otimes L^{2}(\mathbb {R} ^{3},\mu _{L})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊗</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}(SO(3),\mu _{H})\otimes L^{2}(\mathbb {R} ^{3},\mu _{L})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04a749209fe8a140eecbbf884e1bb7b0bf3895bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.624ex; height:3.176ex;" alt="{\displaystyle L^{2}(SO(3),\mu _{H})\otimes L^{2}(\mathbb {R} ^{3},\mu _{L})}"></span></dd></dl> <p>On <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 \mu _{H},\ \mu _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{H},\ \mu _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c46b8c31a0ca76982804206effcb6d255ac2c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.461ex; height:2.176ex;" alt="{\displaystyle \mu _{H},\ \mu _{L}}"></span> són respectivament la <a href="/wiki/Mesura_de_Haar" title="Mesura de Haar">mesura de Haar</a> de <a href="/wiki/Grup_especial_ortogonal" class="mw-redirect" title="Grup especial ortogonal"><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 SO(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SO(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16c677fee782e584fd417726201ce27c567f1e11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.244ex; height:2.843ex;" alt="{\displaystyle SO(3)}"></span></a> i la <a href="/wiki/Mesura_de_Lebesgue" title="Mesura de Lebesgue">mesura de Lebesgue</a> de <a href="/wiki/Espai_euclidi%C3%A0" title="Espai euclidià"><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 \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span></a>. </p><p>Donada la compacitat de <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 SO(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SO(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16c677fee782e584fd417726201ce27c567f1e11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.244ex; height:2.843ex;" alt="{\displaystyle SO(3)}"></span>, l'energia cinètica de rotació pot considerar-se com una suma directa d'operadors actuant sobre espais vectorials de dimensió finita. </p> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&action=edit&section=13" title="Modifica la secció: Bibliografia"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation book" style="font-style:normal" id="CITEREFLandau1991"><span style="font-variant: small-caps;">Landau</span>, L.D.; Lifshitz E.M.. «VI». A: Reverté. <i>Mecánica</i>. 2a edició, 1991, p. 115-157. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/84-291-4080-8" title="Especial:Fonts bibliogràfiques/84-291-4080-8">ISBN 84-291-4080-8</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=Mec%C3%A1nica&rft.atitle=VI&rft.aulast=Landau&rft.aufirst=L.D.&rft.date=1991&rft.edition=2a%C2%A0edici%C3%B3&rft.place=Barcelona&rft.pages=115-157&rft.isbn=84-291-4080-8"><span style="display: none;"> </span></span></li></ul> <div role="navigation" class="navbox" aria-labelledby="Principals_camps_de_la_Física" style="padding:3px"><table class="nowraplinks collapsible collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div class="plainlinks hlist navbar mini"><ul><li class="nv-view"><span typeof="mw:File"><a href="/wiki/Plantilla:F%C3%ADsica" title="Plantilla:Física"><img alt="Vegeu aquesta plantilla" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/18px-Commons-emblem-notice.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/27px-Commons-emblem-notice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/36px-Commons-emblem-notice.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></li></ul></div><div id="Principals_camps_de_la_Física" style="font-size:114%;margin:0 4em">Principals camps de la <a href="/wiki/F%C3%ADsica" title="Física">Física</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"><a href="/wiki/Astrof%C3%ADsica" title="Astrofísica">Astrofísica</a><span style="font-weight:bold;"> ·</span>  <a href="/wiki/Mec%C3%A0nica_cl%C3%A0ssica" title="Mecànica clàssica">Mecànica clàssica</a><span style="font-weight:bold;"> ·</span>  <a href="/wiki/F%C3%ADsica_de_la_mat%C3%A8ria_condensada" title="Física de la matèria condensada">Física de la matèria condensada</a><span style="font-weight:bold;"> ·</span>  <a href="/wiki/Mec%C3%A0nica_dels_medis_continus" title="Mecànica dels medis continus">Mecànica