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<span>Definition</span> </div> </a> <button aria-controls="toc-Definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Definition subsection</span> </button> <ul id="toc-Definition-sublist" class="vector-toc-list"> <li id="toc-A_system_of_particles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_system_of_particles"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>A system of particles</span> </div> </a> <ul id="toc-A_system_of_particles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_continuous_volume" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_continuous_volume"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>A continuous volume</span> </div> </a> <ul id="toc-A_continuous_volume-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Barycentric_coordinates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Barycentric_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Barycentric coordinates</span> </div> </a> <ul id="toc-Barycentric_coordinates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Systems_with_periodic_boundary_conditions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Systems_with_periodic_boundary_conditions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Systems with periodic boundary conditions</span> </div> </a> <ul id="toc-Systems_with_periodic_boundary_conditions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Center_of_gravity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Center_of_gravity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Center of gravity</span> </div> </a> <ul id="toc-Center_of_gravity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_and_angular_momentum" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Linear_and_angular_momentum"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Linear and angular momentum</span> </div> </a> <ul id="toc-Linear_and_angular_momentum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Determination" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Determination"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Determination</span> </div> </a> <button aria-controls="toc-Determination-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Determination subsection</span> </button> <ul id="toc-Determination-sublist" class="vector-toc-list"> <li id="toc-In_two_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_two_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>In two dimensions</span> </div> </a> <ul id="toc-In_two_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_three_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_three_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>In three dimensions</span> </div> </a> <ul id="toc-In_three_dimensions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Engineering_designs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Engineering_designs"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Engineering designs</span> </div> </a> <ul id="toc-Engineering_designs-sublist" class="vector-toc-list"> <li id="toc-Automotive_applications" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Automotive_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.1</span> <span>Automotive applications</span> </div> </a> <ul id="toc-Automotive_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aeronautics" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Aeronautics"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.2</span> <span>Aeronautics</span> </div> </a> <ul id="toc-Aeronautics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Astronomy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Astronomy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Astronomy</span> </div> </a> <ul id="toc-Astronomy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rigging_and_safety" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rigging_and_safety"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Rigging and safety</span> </div> </a> <ul id="toc-Rigging_and_safety-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Body_motion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Body_motion"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Body motion</span> </div> </a> <ul id="toc-Body_motion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optimization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optimization"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Optimization</span> </div> </a> <ul id="toc-Optimization-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" 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="Toggle the table of contents" > <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">Toggle the table of contents</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">Center of mass</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="Go to an article in another language. Available in 60 languages" > <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-60" 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">60 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Massamiddelpunt" title="Massamiddelpunt – Afrikaans" lang="af" hreflang="af" data-title="Massamiddelpunt" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B1%D9%83%D8%B2_%D8%A7%D9%84%D9%83%D8%AA%D9%84%D8%A9" title="مركز الكتلة – Arabic" lang="ar" hreflang="ar" data-title="مركز الكتلة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%B8%E0%A7%8D%E0%A6%A4%E0%A7%81%E0%A6%B0_%E0%A6%AD%E0%A6%B0%E0%A6%95%E0%A7%87%E0%A6%A8%E0%A7%8D%E0%A6%A6%E0%A7%8D%E0%A6%B0" title="বস্তুর ভরকেন্দ্র – Bangla" lang="bn" hreflang="bn" data-title="বস্তুর ভরকেন্দ্র" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A6%D1%8D%D0%BD%D1%82%D1%80_%D0%BC%D0%B0%D1%81" title="Цэнтр мас – Belarusian" lang="be" hreflang="be" data-title="Цэнтр мас" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%8A%D1%80_%D0%BD%D0%B0_%D0%BC%D0%B0%D1%81%D0%B8%D1%82%D0%B5" title="Център на масите – Bulgarian" lang="bg" hreflang="bg" data-title="Център на масите" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Centre_de_massa" title="Centre de massa – Catalan" lang="ca" hreflang="ca" data-title="Centre de massa" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%81%D1%81%C4%83%D1%81%D0%B5%D0%BD_%D1%86%D0%B5%D0%BD%D1%82%D1%80%C4%95" title="Массăсен центрĕ – Chuvash" lang="cv" hreflang="cv" data-title="Массăсен центрĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs badge-Q70894304 mw-list-item" title=""><a href="https://cs.wikipedia.org/wiki/Hmotn%C3%BD_st%C5%99ed" title="Hmotný střed – Czech" lang="cs" hreflang="cs" data-title="Hmotný střed" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Massemidtpunkt" title="Massemidtpunkt – Danish" lang="da" hreflang="da" data-title="Massemidtpunkt" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Massenmittelpunkt" title="Massenmittelpunkt – German" lang="de" hreflang="de" data-title="Massenmittelpunkt" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Massikese" title="Massikese – Estonian" lang="et" hreflang="et" data-title="Massikese" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Centro_de_masas" title="Centro de masas – Spanish" lang="es" hreflang="es" data-title="Centro de masas" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Masocentro" title="Masocentro – Esperanto" lang="eo" hreflang="eo" data-title="Masocentro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Masa-zentro" title="Masa-zentro – Basque" lang="eu" hreflang="eu" data-title="Masa-zentro" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B1%DA%A9%D8%B2_%D8%AC%D8%B1%D9%85" title="مرکز جرم – Persian" lang="fa" hreflang="fa" data-title="مرکز جرم" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Centre_d%27inertie" title="Centre d'inertie – French" lang="fr" hreflang="fr" data-title="Centre d'inertie" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Me%C3%A1chanl%C3%A1r" title="Meáchanlár – Irish" lang="ga" hreflang="ga" data-title="Meáchanlár" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Centro_de_masas" title="Centro de masas – Galician" lang="gl" hreflang="gl" data-title="Centro de masas" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%88%EB%9F%89_%EC%A4%91%EC%8B%AC" title="질량 중심 – Korean" lang="ko" hreflang="ko" data-title="질량 중심" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B6%D5%A1%D5%B6%D5%A3%D5%BE%D5%A1%D5%AE%D5%B6%D5%A5%D6%80%D5%AB_%D5%AF%D5%A5%D5%B6%D5%BF%D6%80%D5%B8%D5%B6" title="Զանգվածների կենտրոն – Armenian" lang="hy" hreflang="hy" data-title="Զանգվածների կենտրոն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%B9%E0%A4%A4%E0%A4%BF-%E0%A4%95%E0%A5%87%E0%A4%A8%E0%A5%8D%E0%A4%A6%E0%A5%8D%E0%A4%B0" title="संहति-केन्द्र – Hindi" lang="hi" hreflang="hi" data-title="संहति-केन्द्र" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Centar_masa" title="Centar masa – Croatian" lang="hr" hreflang="hr" data-title="Centar masa" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pusat_massa" title="Pusat massa – Indonesian" lang="id" hreflang="id" data-title="Pusat massa" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Massami%C3%B0ja" title="Massamiðja – Icelandic" lang="is" hreflang="is" data-title="Massamiðja" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Centro_di_massa" title="Centro di massa – Italian" lang="it" hreflang="it" data-title="Centro di massa" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%9B%D7%96_%D7%9E%D7%A1%D7%94" title="מרכז מסה – Hebrew" lang="he" hreflang="he" data-title="מרכז מסה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-krc mw-list-item"><a href="https://krc.wikipedia.org/wiki/%D0%A1%D0%B0%D1%83%D0%BB%D0%B0%D0%B9_%D0%B0%D1%83%D1%83%D1%80%D0%BB%D1%83%D0%BA%D1%8A%D0%BD%D1%83_%D0%BE%D1%80%D1%82%D0%B0%D1%81%D1%8B" title="Саулай ауурлукъну ортасы – Karachay-Balkar" lang="krc" hreflang="krc" data-title="Саулай ауурлукъну ортасы" data-language-autonym="Къарачай-малкъар" data-language-local-name="Karachay-Balkar" class="interlanguage-link-target"><span>Къарачай-малкъар</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Sant_mas" title="Sant mas – Haitian Creole" lang="ht" hreflang="ht" data-title="Sant mas" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Masas_centrs" title="Masas centrs – Latvian" lang="lv" hreflang="lv" data-title="Masas centrs" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Mas%C4%97s_centras" title="Masės centras – Lithuanian" lang="lt" hreflang="lt" data-title="Masės centras" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Sentro_de_masa" title="Sentro de masa – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Sentro de masa" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%B6megk%C3%B6z%C3%A9ppont" title="Tömegközéppont – Hungarian" lang="hu" hreflang="hu" data-title="Tömegközéppont" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%B6%D0%B8%D1%88%D1%82%D0%B5" title="Тежиште – Macedonian" lang="mk" hreflang="mk" data-title="Тежиште" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B4%BF%E0%B4%A3%E0%B5%8D%E0%B4%A1%E0%B4%95%E0%B5%87%E0%B4%A8%E0%B5%8D%E0%B4%A6%E0%B5%8D%E0%B4%B0%E0%B4%82" title="പിണ്ഡകേന്ദ്രം – Malayalam" lang="ml" hreflang="ml" data-title="പിണ്ഡകേന്ദ്രം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%81%E0%A4%AE%E0%A4%BE%E0%A4%A8_%E0%A4%95%E0%A5%87%E0%A4%82%E0%A4%A6%E0%A5%8D%E0%A4%B0" title="वस्तुमान केंद्र – Marathi" lang="mr" hreflang="mr" data-title="वस्तुमान केंद्र" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Massamiddelpunt" title="Massamiddelpunt – Dutch" lang="nl" hreflang="nl" data-title="Massamiddelpunt" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%87%8D%E5%BF%83" title="重心 – Japanese" lang="ja" hreflang="ja" data-title="重心" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Swaarponkt" title="Swaarponkt – Northern Frisian" lang="frr" hreflang="frr" data-title="Swaarponkt" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Massesentrum" title="Massesentrum – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Massesentrum" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Massesenter" title="Massesenter – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Massesenter" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Massa_markazi" title="Massa markazi – Uzbek" lang="uz" hreflang="uz" data-title="Massa markazi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/%C5%9Arodek_masy" title="Środek masy – Polish" lang="pl" hreflang="pl" data-title="Środek masy" data-language-autonym="Polski" data-language-local-name="Polish" 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/Centro_de_massa" title="Centro de massa – Portuguese" lang="pt" hreflang="pt" data-title="Centro de massa" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Centru_de_mas%C4%83" title="Centru de masă – Romanian" lang="ro" hreflang="ro" data-title="Centru de masă" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%80_%D0%BC%D0%B0%D1%81%D1%81" title="Центр масс – Russian" lang="ru" hreflang="ru" data-title="Центр масс" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Qendra_e_mas%C3%ABs" title="Qendra e masës – Albanian" lang="sq" hreflang="sq" data-title="Qendra e masës" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Center_of_mass" title="Center of mass – Simple English" lang="en-simple" hreflang="en-simple" data-title="Center of mass" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Masno_sredi%C5%A1%C4%8De" title="Masno središče – Slovenian" lang="sl" hreflang="sl" data-title="Masno središče" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%86%D8%A7%D9%88%DB%95%D9%86%D8%AF%DB%8C_%D8%A8%D8%A7%D8%B1%D8%B3%D8%AA%DB%95" title="ناوەندی بارستە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ناوەندی بارستە" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D0%B0%D1%80_%D0%BC%D0%B0%D1%81%D0%B5" title="Центар масе – Serbian" lang="sr" hreflang="sr" data-title="Центар масе" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Centar_masa" title="Centar masa – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Centar masa" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Masscentrum" title="Masscentrum – Swedish" lang="sv" hreflang="sv" data-title="Masscentrum" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AE%BF%E0%AE%B1%E0%AF%88_%E0%AE%AE%E0%AF%88%E0%AE%AF%E0%AE%AE%E0%AF%8D" title="நிறை மையம் – Tamil" lang="ta" hreflang="ta" data-title="நிறை மையம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A8%E0%B8%B9%E0%B8%99%E0%B8%A2%E0%B9%8C%E0%B8%81%E0%B8%A5%E0%B8%B2%E0%B8%87%E0%B8%A1%E0%B8%A7%E0%B8%A5" title="ศูนย์กลางมวล – Thai" lang="th" hreflang="th" data-title="ศูนย์กลางมวล" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C3%BCtle_merkezi" title="Kütle merkezi – Turkish" lang="tr" hreflang="tr" data-title="Kütle merkezi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%80_%D1%96%D0%BD%D0%B5%D1%80%D1%86%D1%96%D1%97" title="Центр інерції – Ukrainian" lang="uk" hreflang="uk" data-title="Центр інерції" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%E1%BB%91i_t%C3%A2m" title="Khối tâm – Vietnamese" lang="vi" hreflang="vi" data-title="Khối tâm" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%B4%A8%E5%BF%83" title="质心 – Wu" lang="wuu" hreflang="wuu" data-title="质心" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%87%8D%E5%BF%83" title="重心 – Cantonese" lang="yue" hreflang="yue" data-title="重心" data-language-autonym="粵語" data-language-local-name="Cantonese" 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</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">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Center_of_gravity&redirect=no" class="mw-redirect" title="Center of gravity">Center of gravity</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Unique point where the weighted relative position of the distributed mass sums to zero</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bird_toy_showing_center_of_gravity.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Bird_toy_showing_center_of_gravity.jpg/220px-Bird_toy_showing_center_of_gravity.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Bird_toy_showing_center_of_gravity.jpg/330px-Bird_toy_showing_center_of_gravity.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Bird_toy_showing_center_of_gravity.jpg/440px-Bird_toy_showing_center_of_gravity.jpg 2x" data-file-width="3000" data-file-height="3000" /></a><figcaption>This toy uses the principles of center of mass to keep balance when sitting on a finger.</figcaption></figure> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, the <b>center of mass</b> of a distribution of <a href="/wiki/Mass" title="Mass">mass</a> in <a href="/wiki/Space" title="Space">space</a> (sometimes referred to as the <b>barycenter</b> or <b>balance point</b>) is the unique point at any given time where the <a href="/wiki/Weight_function" title="Weight function">weighted</a> relative <a href="/wiki/Position_(vector)" class="mw-redirect" title="Position (vector)">position</a> of the distributed mass sums to zero. For a rigid body containing its center of mass, this is the point to which a force may be applied to cause a <a href="/wiki/Linear_acceleration" class="mw-redirect" title="Linear acceleration">linear acceleration</a> without an <a href="/wiki/Angular_acceleration" title="Angular acceleration">angular acceleration</a>. <a href="/wiki/Calculation" title="Calculation">Calculations</a> in <a href="/wiki/Mechanics" title="Mechanics">mechanics</a> are often simplified when formulated with respect to the center of mass. It is a hypothetical point where the entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the particle equivalent of a given object for application of <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>In the case of a single <a href="/wiki/Rigid_body" title="Rigid body">rigid body</a>, the center of mass is fixed in relation to the body, and if the body has uniform <a href="/wiki/Density" title="Density">density</a>, it will be located at the <a href="/wiki/Centroid" title="Centroid">centroid</a>. The center of mass may be located outside the <a href="/wiki/Physical_object" title="Physical object">physical body</a>, as is sometimes the case for <a href="https://en.wiktionary.org/wiki/hollow" class="extiw" title="wikt:hollow">hollow</a> or open-shaped objects, such as a <a href="/wiki/Horseshoe" title="Horseshoe">horseshoe</a>. In the case of a distribution of separate bodies, such as the <a href="/wiki/Planets" class="mw-redirect" title="Planets">planets</a> of the <a href="/wiki/Solar_System" title="Solar System">Solar System</a>, the center of mass may not correspond to the position of any individual member of the system. </p><p>The center of mass is a useful reference point for calculations in <a href="/wiki/Mechanics" title="Mechanics">mechanics</a> that involve masses distributed in space, such as the <a href="/wiki/Momentum" title="Momentum">linear</a> and <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> of planetary bodies and <a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">rigid body dynamics</a>. In <a href="/wiki/Orbital_mechanics" title="Orbital mechanics">orbital mechanics</a>, the equations of motion of planets are formulated as <a href="/wiki/Point_mass" class="mw-redirect" title="Point mass">point masses</a> located at the centers of mass (see <a href="/wiki/Barycenter_(astronomy)" title="Barycenter (astronomy)">Barycenter (astronomy)</a> for details). The <a href="/wiki/Center_of_mass_frame" class="mw-redirect" title="Center of mass frame">center of mass frame</a> is an <a href="/wiki/Inertial_frame" class="mw-redirect" title="Inertial frame">inertial frame</a> in which the center of mass of a system is at rest with respect to the origin of the <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept of center of gravity or <a href="/wiki/Weight" title="Weight">weight</a> was studied extensively by the ancient Greek <a href="/wiki/Mathematician" title="Mathematician">mathematician</a>, <a href="/wiki/Physicist" title="Physicist">physicist</a>, and <a href="/wiki/Engineer" title="Engineer">engineer</a> <a href="/wiki/Archimedes" title="Archimedes">Archimedes of Syracuse</a>. He worked with simplified assumptions about gravity that amount to a uniform field, thus arriving at the mathematical properties of what we now call the center of mass. Archimedes showed that the <a href="/wiki/Torque" title="Torque">torque</a> exerted on a <a href="/wiki/Lever" title="Lever">lever</a> by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. In his work <i><a href="/wiki/On_Floating_Bodies" title="On Floating Bodies">On Floating Bodies</a></i>, Archimedes demonstrated that the orientation of a floating object is the one that makes its center of mass as low as possible. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes.<sup id="cite_ref-FOOTNOTEShore20089–11_2-0" class="reference"><a href="#cite_note-FOOTNOTEShore20089–11-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Other ancient mathematicians who contributed to the theory of the center of mass include <a href="/wiki/Hero_of_Alexandria" title="Hero of Alexandria">Hero of Alexandria</a> and <a href="/wiki/Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus of Alexandria</a>. In the <a href="/wiki/Science_in_the_Renaissance" title="Science in the Renaissance">Renaissance</a> and <a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Early Modern</a> periods, work by <a href="/wiki/Guido_Ubaldi" class="mw-redirect" title="Guido Ubaldi">Guido Ubaldi</a>, <a href="/wiki/Francesco_Maurolico" title="Francesco Maurolico">Francesco Maurolico</a>,<sup id="cite_ref-FOOTNOTEBaron200491–94_3-0" class="reference"><a href="#cite_note-FOOTNOTEBaron200491–94-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Federico_Commandino" title="Federico Commandino">Federico Commandino</a>,<sup id="cite_ref-FOOTNOTEBaron200494–96_4-0" class="reference"><a href="#cite_note-FOOTNOTEBaron200494–96-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Evangelista_Torricelli" title="Evangelista Torricelli">Evangelista Torricelli</a>, <a href="/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevin</a>,<sup id="cite_ref-FOOTNOTEBaron200496–101_5-0" class="reference"><a href="#cite_note-FOOTNOTEBaron200496–101-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Luca_Valerio" title="Luca Valerio">Luca Valerio</a>,<sup id="cite_ref-FOOTNOTEBaron2004101–106_6-0" class="reference"><a href="#cite_note-FOOTNOTEBaron2004101–106-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Jean-Charles_de_la_Faille" class="mw-redirect" title="Jean-Charles de la Faille">Jean-Charles de la Faille</a>, <a href="/wiki/Paul_Guldin" title="Paul Guldin">Paul Guldin</a>,<sup id="cite_ref-FOOTNOTEMancosu199956–61_7-0" class="reference"><a href="#cite_note-FOOTNOTEMancosu199956–61-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/John_Wallis" title="John Wallis">John Wallis</a>, <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a>,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Louis_Carr%C3%A9_(mathematician)" title="Louis Carré (mathematician)">Louis Carré</a>, <a href="/wiki/Pierre_Varignon" title="Pierre Varignon">Pierre Varignon</a>, and <a href="/wiki/Alexis_Clairaut" title="Alexis Clairaut">Alexis Clairaut</a> expanded the concept further.