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class="vector-toc-list"> </ul> </li> <li id="toc-Stone_on_a_string" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stone_on_a_string"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Stone on a string</span> </div> </a> <ul id="toc-Stone_on_a_string-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Earth" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Earth"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Earth</span> </div> </a> <ul id="toc-Earth-sublist" class="vector-toc-list"> <li id="toc-Weight_of_an_object_at_the_poles_and_on_the_equator" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Weight_of_an_object_at_the_poles_and_on_the_equator"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>Weight of an object at the poles and on the equator</span> </div> </a> <ul id="toc-Weight_of_an_object_at_the_poles_and_on_the_equator-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Derivation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Derivation</span> </div> </a> <button aria-controls="toc-Derivation-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 Derivation subsection</span> </button> <ul id="toc-Derivation-sublist" class="vector-toc-list"> <li id="toc-Time_derivatives_in_a_rotating_frame" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time_derivatives_in_a_rotating_frame"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Time derivatives in a rotating frame</span> </div> </a> <ul id="toc-Time_derivatives_in_a_rotating_frame-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Acceleration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Acceleration"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Acceleration</span> </div> </a> <ul id="toc-Acceleration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Force" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Force"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Force</span> </div> </a> <ul id="toc-Force-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Absolute_rotation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Absolute_rotation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Absolute rotation</span> </div> </a> <ul id="toc-Absolute_rotation-sublist" class="vector-toc-list"> </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> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_uses_of_the_term" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_uses_of_the_term"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Other uses of the term</span> </div> </a> <button aria-controls="toc-Other_uses_of_the_term-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 Other uses of the term subsection</span> </button> <ul id="toc-Other_uses_of_the_term-sublist" class="vector-toc-list"> <li id="toc-In_Lagrangian_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_Lagrangian_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>In Lagrangian mechanics</span> </div> </a> <ul id="toc-In_Lagrangian_mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_a_reactive_force" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#As_a_reactive_force"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>As a reactive force</span> </div> </a> <ul id="toc-As_a_reactive_force-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">8</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">9</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">10</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">11</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">Centrifugal force</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 71 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-71" 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">71 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/Middelpuntvliedende_krag" title="Middelpuntvliedende krag – Afrikaans" lang="af" hreflang="af" data-title="Middelpuntvliedende krag" 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%82%D9%88%D8%A9_%D8%B7%D8%B1%D8%AF_%D9%85%D8%B1%D9%83%D8%B2%D9%8A" title="قوة طرد مركزي – Arabic" lang="ar" hreflang="ar" data-title="قوة طرد مركزي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Fuercia_centr%C3%ADfuga" title="Fuercia centrífuga – Asturian" lang="ast" hreflang="ast" data-title="Fuercia centrífuga" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/M%C9%99rk%C9%99zd%C9%99nqa%C3%A7ma_q%C3%BCvv%C9%99si" title="Mərkəzdənqaçma qüvvəsi – Azerbaijani" lang="az" hreflang="az" data-title="Mərkəzdənqaçma qüvvəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%87%E0%A6%A8%E0%A7%8D%E0%A6%A6%E0%A7%8D%E0%A6%B0%E0%A6%AC%E0%A6%BF%E0%A6%AE%E0%A7%81%E0%A6%96%E0%A7%80_%E0%A6%AC%E0%A6%B2" title="কেন্দ্রবিমুখী বল – Bangla" lang="bn" hreflang="bn" data-title="কেন্দ্রবিমুখী বল" 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%B0%D0%B1%D0%B5%D0%B6%D0%BD%D0%B0%D1%8F_%D1%81%D1%96%D0%BB%D0%B0" title="Цэнтрабежная сіла – Belarusian" lang="be" hreflang="be" data-title="Цэнтрабежная сіла" 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%80%D0%BE%D0%B1%D0%B5%D0%B6%D0%BD%D0%B0_%D1%81%D0%B8%D0%BB%D0%B0" title="Центробежна сила – Bulgarian" lang="bg" hreflang="bg" data-title="Центробежна сила" 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/For%C3%A7a_centr%C3%ADfuga" title="Força centrífuga – Catalan" lang="ca" hreflang="ca" data-title="Força centrífuga" 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%92%D0%B0%D1%80%D1%80%D0%B8%D0%BD%D1%87%D0%B5%D0%BD%D0%BB%D0%B5_%D0%B2%C4%83%D0%B9" title="Варринченле вăй – Chuvash" lang="cv" hreflang="cv" data-title="Варринченле вăй" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Odst%C5%99ediv%C3%A1_s%C3%ADla" title="Odstředivá síla – Czech" lang="cs" hreflang="cs" data-title="Odstředivá síla" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Hwarapakati" title="Hwarapakati – Shona" lang="sn" hreflang="sn" data-title="Hwarapakati" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Centrifugalkraft" title="Centrifugalkraft – Danish" lang="da" hreflang="da" data-title="Centrifugalkraft" 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/Zentrifugalkraft" title="Zentrifugalkraft – German" lang="de" hreflang="de" data-title="Zentrifugalkraft" 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/Tsentrifugaalj%C3%B5ud" title="Tsentrifugaaljõud – Estonian" lang="et" hreflang="et" data-title="Tsentrifugaaljõud" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A6%CF%85%CE%B3%CF%8C%CE%BA%CE%B5%CE%BD%CF%84%CF%81%CE%BF%CF%82_%CE%B4%CF%8D%CE%BD%CE%B1%CE%BC%CE%B7" title="Φυγόκεντρος δύναμη – Greek" lang="el" hreflang="el" data-title="Φυγόκεντρος δύναμη" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Fuerza_centr%C3%ADfuga" title="Fuerza centrífuga – Spanish" lang="es" hreflang="es" data-title="Fuerza centrífuga" 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/Centrifuga_forto" title="Centrifuga forto – Esperanto" lang="eo" hreflang="eo" data-title="Centrifuga forto" 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/Indar_zentrifugo" title="Indar zentrifugo – Basque" lang="eu" hreflang="eu" data-title="Indar zentrifugo" 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%86%DB%8C%D8%B1%D9%88%DB%8C_%DA%AF%D8%B1%DB%8C%D8%B2_%D8%A7%D8%B2_%D9%85%D8%B1%DA%A9%D8%B2" title="نیروی گریز از مرکز – Persian" lang="fa" hreflang="fa" data-title="نیروی گریز از مرکز" 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/Force_centrifuge" title="Force centrifuge – French" lang="fr" hreflang="fr" data-title="Force centrifuge" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Forza_centr%C3%ADfuga" title="Forza centrífuga – Galician" lang="gl" hreflang="gl" data-title="Forza centrífuga" 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%9B%90%EC%8B%AC%EB%A0%A5" title="원심력 – Korean" lang="ko" hreflang="ko" data-title="원심력" 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%BF%D5%A5%D5%B6%D5%BF%D6%80%D5%B8%D5%B6%D5%A1%D5%AD%D5%B8%D6%82%D5%B5%D5%BD_%D5%B8%D6%82%D5%AA" title="Կենտրոնախույս ուժ – Armenian" lang="hy" hreflang="hy" data-title="Կենտրոնախույս ուժ" 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%85%E0%A4%AA%E0%A4%95%E0%A5%87%E0%A4%A8%E0%A5%8D%E0%A4%A6%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%AF_%E0%A4%AC%E0%A4%B2" title="अपकेन्द्रिय बल – Hindi" lang="hi" hreflang="hi" data-title="अपकेन्द्रिय बल" 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/Centrifugalna_sila" title="Centrifugalna sila – Croatian" lang="hr" hreflang="hr" data-title="Centrifugalna sila" 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/Gaya_sentrifugal" title="Gaya sentrifugal – Indonesian" lang="id" hreflang="id" data-title="Gaya sentrifugal" 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/Mi%C3%B0fl%C3%B3ttaafl" title="Miðflóttaafl – Icelandic" lang="is" hreflang="is" data-title="Miðflóttaafl" 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/Forza_centrifuga" title="Forza centrifuga – Italian" lang="it" hreflang="it" data-title="Forza centrifuga" 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%9B%D7%95%D7%97_%D7%A6%D7%A0%D7%98%D7%A8%D7%99%D7%A4%D7%95%D7%92%D7%9C%D7%99" title="כוח צנטריפוגלי – Hebrew" lang="he" hreflang="he" data-title="כוח צנטריפוגלי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AA%E1%83%94%E1%83%9C%E1%83%A2%E1%83%A0%E1%83%98%E1%83%93%E1%83%90%E1%83%9C%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%AB%E1%83%90%E1%83%9A%E1%83%90" title="ცენტრიდანული ძალა – Georgian" lang="ka" hreflang="ka" data-title="ცენტრიდანული ძალა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BD%D1%82%D1%80%D0%B4%D0%B5%D0%BD_%D1%82%D0%B5%D0%BF%D0%BA%D1%96%D1%88_%D0%BA%D2%AF%D1%88" title="Центрден тепкіш күш – Kazakh" lang="kk" hreflang="kk" data-title="Центрден тепкіш күш" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Kani_nje" title="Kani nje – Swahili" lang="sw" hreflang="sw" data-title="Kani nje" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/F%C3%B2s_santrifij" title="Fòs santrifij – Haitian Creole" lang="ht" hreflang="ht" data-title="Fòs santrifij" 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-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%9C%E0%BA%BB%E0%BA%99%E0%BA%9A%E0%BA%B1%E0%BA%87%E0%BA%84%E0%BA%B1%E0%BA%9A%E0%BB%83%E0%BA%8A%E0%BB%89_centrifugal" title="ຜົນບັງຄັບໃຊ້ centrifugal – Lao" lang="lo" hreflang="lo" data-title="ຜົນບັງຄັບໃຊ້ centrifugal" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Vis_centrifugalis" title="Vis centrifugalis – Latin" lang="la" hreflang="la" data-title="Vis centrifugalis" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Centrb%C4%93dzes_sp%C4%93ks" title="Centrbēdzes spēks – Latvian" lang="lv" hreflang="lv" data-title="Centrbēdzes spēks" 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/I%C5%A1centrin%C4%97_j%C4%97ga" title="Išcentrinė jėga – Lithuanian" lang="lt" hreflang="lt" data-title="Išcentrinė jėga" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Centrifug%C3%A1lis_er%C5%91" title="Centrifugális erő – Hungarian" lang="hu" hreflang="hu" data-title="Centrifugális erő" 