dels medis continus</a><span style="font-weight:bold;"> ·</span>  <a href="/wiki/Termodin%C3%A0mica" title="Termodinàmica">Termodinàmica</a><span style="font-weight:bold;"> ·</span>  <a href="/wiki/Electromagnetisme" title="Electromagnetisme">Electromagnetisme</a><span style="font-weight:bold;"> ·</span>  <a href="/wiki/Teoria_de_la_relativitat" title="Teoria de la relativitat">Teoria de la relativitat</a><span style="font-weight:bold;"> ·</span>  <a href="/wiki/F%C3%ADsica_de_part%C3%ADcules" title="Física de partícules">Física de partícules</a><span style="font-weight:bold;"> ·</span>  <a href="/wiki/Teoria_qu%C3%A0ntica_de_camps" title="Teoria quàntica de camps">Teoria quàntica de camps</a><span style="font-weight:bold;"> ·</span>  <a href="/wiki/Mec%C3%A0nica_qu%C3%A0ntica" title="Mecànica quàntica">Mecànica quàntica</a><span style="font-weight:bold;"> ·</span>  <a href="/wiki/Mec%C3%A0nica_estad%C3%ADstica" title="Mecànica estadística">Mecànica estadística</a><span style="font-weight:bold;"> ·</span>  <a href="/wiki/F%C3%ADsica_te%C3%B2rica" title="Física teòrica">Física teòrica</a></div></td></tr></tbody></table></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=Dinàmica_del_sòlid_rígid&oldid=34256409">https://ca.wikipedia.org/w/index.php?title=Dinàmica_del_sòlid_rígid&oldid=34256409</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">Categoria</a>: <ul><li><a href="/wiki/Categoria:Mec%C3%A0nica" title="Categoria:Mecànica">Mecànica</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 18 nov 2024 a les 12:23.</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. Vegeu les <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use/ca">Condicions d'ús</a>. Wikipedia® (Viquipèdia™) és una <a href="/wiki/Marca_comercial" title="Marca comercial">marca registrada</a> de <a rel="nofollow" class="external text" href="https://www.wikimediafoundation.org">Wikimedia Foundation, Inc</a>.<br /></li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Política de privadesa</a></li> <li id="footer-places-about"><a href="/wiki/Viquip%C3%A8dia:Quant_a_la_Viquip%C3%A8dia">Quant al projecte Viquipèdia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Viquip%C3%A8dia:Av%C3%ADs_d%27exempci%C3%B3_de_responsabilitat">Descàrrec de responsabilitat</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Codi de conducta</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Desenvolupadors</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/ca.wikipedia.org">Estadístiques</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Declaració de cookies</a></li> <li id="footer-places-mobileview"><a href="//ca.m.wikipedia.org/w/index.php?title=Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Versió per a mòbils</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-dnxpp","wgBackendResponseTime":139,"wgPageParseReport":{"limitreport":{"cputime":"0.138","walltime":"0.273","ppvisitednodes":{"value":1462,"limit":1000000},"postexpandincludesize":{"value":43217,"limit":2097152},"templateargumentsize":{"value":1913,"limit":2097152},"expansiondepth":{"value":10,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":2125,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 80.463 1 -total"," 46.14% 37.127 1 Plantilla:Mecànica_clàssica"," 42.13% 33.901 1 Plantilla:Barra_lateral_amb_llistes_plegables"," 21.45% 17.262 1 Plantilla:Ref-llibre"," 13.47% 10.837 20 Plantilla:Equació"," 12.14% 9.772 1 Plantilla:If_both"," 11.28% 9.076 1 Plantilla:Física"," 7.54% 6.070 1 Plantilla:Caixa_de_navegació"," 6.84% 5.502 63 Plantilla:·w"," 6.34% 5.105 20 Plantilla:Trim"]},"scribunto":{"limitreport-timeusage":{"value":"0.012","limit":"10.000"},"limitreport-memusage":{"value":934242,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-869fdccf5d-l6ts4","timestamp":"20241118115103","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Din\u00e0mica del s\u00f2lid r\u00edgid","url":"https:\/\/ca.wikipedia.org\/wiki\/Din%C3%A0mica_del_s%C3%B2lid_r%C3%ADgid","sameAs":"http:\/\/www.wikidata.org\/entity\/Q2037529","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q2037529","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2011-03-31T22:53:08Z","dateModified":"2024-11-18T11:23:03Z"}</script> </body> </html>

CINXE.COM

Dinàmica del sòlid rígid - Viquipèdia, l'enciclopèdia lliure