<sup id="cite_ref-FOOTNOTEWalton18552_9-0" class="reference"><a href="#cite_note-FOOTNOTEWalton18552-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</a> is reformulated with respect to the center of mass in <a href="/wiki/Euler%27s_laws#Euler's_first_law" class="mw-redirect" title="Euler's laws">Euler's first law</a>.<sup id="cite_ref-FOOTNOTEBeatty200629_10-0" class="reference"><a href="#cite_note-FOOTNOTEBeatty200629-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definition"><span class="anchor" id="Definition_of_center_of_mass"></span>Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=2" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output 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.mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Center_of_mass" title="Special:EditPage/Center of mass">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">September 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The center of mass is the unique point at the center of a distribution of mass in space that has the property that the weighted position <a href="/wiki/Vector_space" title="Vector space">vectors</a> relative to this point sum to zero. In analogy to statistics, the center of mass is the mean location of a distribution of mass in space. </p> <div class="mw-heading mw-heading3"><h3 id="A_system_of_particles">A system of particles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=3" title="Edit section: A system of particles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the case of a system of particles <span class="texhtml"><i>P<sub>i</sub></i>, <i>i</i> = 1, ..., <i>n</i> </span>, each with mass <span class="texhtml mvar" style="font-style:italic;">m<sub>i</sub></span> that are located in space with coordinates <span class="texhtml"><b>r</b><sub><i>i</i></sub>, <i>i</i> = 1, ..., <i>n</i> </span>, the coordinates <b>R</b> of the center of mass satisfy <span class="mwe-math-element" data-qid="Q2945123"><a href="/w/index.php?title=Special:MathWikibase&qid=Q2945123" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )=\mathbf {0} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )=\mathbf {0} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e643bff44c31544a499ed1447133a4ab9798dcfc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.219ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )=\mathbf {0} .}"></a></span> </p><p>Solving this equation for <b>R</b> yields the formula <span class="mwe-math-element" data-qid="Q2945123"><a href="/w/index.php?title=Special:MathWikibase&qid=Q2945123" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ={\sum _{i=1}^{n}m_{i}\mathbf {r} _{i} \over \sum _{i=1}^{n}m_{i}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ={\sum _{i=1}^{n}m_{i}\mathbf {r} _{i} \over \sum _{i=1}^{n}m_{i}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0133216bd42a338b5ac48bd2bfd896098ac8c9e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.068ex; height:6.843ex;" alt="{\displaystyle \mathbf {R} ={\sum _{i=1}^{n}m_{i}\mathbf {r} _{i} \over \sum _{i=1}^{n}m_{i}}.}"></a></span> </p> <div class="mw-heading mw-heading3"><h3 id="A_continuous_volume">A continuous volume</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=4" title="Edit section: A continuous volume"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the mass distribution is continuous with the density ρ(<b>r</b>) within a solid <i>Q</i>, then the integral of the weighted position <a href="/wiki/Coordinate_system" title="Coordinate system">coordinates</a> of the points in this <a href="/wiki/Volume" title="Volume">volume</a> relative to the center of mass <b>R</b> over the volume <b>V</b> is zero, that is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=\mathbf {0} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=\mathbf {0} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d8473e96d6e8819016fd896aa5b1d1e7d60b8ce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.346ex; height:6.009ex;" alt="{\displaystyle \iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=\mathbf {0} .}"></span> </p><p>Solve this equation for the coordinates <b>R</b> to obtain <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ={\frac {1}{M}}\iiint _{Q}\rho (\mathbf {r} )\mathbf {r} \,dV,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>M</mi> </mfrac> </mrow> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mi>ρ</mi> <mo stretchy="false">(</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> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ={\frac {1}{M}}\iiint _{Q}\rho (\mathbf {r} )\mathbf {r} \,dV,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e5576dab04cdbe3dae732017d954ace701fe1f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.638ex; height:6.009ex;" alt="{\displaystyle \mathbf {R} ={\frac {1}{M}}\iiint _{Q}\rho (\mathbf {r} )\mathbf {r} \,dV,}"></span> Where <b>M</b> is the total mass in the volume. </p><p>If a continuous mass distribution has uniform <a href="/wiki/Density" title="Density">density</a>, which means that <i>ρ</i> is constant, then the center of mass is the same as the <a href="/wiki/Centroid" title="Centroid">centroid</a> of the volume.<sup id="cite_ref-FOOTNOTELevi200985_11-0" class="reference"><a href="#cite_note-FOOTNOTELevi200985-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Barycentric_coordinates">Barycentric coordinates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=5" title="Edit section: Barycentric coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Barycentric_coordinate_system" title="Barycentric coordinate system">Barycentric coordinate system</a></div> <p>The coordinates <b>R</b> of the center of mass of a two-particle system, <i>P</i><sub>1</sub> and <i>P</i><sub>2</sub>, with masses <i>m</i><sub>1</sub> and <i>m</i><sub>2</sub> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ={{m_{1}\mathbf {r} _{1}+m_{2}\mathbf {r} _{2}} \over m_{1}+m_{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ={{m_{1}\mathbf {r} _{1}+m_{2}\mathbf {r} _{2}} \over m_{1}+m_{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3a66536528f7a373021e07ccef325638e5a361" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.927ex; height:5.343ex;" alt="{\displaystyle \mathbf {R} ={{m_{1}\mathbf {r} _{1}+m_{2}\mathbf {r} _{2}} \over m_{1}+m_{2}}.}"></span> </p><p>Let the <a href="/wiki/Percentage" title="Percentage">percentage</a> of the total mass divided between these two <a href="/wiki/Particle" title="Particle">particles</a> vary from 100% <i>P</i><sub>1</sub> and 0% <i>P</i><sub>2</sub> through 50% <i>P</i><sub>1</sub> and 50% <i>P</i><sub>2</sub> to 0% <i>P</i><sub>1</sub> and 100% <i>P</i><sub>2</sub>, then the center of mass <b>R</b> moves along the line from <i>P</i><sub>1</sub> to <i>P</i><sub>2</sub>. The percentages of mass at each point can be viewed as projective coordinates of the point <b>R</b> on this line, and are termed <a href="/wiki/Barycentric_coordinate_system" title="Barycentric coordinate system">barycentric coordinates</a>. Another way of interpreting the process here is the mechanical balancing of moments about an arbitrary point. The numerator gives the total moment that is then balanced by an equivalent total force at the center of mass. This can be generalized to three points and four points to define projective coordinates in the plane, and in space, respectively. </p> <div class="mw-heading mw-heading3"><h3 id="Systems_with_periodic_boundary_conditions"><span class="anchor" id="Cluster_straddling"></span>Systems with periodic boundary conditions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=6" title="Edit section: Systems with periodic boundary conditions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For particles in a system with <a href="/wiki/Periodic_boundary_conditions" title="Periodic boundary conditions">periodic boundary conditions</a> two particles can be neighbours even though they are on opposite sides of the system. This occurs often in <a href="/wiki/Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a> simulations, for example, in which clusters form at random locations and sometimes neighbouring atoms cross the periodic boundary. When a cluster straddles the periodic boundary, a naive calculation of the center of mass will be incorrect. A generalized method for calculating the center of mass for periodic systems is to treat each coordinate, <i>x</i> and <i>y</i> and/or <i>z</i>, as if it were on a circle instead of a line.<sup id="cite_ref-FOOTNOTEBaiBreen2008_12-0" class="reference"><a href="#cite_note-FOOTNOTEBaiBreen2008-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The calculation takes every particle's <i>x</i> coordinate and maps it to an angle, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{i}={\frac {x_{i}}{x_{\max }}}2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> </mfrac> </mrow> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{i}={\frac {x_{i}}{x_{\max }}}2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7ba0a4174d53e7d79dae7045338bc09d96c77fa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.94ex; height:5.009ex;" alt="{\displaystyle \theta _{i}={\frac {x_{i}}{x_{\max }}}2\pi }"></span> where <i>x</i><sub>max</sub> is the system size in the <i>x</i> direction and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\in [0,x_{\max })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\in [0,x_{\max })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d37241e9cf1dec0b09f622063719f3c5c90c98a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.339ex; height:2.843ex;" alt="{\displaystyle x_{i}\in [0,x_{\max })}"></span>. From this angle, two new points <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 (\xi _{i},\zeta _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\xi _{i},\zeta _{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982113878681ce199017861e9a30934f82cfb4e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.48ex; height:2.843ex;" alt="{\displaystyle (\xi _{i},\zeta _{i})}"></span> can be generated, which can be weighted by the mass of the particle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span> for the center of mass or given a value of 1 for the geometric center: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\xi _{i}&=\cos(\theta _{i})\\\zeta _{i}&=\sin(\theta _{i})\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>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\xi _{i}&=\cos(\theta _{i})\\\zeta _{i}&=\sin(\theta _{i})\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3b804b338a9e07bda1fb4cac5ec507ed43396d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.479ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\xi _{i}&=\cos(\theta _{i})\\\zeta _{i}&=\sin(\theta _{i})\end{aligned}}}"></span> </p><p>In the <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 (\xi ,\zeta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>,</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\xi ,\zeta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/757f60171c94aa9d933bec6b1c87dc8a46480b3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.968ex; height:2.843ex;" alt="{\displaystyle (\xi ,\zeta )}"></span> plane, these coordinates lie on a circle of radius 1. From the collection of <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 \xi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf3bf8407299b66e55ad17ab166e08a93d3466c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.818ex; height:2.509ex;" alt="{\displaystyle \xi _{i}}"></span> and <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 \zeta _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aab3f49d01957ab74f9581cd34ba72cc7efc4bdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.818ex; height:2.509ex;" alt="{\displaystyle \zeta _{i}}"></span> values from all the particles, the averages <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 {\overline {\xi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ξ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\xi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9e912d39107dabc8aa6324c3da0c7ab163080d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.153ex; height:3.343ex;" alt="{\displaystyle {\overline {\xi }}}"></span> and <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 {\overline {\zeta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ζ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\zeta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe74402c34e2c1dcf75d1f8eb9293d5bed15bd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.26ex; height:3.343ex;" alt="{\displaystyle {\overline {\zeta }}}"></span> are calculated. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\overline {\xi }}&={\frac {1}{M}}\sum _{i=1}^{n}m_{i}\xi _{i},\\{\overline {\zeta }}&={\frac {1}{M}}\sum _{i=1}^{n}m_{i}\zeta _{i},\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> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ξ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>M</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ζ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>M</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {\xi }}&={\frac {1}{M}}\sum _{i=1}^{n}m_{i}\xi _{i},\\{\overline {\zeta }}&={\frac {1}{M}}\sum _{i=1}^{n}m_{i}\zeta _{i},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b67ddbf8113ccdc7e4a719fc3e3e050ef5c74273" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:17.822ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}{\overline {\xi }}&={\frac {1}{M}}\sum _{i=1}^{n}m_{i}\xi _{i},\\{\overline {\zeta }}&={\frac {1}{M}}\sum _{i=1}^{n}m_{i}\zeta _{i},\end{aligned}}}"></span> where <span class="texhtml mvar" style="font-style:italic;">M</span> is the sum of the masses of all of the particles. </p><p>These values are mapped back into a new angle, <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 {\overline {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58fd74f73194fc5ad23e379608ae568853f6538e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.205ex; height:3.009ex;" alt="{\displaystyle {\overline {\theta }}}"></span>, from which the <i>x</i> coordinate of the center of mass can be obtained: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\overline {\theta }}&=\operatorname {atan2} \left(-{\overline {\zeta }},-{\overline {\xi }}\right)+\pi \\x_{\text{com}}&=x_{\max }{\frac {\overline {\theta }}{2\pi }}\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> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>atan2</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ζ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ξ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>π</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>com</mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mover> <mi>θ</mi> <mo accent="false">¯</mo> </mover> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\overline {\theta }}&=\operatorname {atan2} \left(-{\overline {\zeta }},-{\overline {\xi }}\right)+\pi \\x_{\text{com}}&=x_{\max }{\frac {\overline {\theta }}{2\pi }}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/754bd2d58856f4c60929071d4cf70269d1d42087" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:28.028ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}{\overline {\theta }}&=\operatorname {atan2} \left(-{\overline {\zeta }},-{\overline {\xi }}\right)+\pi \\x_{\text{com}}&=x_{\max }{\frac {\overline {\theta }}{2\pi }}\end{aligned}}}"></span> </p><p>The process can be repeated for all dimensions of the system to determine the complete center of mass. The utility of the algorithm is that it allows the mathematics to determine where the "best" center of mass is, instead of guessing or using <a href="/wiki/Cluster_analysis" title="Cluster analysis">cluster analysis</a> to "unfold" a cluster straddling the periodic boundaries. If both average values are zero, <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 \left({\overline {\xi }},{\overline {\zeta }}\right)=(0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ξ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ζ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\overline {\xi }},{\overline {\zeta }}\right)=(0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4453af58b6b5ea2279edf061bf50550a1d362c3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.489ex; height:4.843ex;" alt="{\displaystyle \left({\overline {\xi }},{\overline {\zeta }}\right)=(0,0)}"></span>, then <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 {\overline {\theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>θ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58fd74f73194fc5ad23e379608ae568853f6538e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.205ex; height:3.009ex;" alt="{\displaystyle {\overline {\theta }}}"></span> is undefined. This is a correct result, because it only occurs when all particles are exactly evenly spaced. In that condition, their <i>x</i> coordinates are mathematically identical in a <a href="/wiki/Periodic_boundary_conditions#Practical_implementation:_continuity_and_the_minimum_image_convention" title="Periodic boundary conditions">periodic system</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Center_of_gravity">Center of gravity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=7" title="Edit section: Center of gravity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Center_of_mass" title="Special:EditPage/Center of mass">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">September 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Center of gravity" redirects here. For other uses, see <a href="/wiki/Center_of_gravity_(disambiguation)" class="mw-disambig" title="Center of gravity (disambiguation)">Center of gravity (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Centers_of_gravity_in_non-uniform_fields" title="Centers of gravity in non-uniform fields">Centers of gravity in non-uniform fields</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:CoG_stable.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/CoG_stable.svg/220px-CoG_stable.svg.png" decoding="async" width="220" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/CoG_stable.svg/330px-CoG_stable.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/CoG_stable.svg/440px-CoG_stable.svg.png 2x" data-file-width="293" data-file-height="298" /></a><figcaption>Diagram of an educational toy that balances on a point: the center of mass (C) settles below its support (P)</figcaption></figure> <p>A body's center of gravity is the point around which the <a href="/wiki/Resultant_force" title="Resultant force">resultant torque</a> due to gravity forces vanishes.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Where a gravity field can be considered to be uniform, the mass-center and the center-of-gravity will be the same. However, for satellites in orbit around a planet, in the absence of other torques being applied to a satellite, the slight variation (gradient) in gravitational field between closer-to and further-from the planet (stronger and weaker gravity respectively) can lead to a torque that will tend to align the satellite such that its long axis is vertical. In such a case, it is important to make the distinction between the center-of-gravity and the mass-center.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Any horizontal offset between the two will result in an applied torque. </p><p>The mass-center is a fixed property for a given rigid body (e.g. with no slosh or articulation), whereas the center-of-gravity may, in addition, depend upon its orientation in a non-uniform gravitational field. In the latter case, the center-of-gravity will always be located somewhat closer to the main attractive body as compared to the mass-center, and thus will change its position in the body of interest as its orientation is changed. </p><p>In the study of the dynamics of aircraft, vehicles and vessels, forces and moments need to be resolved relative to the mass center. That is true independent of whether gravity itself is a consideration. Referring to the mass-center as the center-of-gravity is something of a colloquialism, but it is in common usage and when gravity gradient effects are negligible, center-of-gravity and mass-center are the same and are used interchangeably. </p><p>In physics the benefits of using the center of mass to model a mass distribution can be seen by considering the <a href="/wiki/Resultant_force" title="Resultant force">resultant</a> of the gravity forces on a continuous body. Consider a body <i>Q</i> of volume <i>V</i> with density <i>ρ</i>(<b>r</b>) at each point <b>r</b> in the volume. In a parallel gravity field the force <b>f</b> at each point <b>r</b> is given by, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} (\mathbf {r} )=-dm\,g\mathbf {\hat {k}} =-\rho (\mathbf {r} )\,dV\,g\mathbf {\hat {k}} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>d</mi> <mi>m</mi> <mspace width="thinmathspace" /> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">k</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mspace width="thinmathspace" /> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">k</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} (\mathbf {r} )=-dm\,g\mathbf {\hat {k}} =-\rho (\mathbf {r} )\,dV\,g\mathbf {\hat {k}} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13adc6788b509235692d33a5a3103fe15f2aa2a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.013ex; height:3.343ex;" alt="{\displaystyle \mathbf {f} (\mathbf {r} )=-dm\,g\mathbf {\hat {k}} =-\rho (\mathbf {r} )\,dV\,g\mathbf {\hat {k}} ,}"></span> where <i>dm</i> is the mass at the point <b>r</b>, <i>g</i> is the acceleration of gravity, and <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="{\textstyle \mathbf {\hat {k}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">k</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {\hat {k}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0528c923b3a049b3bdf5a219e9698a98ffd7a85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.843ex;" alt="{\textstyle \mathbf {\hat {k}} }"></span> is a unit vector defining the vertical direction. </p><p>Choose a reference point <b>R</b> in the volume and compute the <a href="/wiki/Resultant_force" title="Resultant force">resultant force</a> and torque at this point, <span class="mwe-math-element" data-qid="Q11402"><a href="/w/index.php?title=Special:MathWikibase&qid=Q11402" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =\iiint _{Q}\mathbf {f} (\mathbf {r} )\,dV=\iiint _{Q}\rho (\mathbf {r} )\,dV\left(-g\mathbf {\hat {k}} \right)=-Mg\mathbf {\hat {k}} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>=</mo> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">k</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>M</mi> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">k</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\iiint _{Q}\mathbf {f} (\mathbf {r} )\,dV=\iiint _{Q}\rho (\mathbf {r} )\,dV\left(-g\mathbf {\hat {k}} \right)=-Mg\mathbf {\hat {k}} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35150e914ae0f183d3038f5ba62708747a761b97" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.994ex; height:6.009ex;" alt="{\displaystyle \mathbf {F} =\iiint _{Q}\mathbf {f} (\mathbf {r} )\,dV=\iiint _{Q}\rho (\mathbf {r} )\,dV\left(-g\mathbf {\hat {k}} \right)=-Mg\mathbf {\hat {k}} ,}"></a></span> and <span class="mwe-math-element" data-qid="Q48103"><a href="/w/index.php?title=Special:MathWikibase&qid=Q48103" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} =\iiint _{Q}(\mathbf {r} -\mathbf {R} )\times \mathbf {f} (\mathbf {r} )\,dV=\iiint _{Q}(\mathbf {r} -\mathbf {R} )\times \left(-g\rho (\mathbf {r} )\,dV\,\mathbf {\hat {k}} \right)=\left(\iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV\right)\times \left(-g\mathbf {\hat {k}} \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <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> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>=</mo> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <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> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>g</mi> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">k</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>V</mi> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">k</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} =\iiint _{Q}(\mathbf {r} -\mathbf {R} )\times \mathbf {f} (\mathbf {r} )\,dV=\iiint _{Q}(\mathbf {r} -\mathbf {R} )\times \left(-g\rho (\mathbf {r} )\,dV\,\mathbf {\hat {k}} \right)=\left(\iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV\right)\times \left(-g\mathbf {\hat {k}} \right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fbac1209e9c29d79d8bbe97b2a9589c27662005" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:100.833ex; height:6.343ex;" alt="{\displaystyle \mathbf {T} =\iiint _{Q}(\mathbf {r} -\mathbf {R} )\times \mathbf {f} (\mathbf {r} )\,dV=\iiint _{Q}(\mathbf {r} -\mathbf {R} )\times \left(-g\rho (\mathbf {r} )\,dV\,\mathbf {\hat {k}} \right)=\left(\iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV\right)\times \left(-g\mathbf {\hat {k}} \right).