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%A6%D0%B5%D0%BD%D1%82%D1%80%D0%B8%D1%84%D1%83%D0%B3%D0%B0%D0%BB%D0%BD%D0%B0_%D1%81%D0%B8%D0%BB%D0%B0" title="Центрифугална сила – Macedonian" lang="mk" hreflang="mk" data-title="Центрифугална сила" 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%85%E0%B4%AA%E0%B4%95%E0%B5%87%E0%B4%A8%E0%B5%8D%E0%B4%A6%E0%B5%8D%E0%B4%B0%E0%B4%AC%E0%B4%B2%E0%B4%82" title="അപകേന്ദ്രബലം – Malayalam" lang="ml" hreflang="ml" data-title="അപകേന്ദ്രബലം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Daya_emparan" title="Daya emparan – Malay" lang="ms" hreflang="ms" data-title="Daya emparan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%97%E1%80%9F%E1%80%AD%E1%80%AF%E1%80%81%E1%80%BD%E1%80%AC%E1%80%A1%E1%80%AC%E1%80%B8" title="ဗဟိုခွာအား – Burmese" lang="my" hreflang="my" data-title="ဗဟိုခွာအား" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Middelpuntvliedende_kracht" title="Middelpuntvliedende kracht – Dutch" lang="nl" hreflang="nl" data-title="Middelpuntvliedende kracht" 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%81%A0%E5%BF%83%E5%8A%9B" title="遠心力 – Japanese" lang="ja" hreflang="ja" data-title="遠心力" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Sentrifugalkraft" title="Sentrifugalkraft – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Sentrifugalkraft" 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/Sentrifugalkraft" title="Sentrifugalkraft – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Sentrifugalkraft" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/F%C3%B2r%C3%A7a_centrifuga" title="Fòrça centrifuga – Occitan" lang="oc" hreflang="oc" data-title="Fòrça centrifuga" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Si%C5%82a_od%C5%9Brodkowa" title="Siła odśrodkowa – Polish" lang="pl" hreflang="pl" data-title="Siła odśrodkowa" 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/For%C3%A7a_inercial_centr%C3%ADfuga" title="Força inercial centrífuga – Portuguese" lang="pt" hreflang="pt" data-title="Força inercial centrífuga" 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/For%C8%9Ba_centrifug%C4%83" title="Forța centrifugă – Romanian" lang="ro" hreflang="ro" data-title="Forța centrifugă" 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%BE%D0%B1%D0%B5%D0%B6%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D0%BB%D0%B0" title="Центробежная сила – Russian" lang="ru" hreflang="ru" data-title="Центробежная сила" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Forza_cintr%C3%ACfuga" title="Forza cintrìfuga – Sicilian" lang="scn" hreflang="scn" data-title="Forza cintrìfuga" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Centrifugal_force" title="Centrifugal force – Simple English" lang="en-simple" hreflang="en-simple" data-title="Centrifugal force" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Odstrediv%C3%A1_sila" title="Odstredivá sila – Slovak" lang="sk" hreflang="sk" data-title="Odstredivá sila" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Centrifugalna_sila" title="Centrifugalna sila – Slovenian" lang="sl" hreflang="sl" data-title="Centrifugalna sila" 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/%DA%BE%DB%8E%D8%B2%DB%8C_%D8%AF%DA%98%DB%95%D9%86%D8%A7%D9%88%DB%95%D9%86%D8%AF" title="ھێزی دژەناوەند – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ھێزی دژەناوەند" 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%D1%80%D0%B8%D1%84%D1%83%D0%B3%D0%B0%D0%BB%D0%BD%D0%B0_%D1%81%D0%B8%D0%BB%D0%B0" title="Центрифугална сила – Serbian" lang="sr" hreflang="sr" data-title="Центрифугална сила" 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/Centrifugalna_sila" title="Centrifugalna sila – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Centrifugalna sila" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Keskipakoisvoima" title="Keskipakoisvoima – Finnish" lang="fi" hreflang="fi" data-title="Keskipakoisvoima" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Centrifugalkraft" title="Centrifugalkraft – Swedish" lang="sv" hreflang="sv" data-title="Centrifugalkraft" 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%AE%E0%AF%88%E0%AE%AF%E0%AE%B5%E0%AE%BF%E0%AE%B2%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%81_%E0%AE%B5%E0%AE%BF%E0%AE%9A%E0%AF%88" title="மையவிலக்கு விசை – Tamil" lang="ta" hreflang="ta" data-title="மையவிலக்கு விசை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D2%AE%D0%B7%D3%99%D0%BA%D1%82%D3%99%D0%BD_%D0%BA%D1%83%D1%83_%D0%BA%D3%A9%D1%87%D0%B5" title="Үзәктән куу көче – Tatar" lang="tt" hreflang="tt" data-title="Үзәктән куу көче" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%85%E0%B0%AA%E0%B0%95%E0%B1%87%E0%B0%82%E0%B0%A6%E0%B1%8D%E0%B0%B0%E0%B0%AC%E0%B0%B2%E0%B0%82" title="అపకేంద్రబలం – Telugu" lang="te" hreflang="te" data-title="అపకేంద్రబలం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%A3%E0%B8%87%E0%B8%AB%E0%B8%99%E0%B8%B5%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" title="แรงหนีศูนย์กลาง – Thai" lang="th" hreflang="th" data-title="แรงหนีศูนย์กลาง" 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/Merkezka%C3%A7_kuvveti" title="Merkezkaç kuvveti – Turkish" lang="tr" hreflang="tr" data-title="Merkezkaç kuvveti" 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%92%D1%96%D0%B4%D1%86%D0%B5%D0%BD%D1%82%D1%80%D0%BE%D0%B2%D0%B0_%D1%81%D0%B8%D0%BB%D0%B0" title="Відцентрова сила – Ukrainian" lang="uk" hreflang="uk" data-title="Відцентрова сила" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%B1%DA%A9%D8%B2_%DA%AF%D8%B1%DB%8C%D8%B2_%D9%82%D9%88%D8%AA" title="مرکز گریز قوت – Urdu" lang="ur" hreflang="ur" data-title="مرکز گریز قوت" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%E1%BB%B1c_ly_t%C3%A2m" title="Lực ly tâm – Vietnamese" lang="vi" hreflang="vi" data-title="Lực ly 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/%E7%A6%BB%E5%BF%83%E5%8A%9B" title="离心力 – Wu" lang="wuu" hreflang="wuu" data-title="离心力" 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%9B%A2%E5%BF%83%E5%8A%9B" title="離心力 – Cantonese" lang="yue" hreflang="yue" data-title="離心力" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%9B%A2%E5%BF%83%E5%8A%9B" title="離心力 – Chinese" lang="zh" hreflang="zh" data-title="離心力" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q178733#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li 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.