}"></a></span> </p><p>If the reference point <b>R</b> is chosen so that it is the center of mass, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>V</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd1caf898a5cd39dd4f95e88d79e8b19d26d8ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.171ex; height:6.009ex;" alt="{\displaystyle \iiint _{Q}\rho (\mathbf {r} )\left(\mathbf {r} -\mathbf {R} \right)dV=0,}"></span> which means the resultant torque <span class="nowrap"><b>T</b> = 0</span>. Because the resultant torque is zero the body will move as though it is a particle with its mass concentrated at the center of mass. </p><p>By selecting the center of gravity as the reference point for a rigid body, the gravity forces will not cause the body to rotate, which means the weight of the body can be considered to be concentrated at the center of mass. </p> <div class="mw-heading mw-heading2"><h2 id="Linear_and_angular_momentum">Linear and angular momentum</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=8" title="Edit section: Linear and angular momentum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The linear and angular momentum of a collection of particles can be simplified by measuring the position and velocity of the particles relative to the center of mass. Let the system of particles <i>P<sub>i</sub></i>, <i>i</i> = 1, ..., <i>n</i> of masses <i>m<sub>i</sub></i> be located at the coordinates <b>r</b><sub><i>i</i></sub> with velocities <b>v</b><sub><i>i</i></sub>. Select a reference point <b>R</b> and compute the relative position and velocity vectors, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{i}=(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {R} ,\quad \mathbf {v} _{i}={\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {v} .}"> <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>i</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{i}=(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {R} ,\quad \mathbf {v} _{i}={\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {v} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55e26f21e31402842a08dbae6c80c30240351f5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.409ex; height:5.509ex;" alt="{\displaystyle \mathbf {r} _{i}=(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {R} ,\quad \mathbf {v} _{i}={\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\mathbf {v} .}"></span> </p><p>The total linear momentum and angular momentum of the system are <span class="mwe-math-element" data-qid="Q41273"><a href="/w/index.php?title=Special:MathWikibase&qid=Q41273" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} ={\frac {d}{dt}}\left(\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\right)+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {v} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} ={\frac {d}{dt}}\left(\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\right)+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {v} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dec69a81a8de908785f4257015a6cb7ba6d11c3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.229ex; height:7.509ex;" alt="{\displaystyle \mathbf {p} ={\frac {d}{dt}}\left(\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\right)+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {v} ,}"></a></span> and <span class="mwe-math-element" data-qid="Q161254"><a href="/w/index.php?title=Special:MathWikibase&qid=Q161254" style="color:inherit;"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\left(\sum _{i=1}^{n}m_{i}\right)\left[\mathbf {R} \times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+(\mathbf {r} _{i}-\mathbf {R} )\times \mathbf {v} \right]+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {R} \times \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\left(\sum _{i=1}^{n}m_{i}\right)\left[\mathbf {R} \times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+(\mathbf {r} _{i}-\mathbf {R} )\times \mathbf {v} \right]+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {R} \times \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d6d3b84ccbaeb3ade8c86e43b70cd5b864c6ff2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:101.758ex; height:7.509ex;" alt="{\displaystyle \mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\left(\sum _{i=1}^{n}m_{i}\right)\left[\mathbf {R} \times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+(\mathbf {r} _{i}-\mathbf {R} )\times \mathbf {v} \right]+\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {R} \times \mathbf {v} }"></a></span> </p><p>If <b>R</b> is chosen as the center of mass these equations simplify to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} =m\mathbf {v} ,\quad \mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\sum _{i=1}^{n}m_{i}\mathbf {R} \times \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} =m\mathbf {v} ,\quad \mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\sum _{i=1}^{n}m_{i}\mathbf {R} \times \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05a0cacada8ec0b23c2ae41683c477316d6539ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:61.199ex; height:6.843ex;" alt="{\displaystyle \mathbf {p} =m\mathbf {v} ,\quad \mathbf {L} =\sum _{i=1}^{n}m_{i}(\mathbf {r} _{i}-\mathbf {R} )\times {\frac {d}{dt}}(\mathbf {r} _{i}-\mathbf {R} )+\sum _{i=1}^{n}m_{i}\mathbf {R} \times \mathbf {v} }"></span> where <i>m</i> is the total mass of all the particles, <b>p</b> is the linear momentum, and <b>L</b> is the angular momentum. </p><p>The <a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">law of conservation of momentum</a> predicts that for any system not subjected to external forces the momentum of the system will remain constant, which means the center of mass will move with constant velocity. This applies for all systems with classical internal forces, including magnetic fields, electric fields, chemical reactions, and so on. More formally, this is true for any internal forces that cancel in accordance with <a href="/wiki/Newton%27s_Third_Law" class="mw-redirect" title="Newton's Third Law">Newton's Third Law</a>.<sup id="cite_ref-FOOTNOTEKleppnerKolenkow1973117_15-0" class="reference"><a href="#cite_note-FOOTNOTEKleppnerKolenkow1973117-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Determination">Determination<span class="anchor" id="Locating"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=9" title="Edit section: Determination"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Centroid#Determination" title="Centroid">Centroid § Determination</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Center_gravity_2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Center_gravity_2.svg/220px-Center_gravity_2.svg.png" decoding="async" width="220" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Center_gravity_2.svg/330px-Center_gravity_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Center_gravity_2.svg/440px-Center_gravity_2.svg.png 2x" data-file-width="210" data-file-height="248" /></a><figcaption>Plumb line method</figcaption></figure> <p>The experimental determination of a body's center of mass makes use of gravity forces on the body and is based on the fact that the center of mass is the same as the center of gravity in the parallel gravity field near the earth's surface. </p><p>The center of mass of a body with an axis of symmetry and constant density must lie on this axis. Thus, the center of mass of a circular cylinder of constant density has its center of mass on the axis of the cylinder. In the same way, the center of mass of a spherically symmetric body of constant density is at the center of the sphere. In general, for any symmetry of a body, its center of mass will be a fixed point of that symmetry.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="In_two_dimensions">In two dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=10" title="Edit section: In two dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An experimental method for locating the center of mass is to suspend the object from two locations and to drop <a href="/wiki/Plumb_line" class="mw-redirect" title="Plumb line">plumb lines</a> from the suspension points. The intersection of the two lines is the center of mass.<sup id="cite_ref-FOOTNOTEKleppnerKolenkow1973119–120_17-0" class="reference"><a href="#cite_note-FOOTNOTEKleppnerKolenkow1973119–120-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The shape of an object might already be mathematically determined, but it may be too complex to use a known formula. In this case, one can subdivide the complex shape into simpler, more elementary shapes, whose centers of mass are easy to find. If the total mass and center of mass can be determined for each area, then the center of mass of the whole is the weighted average of the centers.<sup id="cite_ref-FOOTNOTEFeynmanLeightonSands196319.1–19.2_18-0" class="reference"><a href="#cite_note-FOOTNOTEFeynmanLeightonSands196319.1–19.2-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> This method can even work for objects with holes, which can be accounted for as negative masses.<sup id="cite_ref-FOOTNOTEHamill200920–21_19-0" class="reference"><a href="#cite_note-FOOTNOTEHamill200920–21-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>A direct development of the <a href="/wiki/Planimeter" title="Planimeter">planimeter</a> known as an integraph, or integerometer, can be used to establish the position of the <a href="/wiki/Centroid" title="Centroid">centroid</a> or center of mass of an irregular two-dimensional shape. This method can be applied to a shape with an irregular, smooth or complex boundary where other methods are too difficult. It was regularly used by ship builders to compare with the required <a href="/wiki/Displacement_(ship)" title="Displacement (ship)">displacement</a> and <a href="/wiki/Center_of_buoyancy" class="mw-redirect" title="Center of buoyancy">center of buoyancy</a> of a ship, and ensure it would not capsize.