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">Not to be confused with <a href="/wiki/Centripetal_force" title="Centripetal force">Centripetal force</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Corioliskraftanimation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b6/Corioliskraftanimation.gif" decoding="async" width="200" height="283" class="mw-file-element" data-file-width="200" data-file-height="283" /></a><figcaption>In the inertial frame of reference (upper part of the picture), the black ball moves in a straight line. 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class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Acceleration" title="Acceleration">Acceleration</a></li> <li><a href="/wiki/Angular_momentum" title="Angular momentum">Angular momentum</a></li> <li><a href="/wiki/Couple_(mechanics)" title="Couple (mechanics)">Couple</a></li> <li><a href="/wiki/D%27Alembert%27s_principle" title="D'Alembert's principle">D'Alembert's principle</a></li> <li><a href="/wiki/Energy" title="Energy">Energy</a> <ul><li><a href="/wiki/Kinetic_energy#Newtonian_kinetic_energy" title="Kinetic energy">kinetic</a></li> <li><a href="/wiki/Potential_energy" title="Potential energy">potential</a></li></ul></li> <li><a href="/wiki/Force" title="Force">Force</a></li> <li><a href="/wiki/Frame_of_reference" title="Frame of reference">Frame of reference</a></li> <li><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial frame of reference</a></li> <li><a href="/wiki/Impulse_(physics)" title="Impulse (physics)">Impulse</a></li> <li><span class="nowrap"><a href="/wiki/Inertia" title="Inertia">Inertia</a> / <a href="/wiki/Moment_of_inertia" title="Moment of inertia">Moment of inertia</a></span></li> <li><a href="/wiki/Mass" title="Mass">Mass</a></li> <li><br /><a href="/wiki/Mechanical_power_(physics)" class="mw-redirect" title="Mechanical power (physics)">Mechanical power</a></li> <li><a href="/wiki/Work_(physics)" title="Work (physics)">Mechanical work</a></li> <li><br /><a href="/wiki/Moment_(physics)" title="Moment (physics)">Moment</a></li> <li><a href="/wiki/Momentum" title="Momentum">Momentum</a></li> <li><a href="/wiki/Space" title="Space">Space</a></li> <li><a href="/wiki/Speed" title="Speed">Speed</a></li> <li><a href="/wiki/Time" title="Time">Time</a></li> <li><a href="/wiki/Torque" title="Torque">Torque</a></li> <li><a href="/wiki/Velocity" title="Velocity">Velocity</a></li> <li><a href="/wiki/Virtual_work" title="Virtual work">Virtual work</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Formulations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"> <ul><li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></b></div></li> <li><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><b><a href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</a></b> <div class="plainlist"><ul><li><a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a></li><li><a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></li><li><a href="/wiki/Routhian_mechanics" title="Routhian mechanics">Routhian mechanics</a></li><li><a href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</a></li><li><a href="/wiki/Appell%27s_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</a></li><li><a href="/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</a></li></ul></div></div></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Core topics</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Damping" title="Damping">Damping</a></li> <li><a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">Displacement</a></li> <li><a href="/wiki/Equations_of_motion" title="Equations of motion">Equations of motion</a></li> <li><a href="/wiki/Euler%27s_laws_of_motion" title="Euler's laws of motion"><span class="wrap">Euler's laws of motion</span></a></li> <li><a href="/wiki/Fictitious_force" title="Fictitious force">Fictitious force</a></li> <li><a href="/wiki/Friction" title="Friction">Friction</a></li> <li><a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">Harmonic oscillator</a></li></ul> </div> <ul><li><span class="nowrap"><a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">Inertial</a> / <a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">Non-inertial reference frame</a></span></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Motion" title="Motion">Motion</a> (<a href="/wiki/Linear_motion" title="Linear motion">linear</a>)</li> <li><a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation"><span class="wrap">Newton's law of universal gravitation</span></a></li> <li><a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li> <li><a href="/wiki/Relative_velocity" title="Relative velocity">Relative velocity</a></li> <li><a href="/wiki/Rigid_body" title="Rigid body">Rigid body</a> <ul><li><a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">dynamics</a></li> <li><a href="/wiki/Euler%27s_equations_(rigid_body_dynamics)" title="Euler's equations (rigid body dynamics)">Euler's equations</a></li></ul></li> <li><a href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</a></li> <li><a href="/wiki/Vibration" title="Vibration">Vibration</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)"><a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">Rotation</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Circular_motion" title="Circular motion">Circular motion</a></li> <li><a href="/wiki/Rotating_reference_frame" title="Rotating reference frame">Rotating reference frame</a></li> <li><a href="/wiki/Centripetal_force" title="Centripetal force">Centripetal force</a></li> <li><a class="mw-selflink selflink">Centrifugal force</a> <ul><li><a href="/wiki/Reactive_centrifugal_force" title="Reactive centrifugal force">reactive</a></li></ul></li> <li><a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a></li> <li><a href="/wiki/Pendulum_(mechanics)" title="Pendulum (mechanics)">Pendulum</a></li> <li><a href="/wiki/Tangential_speed" title="Tangential speed">Tangential speed</a></li> <li><a href="/wiki/Rotational_frequency" title="Rotational frequency">Rotational frequency</a></li></ul> </div> <ul><li><a href="/wiki/Angular_acceleration" title="Angular acceleration">Angular acceleration</a> / <a href="/wiki/Angular_displacement" title="Angular displacement">displacement</a> / <a href="/wiki/Angular_frequency" title="Angular frequency">frequency</a> / <a href="/wiki/Angular_velocity" title="Angular velocity">velocity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="border-bottom: 1px solid black;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler</a></li> <li><a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Jeremiah_Horrocks" title="Jeremiah Horrocks">Horrocks</a></li> <li><a href="/wiki/Edmond_Halley" title="Edmond Halley">Halley</a></li> <li><a href="/wiki/Pierre_Louis_Maupertuis" title="Pierre Louis Maupertuis">Maupertuis</a></li> <li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a></li> <li><a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">d'Alembert</a></li> <li><a href="/wiki/Alexis_Clairaut" title="Alexis Clairaut">Clairaut</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a></li> <li><a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li> <li><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a></li> <li><a href="/wiki/Carl_Gustav_Jacob_Jacobi" title="Carl Gustav Jacob Jacobi">Jacobi</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li> <li><a href="/wiki/Edward_Routh" title="Edward Routh">Routh</a></li> <li><a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Liouville</a></li> <li><a href="/wiki/Paul_%C3%89mile_Appell" title="Paul Émile Appell">Appell</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Bernard_Koopman" title="Bernard Koopman">Koopman</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-below hlist" style="background-color: transparent; border-color: #A2B8BF"> <ul><li><span class="nowrap"><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/14px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="14" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/21px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/28px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span> </span><a href="/wiki/Portal:Physics" title="Portal:Physics">Physics portal</a></span></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Classical_mechanics" title="Category:Classical mechanics">Category</a></span></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Classical_mechanics" title="Template:Classical mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Classical_mechanics" title="Template talk:Classical mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Classical_mechanics" title="Special:EditPage/Template:Classical mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Centrifugal force</b> is a <a href="/wiki/Fictitious_force" title="Fictitious force">fictitious force</a> in <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a> (also called an "inertial" or "pseudo" force) that appears to act on all objects when viewed in a <a href="/wiki/Rotating_frame_of_reference" class="mw-redirect" title="Rotating frame of reference">rotating frame of reference</a>. It appears to be directed radially away from the <a href="/wiki/Rotation_around_a_fixed_axis" title="Rotation around a fixed axis">axis of rotation</a> of the frame. The magnitude of the centrifugal force <i>F</i> on an object of <a href="/wiki/Mass" title="Mass">mass</a> <i>m</i> at the distance <i>r</i> from the axis of a rotating frame of reference with <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> <span class="texhtml mvar" style="font-style:italic;">ω</span> 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 F=m\omega ^{2}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=m\omega ^{2}r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b65f1cf00f8d9f05bc9381a42fba4d443ba659b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.428ex; height:2.676ex;" alt="{\displaystyle F=m\omega ^{2}r}"></span> </p><p>This fictitious force is often applied to rotating devices, such as <a href="/wiki/Centrifuge" title="Centrifuge">centrifuges</a>, <a href="/wiki/Centrifugal_pump" title="Centrifugal pump">centrifugal pumps</a>, <a href="/wiki/Centrifugal_governor" title="Centrifugal governor">centrifugal governors</a>, and <a href="/wiki/Centrifugal_clutch" title="Centrifugal clutch">centrifugal