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTESangwin20067_21-0" class="reference"><a href="#cite_note-FOOTNOTESangwin20067-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="In_three_dimensions">In three dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=11" title="Edit section: In three dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An experimental method to locate the three-dimensional coordinates of the center of mass begins by supporting the object at three points and measuring the forces, <b>F</b><sub>1</sub>, <b>F</b><sub>2</sub>, and <b>F</b><sub>3</sub> that resist the weight of the object, <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 {W} =-W\mathbf {\hat {k}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">W</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">k</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {W} =-W\mathbf {\hat {k}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1941b92b90ae7ac91f4a373d3c3fb7c8933f73e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.516ex; height:3.009ex;" alt="{\displaystyle \mathbf {W} =-W\mathbf {\hat {k}} }"></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 {\hat {k}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">k</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {k}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.843ex;" alt="{\displaystyle \mathbf {\hat {k}} }"></span> is the unit vector in the vertical direction). Let <b>r</b><sub>1</sub>, <b>r</b><sub>2</sub>, and <b>r</b><sub>3</sub> be the position coordinates of the support points, then the coordinates <b>R</b> of the center of mass satisfy the condition that the resultant torque is zero, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} =(\mathbf {r} _{1}-\mathbf {R} )\times \mathbf {F} _{1}+(\mathbf {r} _{2}-\mathbf {R} )\times \mathbf {F} _{2}+(\mathbf {r} _{3}-\mathbf {R} )\times \mathbf {F} _{3}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} =(\mathbf {r} _{1}-\mathbf {R} )\times \mathbf {F} _{1}+(\mathbf {r} _{2}-\mathbf {R} )\times \mathbf {F} _{2}+(\mathbf {r} _{3}-\mathbf {R} )\times \mathbf {F} _{3}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b6ba2040e93d462f58cf3f6b0eb01a1ec449dc5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.706ex; height:2.843ex;" alt="{\displaystyle \mathbf {T} =(\mathbf {r} _{1}-\mathbf {R} )\times \mathbf {F} _{1}+(\mathbf {r} _{2}-\mathbf {R} )\times \mathbf {F} _{2}+(\mathbf {r} _{3}-\mathbf {R} )\times \mathbf {F} _{3}=0,}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} \times \left(-W\mathbf {\hat {k}} \right)=\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\mathbf {r} _{3}\times \mathbf {F} _{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">k</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} \times \left(-W\mathbf {\hat {k}} \right)=\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\mathbf {r} _{3}\times \mathbf {F} _{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a55163a6e51ed601a98ca3030499311c39b3bf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.901ex; height:4.843ex;" alt="{\displaystyle \mathbf {R} \times \left(-W\mathbf {\hat {k}} \right)=\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\mathbf {r} _{3}\times \mathbf {F} _{3}.}"></span> </p><p>This equation yields the coordinates of the center of mass <b>R</b>* in the horizontal plane as, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{*}=-{\frac {1}{W}}\mathbf {\hat {k}} \times (\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\mathbf {r} _{3}\times \mathbf {F} _{3}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>W</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">k</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{*}=-{\frac {1}{W}}\mathbf {\hat {k}} \times (\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\mathbf {r} _{3}\times \mathbf {F} _{3}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0e87d346eb0edf44a1888867fe2a94f135da290" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:46.825ex; height:5.343ex;" alt="{\displaystyle \mathbf {R} ^{*}=-{\frac {1}{W}}\mathbf {\hat {k}} \times (\mathbf {r} _{1}\times \mathbf {F} _{1}+\mathbf {r} _{2}\times \mathbf {F} _{2}+\mathbf {r} _{3}\times \mathbf {F} _{3}).}"></span> </p><p>The center of mass lies on the vertical line <b>L</b>, given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {L} (t)=\mathbf {R} ^{*}+t\mathbf {\hat {k}} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">L</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>+</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">k</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {L} (t)=\mathbf {R} ^{*}+t\mathbf {\hat {k}} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1904c1c501300d34dcb1a4e8947ac150871187a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.151ex; height:3.343ex;" alt="{\displaystyle \mathbf {L} (t)=\mathbf {R} ^{*}+t\mathbf {\hat {k}} .}"></span> </p><p>The three-dimensional coordinates of the center of mass are determined by performing this experiment twice with the object positioned so that these forces are measured for two different horizontal planes through the object. The center of mass will be the intersection of the two lines <b>L</b><sub>1</sub> and <b>L</b><sub>2</sub> obtained from the two experiments. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=12" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Engineering_designs">Engineering designs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=13" title="Edit section: Engineering designs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Automotive_applications">Automotive applications</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=14" title="Edit section: Automotive applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Center_of_mass" title="Special:EditPage/Center of mass">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">September 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Engineers try to design a <a href="/wiki/Sports_car" title="Sports car">sports car</a> so that its center of mass is lowered to make the car <a href="/wiki/Car_handling" class="mw-redirect" title="Car handling">handle</a> better, which is to say, maintain traction while executing relatively sharp turns. </p><p>The characteristic low profile of the U.S. military <a href="/wiki/Humvee" title="Humvee">Humvee</a> was designed in part to allow it to tilt farther than taller vehicles without <a href="/wiki/Vehicle_rollover" title="Vehicle rollover">rolling over</a>, by ensuring its low center of mass stays over the space bounded by the four wheels even at angles far from the <a href="/wiki/Vertical_and_horizontal" title="Vertical and horizontal">horizontal</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Aeronautics">Aeronautics</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=15" title="Edit section: Aeronautics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Center_of_gravity_of_an_aircraft" title="Center of gravity of an aircraft">Center of gravity of an aircraft</a></div> <p>The center of mass is an important point on an <a href="/wiki/Aircraft" title="Aircraft">aircraft</a>, which significantly affects the stability of the aircraft. To ensure the aircraft is stable enough to be safe to fly, the center of mass must fall within specified limits. If the center of mass is ahead of the <a href="/wiki/Center_of_gravity_of_an_aircraft" title="Center of gravity of an aircraft">forward limit</a>, the aircraft will be less maneuverable, possibly to the point of being unable to rotate for takeoff or flare for landing.<sup id="cite_ref-FOOTNOTEFederal_Aviation_Administration20071.4_22-0" class="reference"><a href="#cite_note-FOOTNOTEFederal_Aviation_Administration20071.4-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> If the center of mass is behind the aft limit, the aircraft will be more maneuverable, but also less stable, and possibly unstable enough so as to be impossible to fly. The moment arm of the <a href="/wiki/Elevator_(aircraft)" class="mw-redirect" title="Elevator (aircraft)">elevator</a> will also be reduced, which makes it more difficult to recover from a <a href="/wiki/Stall_(flight)" class="mw-redirect" title="Stall (flight)">stalled</a> condition.<sup id="cite_ref-FOOTNOTEFederal_Aviation_Administration20071.3_23-0" class="reference"><a href="#cite_note-FOOTNOTEFederal_Aviation_Administration20071.3-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>For <a href="/wiki/Helicopter" title="Helicopter">helicopters</a> in <a href="/wiki/Hover_(helicopter)" class="mw-redirect" title="Hover (helicopter)">hover</a>, the center of mass is always directly below the <a href="/wiki/Rotorhead" title="Rotorhead">rotorhead</a>. In forward flight, the center of mass will move forward to balance the negative pitch torque produced by applying <a href="/wiki/Helicopter_flight_controls#Cyclic" title="Helicopter flight controls">cyclic</a> control to propel the helicopter forward; consequently a cruising helicopter flies "nose-down" in level flight.<sup id="cite_ref-Helicopter_Centre_Of_Mass_24-0" class="reference"><a href="#cite_note-Helicopter_Centre_Of_Mass-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Astronomy"><span class="anchor" id="Barycenter_in_astronomy"></span><span class="anchor" id="Barycenter_in_astrophysics_and_astronomy"></span><span class="anchor" id="Sun-Jupiter_barycenter"></span> Astronomy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=16" title="Edit section: Astronomy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Barycenter" class="mw-redirect" title="Barycenter">Barycenter</a></div><figure typeof="mw:File/Thumb"><a href="/wiki/File:Orbit3.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Orbit3.gif/180px-Orbit3.gif" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/59/Orbit3.gif 1.5x" data-file-width="200" data-file-height="200" /></a><figcaption>Two bodies orbiting their <a href="/wiki/Barycenter" class="mw-redirect" title="Barycenter">barycenter</a> (red cross)</figcaption></figure> <p>The center of mass plays an important role in astronomy and astrophysics, where it is commonly referred to as the <i>barycenter</i>. The barycenter is the point between two objects where they balance each other; it is the center of mass where two or more celestial bodies <a href="/wiki/Orbit" title="Orbit">orbit</a> each other. When a <a href="/wiki/Natural_satellite" title="Natural satellite">moon</a> orbits a <a href="/wiki/Planet" title="Planet">planet</a>, or a planet orbits a <a href="/wiki/Star" title="Star">star</a>, both bodies are actually orbiting a point that lies away from the center of the primary (larger) body.<sup id="cite_ref-FOOTNOTEMurrayDermott199945–47_25-0" class="reference"><a href="#cite_note-FOOTNOTEMurrayDermott199945–47-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> For example, the Moon does not orbit the exact center of the <a href="/wiki/Earth" title="Earth">Earth</a>, but a point on a line between the center of the Earth and the Moon, approximately 1,710 km (1,062 miles) below the surface of the Earth, where their respective masses balance. This is the point about which the Earth and Moon orbit as they travel around the <a href="/wiki/Sun" title="Sun">Sun</a>. If the masses are more similar, e.g., <a href="/wiki/Charon_(moon)#Orbit" title="Charon (moon)">Pluto and Charon</a>, the barycenter will fall outside both bodies. </p> <div class="mw-heading mw-heading3"><h3 id="Rigging_and_safety">Rigging and safety</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=17" title="Edit section: Rigging and safety"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Knowing the location of the center of gravity when <a href="/wiki/Rigging_(material_handling)" title="Rigging (material handling)">rigging</a> is crucial, possibly resulting in severe injury or death if assumed incorrectly. A center of gravity that is at or above the lift point will most likely result in a tip-over incident. In general, the further the center of gravity below the pick point, the safer the lift. There are other things to consider, such as shifting loads, strength of the load and mass, distance between pick points, and number of pick points. Specifically, when selecting lift points, it is very important to place the center of gravity at the center and well below the lift points.