clutches</a>, and in <a href="/wiki/Centrifugal_railway" title="Centrifugal railway">centrifugal railways</a>, <a href="/wiki/Planetary_orbit" class="mw-redirect" title="Planetary orbit">planetary orbits</a> and <a href="/wiki/Banked_curve" class="mw-redirect" title="Banked curve">banked curves</a>, when they are analyzed in a <a href="/wiki/Inertial_reference_frame" class="mw-redirect" title="Inertial reference frame">non–inertial reference frame</a> such as a rotating coordinate system. </p><p>The term has sometimes also been used for the <i><a href="/wiki/Reactive_centrifugal_force" title="Reactive centrifugal force">reactive centrifugal force</a></i>, a real frame-independent Newtonian force that exists as a reaction to a <a href="/wiki/Centripetal_force" title="Centripetal force">centripetal force</a> in some scenarios. </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=Centrifugal_force&action=edit&section=1" title="Edit section: History"><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/History_of_centrifugal_and_centripetal_forces" title="History of centrifugal and centripetal forces">History of centrifugal and centripetal forces</a></div> <p>From 1659, the <a href="/wiki/Neo-Latin" title="Neo-Latin">Neo-Latin</a> term <i>vi centrifuga</i> ("centrifugal force") is attested in <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a>' notes and letters.<sup id="cite_ref-yoeder_1-0" class="reference"><a href="#cite_note-yoeder-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Yoder2013_2-0" class="reference"><a href="#cite_note-Yoder2013-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Note, that in Latin <i lang="la"><a href="https://en.wiktionary.org/wiki/centrum#Latin" class="extiw" title="wikt:centrum">centrum</a></i> means "center" and <i lang="la"><a href="https://en.wiktionary.org/wiki/%E2%80%91fugus#Latin" class="extiw" title="wikt:‑fugus">‑fugus</a></i> (from <i lang="la"><a href="https://en.wiktionary.org/wiki/fugio#Latin" class="extiw" title="wikt:fugio">fugiō</a></i>) means "fleeing, avoiding". Thus, <i>centrifugus</i> means "fleeing from the center" in a <a href="/wiki/Literal_translation" title="Literal translation">literal translation</a>. </p><p>In 1673, in <i><a href="/wiki/Horologium_Oscillatorium" title="Horologium Oscillatorium">Horologium Oscillatorium</a></i>, Huygens writes (as translated by <a href="/wiki/Richard_J._Blackwell" title="Richard J. Blackwell">Richard J. Blackwell</a>):<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <blockquote> <p>There is another kind of oscillation in addition to the one we have examined up to this point; namely, a motion in which a suspended weight is moved around through the circumference of a circle. From this we were led to the construction of another clock at about the same time we invented the first one. [...] I originally intended to publish here a lengthy description of these clocks, along with matters pertaining to circular motion and <b>centrifugal force</b><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup>, as it might be called, a subject about which I have more to say than I am able to do at present. But, in order that those interested in these things can sooner enjoy these new and not useless speculations, and in order that their publication not be prevented by some accident, I have decided, contrary to my plan, to add this fifth part [...]. </p> </blockquote> <p>The same year, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> received Huygens work via <a href="/wiki/Henry_Oldenburg" title="Henry Oldenburg">Henry Oldenburg</a> and replied "I pray you return [Mr. Huygens] my humble thanks [...] I am glad we can expect another discourse of the <i>vis centrifuga</i>, which speculation may prove of good use in <a href="/wiki/Natural_philosophy" title="Natural philosophy">natural philosophy</a> and <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, as well as <a href="/wiki/Mechanics" title="Mechanics">mechanics</a>".<sup id="cite_ref-yoeder_1-1" class="reference"><a href="#cite_note-yoeder-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>In 1687, in <i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a></i>, Newton further develops <i>vis centrifuga</i> ("centrifugal force"). Around this time, the concept is also further evolved by Newton, <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>, and <a href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</a>. </p><p>In the late 18th century, the modern conception of the centrifugal force evolved as a "<a href="/wiki/Fictitious_force" title="Fictitious force">fictitious force</a>" arising in a rotating reference.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (December 2012)">citation needed</span></a></i>]</sup> </p><p>Centrifugal force has also played a role in debates in <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> about detection of absolute motion. Newton suggested two arguments to answer the question of whether <a href="/wiki/Absolute_rotation" title="Absolute rotation">absolute rotation</a> can be detected: the rotating <a href="/wiki/Bucket_argument" title="Bucket argument">bucket argument</a>, and the <a href="/wiki/Rotating_spheres" title="Rotating spheres">rotating spheres</a> argument.<sup id="cite_ref-Newton_6-0" class="reference"><a href="#cite_note-Newton-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> According to Newton, in each scenario the centrifugal force would be observed in the object's local frame (the frame where the object is stationary) only if the frame were rotating with respect to absolute space. </p><p>Around 1883, <a href="/wiki/Mach%27s_principle" title="Mach's principle">Mach's principle</a> was proposed where, instead of absolute rotation, the motion of the distant stars relative to the local inertial frame gives rise through some (hypothetical) physical law to the centrifugal force and other inertia effects. Today's view is based upon the idea of an inertial frame of reference, which privileges observers for which the laws of physics take on their simplest form, and in particular, frames that do not use centrifugal forces in their equations of motion in order to describe motions correctly. </p><p>Around 1914, the analogy between centrifugal force (sometimes used to create <a href="/wiki/Artificial_gravity" title="Artificial gravity">artificial gravity</a>) and gravitational forces led to the <a href="/wiki/Equivalence_principle" title="Equivalence principle">equivalence principle</a> of <a href="/wiki/General_relativity" title="General relativity">general relativity</a>.<sup id="cite_ref-Barbour_7-0" class="reference"><a href="#cite_note-Barbour-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Eriksson_8-0" class="reference"><a href="#cite_note-Eriksson-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=2" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Centrifugal force is an outward force apparent in a <a href="/wiki/Rotating_reference_frame" title="Rotating reference frame">rotating reference frame</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Taylor1_11-0" class="reference"><a href="#cite_note-Taylor1-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> It does not exist when a system is described relative to an <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">inertial frame of reference</a>. </p><p>All measurements of position and velocity must be made relative to some frame of reference. For example, an analysis of the motion of an object in an airliner in flight could be made relative to the airliner, to the surface of the Earth, or even to the Sun.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> A reference frame that is at rest (or one that moves with no rotation and at constant velocity) relative to the "<a href="/wiki/Fixed_stars" title="Fixed stars">fixed stars</a>" is generally taken to be an inertial frame. Any system can be analyzed in an inertial frame (and so with no centrifugal force). However, it is often more convenient to describe a rotating system by using a rotating frame—the calculations are simpler, and descriptions more intuitive. When this choice is made, fictitious forces, including the centrifugal force, arise. </p><p>In a reference frame rotating about an axis through its origin, all objects, regardless of their state of motion, appear to be under the influence of a radially (from the axis of rotation) outward force that is proportional to their mass, to the distance from the axis of rotation of the frame, and to the square of the <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> of the frame.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> This is the centrifugal force. As humans usually experience centrifugal force from within the rotating reference frame, e.g. on a merry-go-round or vehicle, this is much more well-known than centripetal force. </p><p>Motion relative to a rotating frame results in another fictitious force: the <a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a>. If the rate of rotation of the frame changes, a third fictitious force (the <a href="/wiki/Euler_force" title="Euler force">Euler force</a>) is required. These fictitious forces are necessary for the formulation of correct equations of motion in a rotating reference frame<sup id="cite_ref-Fetter_16-0" class="reference"><a href="#cite_note-Fetter-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Marsden_17-0" class="reference"><a href="#cite_note-Marsden-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> and allow Newton's laws to be used in their normal form in such a frame (with one exception: the fictitious forces do not obey Newton's third law: they have no equal and opposite counterparts).<sup id="cite_ref-Fetter_16-1" class="reference"><a href="#cite_note-Fetter-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Newton's third law requires the counterparts to exist within the same frame of reference, hence centrifugal and centripetal force, which do not, are not action and reaction (as is sometimes erroneously contended). </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Vehicle_driving_round_a_curve">Vehicle driving round a curve</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=4" title="Edit section: Vehicle driving round a curve"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A common experience that gives rise to the idea of a centrifugal force is encountered by passengers riding in a vehicle, such as a car, that is changing direction. If a car is traveling at a constant speed along a straight road, then a passenger inside is not accelerating and, according to <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's second law of motion</a>, the net force acting on them is therefore zero (all forces acting on them cancel each other out). If the car enters a curve that bends to the left, the passenger experiences an apparent force that seems to be pulling them towards the right. This is the fictitious centrifugal force. It is needed within the passengers' local frame of reference to explain their sudden tendency to start accelerating to the right relative to the car—a tendency which they must resist by applying a rightward force to the car (for instance, a frictional force against the seat) in order to remain in a fixed position inside. Since they push the seat toward the right, Newton's third law says that the seat pushes them towards the left. The centrifugal force must be included in the passenger's reference frame (in which the passenger remains at rest): it counteracts the leftward force applied to the passenger by the seat, and explains why this otherwise unbalanced force does not cause them to accelerate.