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Body_motion">Body motion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=18" title="Edit section: Body motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Kinesiology" title="Kinesiology">Kinesiology</a></div><p><span class="anchor" id="Kinesiology"></span> </p><p>The center of mass of the adult human body is 10 cm above the <a href="/wiki/Trochanter" title="Trochanter">trochanter</a> (the femur joins the hip).<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> In kinesiology and biomechanics, the center of mass is an important parameter that assists people in understanding their human locomotion. Typically, a human's center of mass is detected with one of two methods: the reaction board method is a static analysis that involves the person lying down on that instrument, and use of their <a href="/wiki/Static_equilibrium" class="mw-redirect" title="Static equilibrium">static equilibrium</a> equation to find their center of mass; the segmentation method relies on a mathematical solution based on the <a href="/wiki/Physical_law" class="mw-redirect" title="Physical law">physical principle</a> that the <a href="/wiki/Summation" title="Summation">summation</a> of the <a href="/wiki/Torque" title="Torque">torques</a> of individual body sections, <a href="/wiki/Relative_motion" class="mw-redirect" title="Relative motion">relative to</a> a specified <a href="/wiki/Axis_of_rotation" class="mw-redirect" title="Axis of rotation">axis</a>, must equal the torque of the whole system that constitutes the body, measured relative to the same axis.<sup id="cite_ref-FOOTNOTEVint20031–11_28-0" class="reference"><a href="#cite_note-FOOTNOTEVint20031–11-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Optimization">Optimization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=19" title="Edit section: Optimization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Center-of-gravity_method" title="Center-of-gravity method">Center-of-gravity method</a> is a method for convex optimization, which uses the center-of-gravity of the feasible region. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span 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mechanics)">Center of pressure (fluid mechanics)</a></li> <li><a href="/wiki/Center_of_pressure_(terrestrial_locomotion)" title="Center of pressure (terrestrial locomotion)">Center of pressure (terrestrial locomotion)</a></li> <li><a href="/wiki/Centroid" title="Centroid">Centroid</a></li> <li><a href="/wiki/Circumcenter_of_mass" title="Circumcenter of mass">Circumcenter of mass</a></li> <li><a href="/wiki/Expected_value" title="Expected value">Expected value</a></li> <li><a href="/wiki/Mass_point_geometry" title="Mass point geometry">Mass point geometry</a></li> <li><a href="/wiki/Metacentric_height" title="Metacentric height">Metacentric height</a></li> <li><a href="/wiki/Roll_center" title="Roll center">Roll center</a></li> <li><a href="/wiki/Weight_distribution" title="Weight distribution">Weight distribution</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a 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no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://phys.libretexts.org/Bookshelves/University_Physics/Book:_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/10:_Linear_Momentum_and_the_Center_of_Mass/10.03:_The_center_of_mass">"10.3: The center of mass"</a>. <i>Physics LibreTexts</i>. 17 September 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-10-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Physics+LibreTexts&rft.atitle=10.3%3A+The+center+of+mass&rft.date=2019-09-17&rft_id=https%3A%2F%2Fphys.libretexts.org%2FBookshelves%2FUniversity_Physics%2FBook%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_%28Martin_Neary_Rinaldo_and_Woodman%29%2F10%3A_Linear_Momentum_and_the_Center_of_Mass%2F10.03%3A_The_center_of_mass&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEShore20089–11-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEShore20089–11_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFShore2008">Shore 2008</a>, pp. 9–11.</span> </li> <li id="cite_note-FOOTNOTEBaron200491–94-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBaron200491–94_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBaron2004">Baron 2004</a>, pp. 91–94.</span> </li> <li id="cite_note-FOOTNOTEBaron200494–96-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBaron200494–96_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBaron2004">Baron 2004</a>, pp. 94–96.</span> </li> <li id="cite_note-FOOTNOTEBaron200496–101-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBaron200496–101_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBaron2004">Baron 2004</a>, pp. 96–101.</span> </li> <li id="cite_note-FOOTNOTEBaron2004101–106-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBaron2004101–106_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBaron2004">Baron 2004</a>, pp. 101–106.</span> </li> <li id="cite_note-FOOTNOTEMancosu199956–61-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMancosu199956–61_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMancosu1999">Mancosu 1999</a>, pp. 56–61.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErlichson1996" class="citation journal cs1">Erlichson, H. 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(1962) <i>Physics</i>, 9-1 “Center of Mass”, Wiley</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Resnick, R. and Halliday, D. (1962) <i>Physics</i>, 14-3 “Center of Gravity”, Wiley</span> </li> <li id="cite_note-FOOTNOTEKleppnerKolenkow1973117-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKleppnerKolenkow1973117_15-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKleppnerKolenkow1973">Kleppner & Kolenkow 1973</a>, p. 117.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/I_19.html#Ch19-S1-p3">The Feynman Lectures on Physics Vol. I Ch. 19: Center of Mass; Moment of Inertia</a></span> </li> <li id="cite_note-FOOTNOTEKleppnerKolenkow1973119–120-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKleppnerKolenkow1973119–120_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKleppnerKolenkow1973">Kleppner & Kolenkow 1973</a>, pp. 119–120.</span> </li> <li id="cite_note-FOOTNOTEFeynmanLeightonSands196319.1–19.2-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFeynmanLeightonSands196319.1–19.2_18-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFeynmanLeightonSands1963">Feynman, Leighton & Sands 1963</a>, pp. 19.1–19.2.</span> </li> <li id="cite_note-FOOTNOTEHamill200920–21-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHamill200920–21_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHamill2009">Hamill 2009</a>, pp. 20–21.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.ebooksread.com/authors-eng/amos-lowrey-ayre/the-theory-and-design-of-british-shipbuilding-hci/page-3-the-theory-and-design-of-british-shipbuilding-hci.shtml">"The theory and design of British shipbuilding"</a>. <i>Amos Lowrey Ayre</i>. p. 3<span class="reference-accessdate">. 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Retrieved <span class="nowrap">2013-11-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Helicopter+Aerodynamics&rft.pages=82&rft_id=http%3A%2F%2Fwww.ultraligero.net%2FCursos%2Fhelicoptero%2FIntroduccion_a_la_aerodinamica_del%2520_helicoptero.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEMurrayDermott199945–47-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMurrayDermott199945–47_25-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMurrayDermott1999">Murray & Dermott 1999</a>, pp. 45–47.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.fema.gov/pdf/emergency/usr/module4.pdf">"Structural Collapse Technician: Module 4 - Lifting and Rigging"</a> <span class="cs1-format">(PDF)</span>. <i>FEMA.gov</i><span class="reference-accessdate">. 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"Calculating Center of Mass in an Unbounded 2D Environment". <i>Journal of Graphics, GPU, and Game Tools</i>. <b>13</b> (4): 53–60. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F2151237X.2008.10129266">10.1080/2151237X.2008.10129266</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:40807367">40807367</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Graphics%2C+GPU%2C+and+Game+Tools&rft.atitle=Calculating+Center+of+Mass+in+an+Unbounded+2D+Environment&rft.volume=13&rft.issue=4&rft.pages=53-60&rft.date=2008&rft_id=info%3Adoi%2F10.1080%2F2151237X.2008.10129266&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A40807367%23id-name%3DS2CID&rft.aulast=Bai&rft.aufirst=Linge&rft.au=Breen%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaron2004" class="citation cs2"><a href="/wiki/Margaret_Baron" title="Margaret Baron">Baron, Margaret E.</a> (2004) [1969], <i>The Origins of the Infinitesimal Calculus</i>, Courier Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-49544-6" title="Special:BookSources/978-0-486-49544-6"><bdi>978-0-486-49544-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Origins+of+the+Infinitesimal+Calculus&rft.pub=Courier+Dover+Publications&rft.date=2004&rft.isbn=978-0-486-49544-6&rft.aulast=Baron&rft.aufirst=Margaret+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeatty2006" class="citation cs2">Beatty, Millard F. 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(2001) [1964], <i>Statics</i>, Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-42005-9" title="Special:BookSources/978-0-486-42005-9"><bdi>978-0-486-42005-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statics&rft.pub=Dover&rft.date=2001&rft.isbn=978-0-486-42005-9&rft.aulast=Goodman&rft.aufirst=Lawrence+E.&rft.au=Warner%2C+William+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHamill2009" class="citation cs2">Hamill, Patrick (2009), <i>Intermediate Dynamics</i>, Jones & Bartlett Learning, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7637-5728-1" title="Special:BookSources/978-0-7637-5728-1"><bdi>978-0-7637-5728-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Intermediate+Dynamics&rft.pub=Jones+%26+Bartlett+Learning&rft.date=2009&rft.isbn=978-0-7637-5728-1&rft.aulast=Hamill&rft.aufirst=Patrick&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJongRogers1995" class="citation cs2">Jong, I. G.; Rogers, B. G. (1995), <i>Engineering Mechanics: Statics</i>, Saunders College Publishing, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-03-026309-5" title="Special:BookSources/978-0-03-026309-5"><bdi>978-0-03-026309-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics%3A+Statics&rft.pub=Saunders+College+Publishing&rft.date=1995&rft.isbn=978-0-03-026309-5&rft.aulast=Jong&rft.aufirst=I.+G.&rft.au=Rogers%2C+B.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleppnerKolenkow1973" class="citation cs2"><a href="/wiki/Daniel_Kleppner" title="Daniel Kleppner">Kleppner, Daniel</a>; <a href="/wiki/Robert_J._Kolenkow" title="Robert J. Kolenkow">Kolenkow, Robert</a> (1973), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontome00dani"><i>An Introduction to Mechanics</i></a></span> (2nd ed.), McGraw-Hill, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-035048-9" title="Special:BookSources/978-0-07-035048-9"><bdi>978-0-07-035048-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Mechanics&rft.edition=2nd&rft.pub=McGraw-Hill&rft.date=1973&rft.isbn=978-0-07-035048-9&rft.aulast=Kleppner&rft.aufirst=Daniel&rft.au=Kolenkow%2C+Robert&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontome00dani&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevi2009" class="citation cs2">Levi, Mark (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2Jp3FKRcZbEC&q=%22center+of+mass%22"><i>The Mathematical Mechanic: Using Physical Reasoning to Solve Problems</i></a>, Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-14020-9" title="Special:BookSources/978-0-691-14020-9"><bdi>978-0-691-14020-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical+Mechanic%3A+Using+Physical+Reasoning+to+Solve+Problems&rft.pub=Princeton+University+Press&rft.date=2009&rft.isbn=978-0-691-14020-9&rft.aulast=Levi&rft.aufirst=Mark&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2Jp3FKRcZbEC%26q%3D%2522center%2Bof%2Bmass%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMancosu1999" class="citation cs2">Mancosu, Paolo (1999), <i>Philosophy of mathematics and mathematical practice in the seventeenth century</i>, Oxford University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-513244-1" title="Special:BookSources/978-0-19-513244-1"><bdi>978-0-19-513244-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Philosophy+of+mathematics+and+mathematical+practice+in+the+seventeenth+century&rft.pub=Oxford+University+Press&rft.date=1999&rft.isbn=978-0-19-513244-1&rft.aulast=Mancosu&rft.aufirst=Paolo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMillikan1902" class="citation cs2"><a href="/wiki/Robert_Andrews_Millikan" title="Robert Andrews Millikan">Millikan, Robert Andrews</a> (1902), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=X0tBAAAAYAAJ"><i>Mechanics, molecular physics and heat: a twelve weeks' college course</i></a>, Chicago: Scott, Foresman and Company<span class="reference-accessdate">, retrieved <span class="nowrap">2011-05-25</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics%2C+molecular+physics+and+heat%3A+a+twelve+weeks%27+college+course&rft.place=Chicago&rft.pub=Scott%2C+Foresman+and+Company&rft.date=1902&rft.aulast=Millikan&rft.aufirst=Robert+Andrews&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DX0tBAAAAYAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurrayDermott1999" class="citation cs2">Murray, Carl; Dermott, Stanley (1999), <i>Solar System Dynamics</i>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-57295-8" title="Special:BookSources/978-0-521-57295-8"><bdi>978-0-521-57295-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Solar+System+Dynamics&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=978-0-521-57295-8&rft.aulast=Murray&rft.aufirst=Carl&rft.au=Dermott%2C+Stanley&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'Donnell2015" class="citation cs2">O'Donnell, Peter J. (2015), <i>Essential Dynamics and Relativity</i>, CRC Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-466-58839-4" title="Special:BookSources/978-1-466-58839-4"><bdi>978-1-466-58839-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essential+Dynamics+and+Relativity&rft.pub=CRC+Press&rft.date=2015&rft.isbn=978-1-466-58839-4&rft.aulast=O%27Donnell&rft.aufirst=Peter+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPollardFletcher2005" class="citation cs2">Pollard, David D.; Fletcher, Raymond C. (2005), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/trent_0116405531629"><i>Fundamentals of Structural Geology</i></a></span>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-83927-3" title="Special:BookSources/978-0-521-83927-3"><bdi>978-0-521-83927-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Structural+Geology&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=978-0-521-83927-3&rft.aulast=Pollard&rft.aufirst=David+D.&rft.au=Fletcher%2C+Raymond+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftrent_0116405531629&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPytelKiusalaas2010" class="citation cs2">Pytel, Andrew; Kiusalaas, Jaan (2010), <i>Engineering Mechanics: Statics</i>, vol. 1 (3rd ed.), Cengage Learning, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-29559-4" title="Special:BookSources/978-0-495-29559-4"><bdi>978-0-495-29559-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics%3A+Statics&rft.edition=3rd&rft.pub=Cengage+Learning&rft.date=2010&rft.isbn=978-0-495-29559-4&rft.aulast=Pytel&rft.aufirst=Andrew&rft.au=Kiusalaas%2C+Jaan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosenGothard2009" class="citation cs2">Rosen, Joe; Gothard, Lisa Quinn (2009), <i>Encyclopedia of Physical Science</i>, Infobase Publishing, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8160-7011-4" title="Special:BookSources/978-0-8160-7011-4"><bdi>978-0-8160-7011-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopedia+of+Physical+Science&rft.pub=Infobase+Publishing&rft.date=2009&rft.isbn=978-0-8160-7011-4&rft.aulast=Rosen&rft.aufirst=Joe&rft.au=Gothard%2C+Lisa+Quinn&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSangwin2006" class="citation cs2">Sangwin, Christopher J. (2006), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111005145208/http://web.mat.bham.ac.uk/C.J.Sangwin/Publications/integrometer.pdf">"Locating the centre of mass by mechanical means"</a> <span class="cs1-format">(PDF)</span>, <i>Journal of the Oughtred Society</i>, <b>15</b> (2), archived from <a rel="nofollow" class="external text" href="http://web.mat.bham.ac.uk/C.J.Sangwin/Publications/integrometer.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2011-10-05<span class="reference-accessdate">, retrieved <span class="nowrap">2011-10-23</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Oughtred+Society&rft.atitle=Locating+the+centre+of+mass+by+mechanical+means&rft.volume=15&rft.issue=2&rft.date=2006&rft.aulast=Sangwin&rft.aufirst=Christopher+J.&rft_id=http%3A%2F%2Fweb.mat.bham.ac.uk%2FC.J.Sangwin%2FPublications%2Fintegrometer.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerwayJewett2006" class="citation cs2">Serway, Raymond A.; Jewett, John W. (2006), <i>Principles of physics: a calculus-based text</i>, vol. 1 (4th ed.), Thomson Learning, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006ppcb.book.....J">2006ppcb.book.....J</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-49143-7" title="Special:BookSources/978-0-534-49143-7"><bdi>978-0-534-49143-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+physics%3A+a+calculus-based+text&rft.edition=4th&rft.pub=Thomson+Learning&rft.date=2006&rft_id=info%3Abibcode%2F2006ppcb.book.....J&rft.isbn=978-0-534-49143-7&rft.aulast=Serway&rft.aufirst=Raymond+A.&rft.au=Jewett%2C+John+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShirleyFairbridge1997" class="citation cs2">Shirley, James H.; Fairbridge, Rhodes Whitmore (1997), <i>Encyclopedia of planetary sciences</i>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-412-06951-2" title="Special:BookSources/978-0-412-06951-2"><bdi>978-0-412-06951-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopedia+of+planetary+sciences&rft.pub=Springer&rft.date=1997&rft.isbn=978-0-412-06951-2&rft.aulast=Shirley&rft.aufirst=James+H.&rft.au=Fairbridge%2C+Rhodes+Whitmore&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShore2008" class="citation cs2">Shore, Steven N. (2008), <i>Forces in Physics: A Historical Perspective</i>, Greenwood Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-313-33303-3" title="Special:BookSources/978-0-313-33303-3"><bdi>978-0-313-33303-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Forces+in+Physics%3A+A+Historical+Perspective&rft.pub=Greenwood+Press&rft.date=2008&rft.isbn=978-0-313-33303-3&rft.aulast=Shore&rft.aufirst=Steven+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSymon1971" class="citation cs2">Symon, Keith R. (1971), <i>Mechanics</i> (3rd ed.), Addison-Wesley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-07392-8" title="Special:BookSources/978-0-201-07392-8"><bdi>978-0-201-07392-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics&rft.edition=3rd&rft.pub=Addison-Wesley&rft.date=1971&rft.isbn=978-0-201-07392-8&rft.aulast=Symon&rft.aufirst=Keith+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTiplerMosca2004" class="citation cs2">Tipler, Paul A.; Mosca, Gene (2004), <i>Physics for Scientists and Engineers</i>, vol. 1A (5th ed.), W. H. Freeman and Company, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-0900-8" title="Special:BookSources/978-0-7167-0900-8"><bdi>978-0-7167-0900-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers&rft.edition=5th&rft.pub=W.+H.+Freeman+and+Company&rft.date=2004&rft.isbn=978-0-7167-0900-8&rft.aulast=Tipler&rft.aufirst=Paul+A.&rft.au=Mosca%2C+Gene&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Pelt2005" class="citation cs2">Van Pelt, Michael (2005), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/spacetourismadve0000vanp"><i>Space Tourism: Adventures in Earth Orbit and Beyond</i></a></span>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-40213-0" title="Special:BookSources/978-0-387-40213-0"><bdi>978-0-387-40213-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Space+Tourism%3A+Adventures+in+Earth+Orbit+and+Beyond&rft.pub=Springer&rft.date=2005&rft.isbn=978-0-387-40213-0&rft.aulast=Van+Pelt&rft.aufirst=Michael&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fspacetourismadve0000vanp&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVint2003" class="citation cs2">Vint, Peter (2003), <a rel="nofollow" class="external text" href="http://www.asu.edu/courses/kin335/documents/CM%20Lab.pdf">"LAB: Center of Mass (Center of Gravity) of the Human Body"</a> <span class="cs1-format">(PDF)</span>, <i>KIN 335 - Biomechanics</i><span class="reference-accessdate">, retrieved <span class="nowrap">2013-10-18</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=KIN+335+-+Biomechanics&rft.atitle=LAB%3A+Center+of+Mass+%28Center+of+Gravity%29+of+the+Human+Body&rft.date=2003&rft.aulast=Vint&rft.aufirst=Peter&rft_id=http%3A%2F%2Fwww.asu.edu%2Fcourses%2Fkin335%2Fdocuments%2FCM%2520Lab.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWalton1855" class="citation cs2">Walton, William (1855), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vY1NAAAAMAAJ"><i>A collection of problems in illustration of the principles of theoretical mechanics</i></a> (2nd ed.), Deighton, Bell & Co.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+collection+of+problems+in+illustration+of+the+principles+of+theoretical+mechanics&rft.edition=2nd&rft.pub=Deighton%2C+Bell+%26+Co.&rft.date=1855&rft.aulast=Walton&rft.aufirst=William&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvY1NAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACenter+of+mass" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Center_of_mass&action=edit&section=23" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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title="Torsion spring">Torsion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Car_suspension" title="Car suspension">Suspension</a> types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Dependent</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Beam_axle" title="Beam axle">Beam axle</a></li> <li><a href="/wiki/De_Dion_tube" class="mw-redirect" title="De Dion tube">De Dion tube</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Semi-independent</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a 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