<sup id="cite_ref-EB_18-0" class="reference"><a href="#cite_note-EB-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> However, it would be apparent to a stationary observer watching from an overpass above that the frictional force exerted on the passenger by the seat is not being balanced; it constitutes a net force to the left, causing the passenger to accelerate toward the inside of the curve, as they must in order to keep moving with the car rather than proceeding in a straight line as they otherwise would. Thus the "centrifugal force" they feel is the result of a "centrifugal tendency" caused by inertia.<sup id="cite_ref-Science_of_Everyday_Things_19-0" class="reference"><a href="#cite_note-Science_of_Everyday_Things-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Similar effects are encountered in aeroplanes and <a href="/wiki/Roller_coaster" title="Roller coaster">roller coasters</a> where the magnitude of the apparent force is often reported in "<a href="/wiki/G-force" title="G-force">G's</a>". </p> <div class="mw-heading mw-heading3"><h3 id="Stone_on_a_string">Stone on a string</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=5" title="Edit section: Stone on a string"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If a stone is whirled round on a string, in a horizontal plane, the only real force acting on the stone in the horizontal plane is applied by the string (gravity acts vertically). There is a net force on the stone in the horizontal plane which acts toward the center. </p><p>In an <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">inertial frame of reference</a>, were it not for this net force acting on the stone, the stone would travel in a straight line, according to <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's first law of motion</a>. In order to keep the stone moving in a circular path, a <a href="/wiki/Centripetal_force" title="Centripetal force">centripetal force</a>, in this case provided by the string, must be continuously applied to the stone. As soon as it is removed (for example if the string breaks) the stone moves in a straight line, as viewed from above. In this inertial frame, the concept of centrifugal force is not required as all motion can be properly described using only real forces and Newton's laws of motion. </p><p>In a frame of reference rotating with the stone around the same axis as the stone, the stone is stationary. However, the force applied by the string is still acting on the stone. If one were to apply Newton's laws in their usual (inertial frame) form, one would conclude that the stone should accelerate in the direction of the net applied force—towards the axis of rotation—which it does not do. The centrifugal force and other fictitious forces must be included along with the real forces in order to apply Newton's laws of motion in the rotating frame. </p> <div class="mw-heading mw-heading3"><h3 id="Earth">Earth</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=6" title="Edit section: Earth"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Earth" title="Earth">Earth</a> constitutes a rotating reference frame because it rotates <a href="/wiki/Earth%27s_rotation" title="Earth's rotation">once every 23 hours and 56 minutes</a> around its axis. Because the rotation is slow, the fictitious forces it produces are often small, and in everyday situations can generally be neglected. Even in calculations requiring high precision, the centrifugal force is generally not explicitly included, but rather lumped in with the <a href="/wiki/Gravitational_force" class="mw-redirect" title="Gravitational force">gravitational force</a>: the strength and direction of the local "<a href="/wiki/Gravity_of_Earth" title="Gravity of Earth">gravity</a>" at any point on the Earth's surface is actually a combination of gravitational and centrifugal forces. However, the fictitious forces can be of arbitrary size. For example, in an Earth-bound reference system (where the earth is represented as stationary), the fictitious force (the net of Coriolis and centrifugal forces) is enormous and is responsible for the <a href="/wiki/Sun" title="Sun">Sun</a> orbiting around the Earth. This is due to the large mass and velocity of the Sun (relative to the Earth). </p> <div class="mw-heading mw-heading4"><h4 id="Weight_of_an_object_at_the_poles_and_on_the_equator">Weight of an object at the poles and on the equator</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=7" title="Edit section: Weight of an object at the poles and on the equator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If an object is weighed with a simple <a href="/wiki/Spring_balance" class="mw-redirect" title="Spring balance">spring balance</a> at one of the Earth's poles, there are two forces acting on the object: the Earth's gravity, which acts in a downward direction, and the equal and opposite <a href="/wiki/Restoring_force" title="Restoring force">restoring force</a> in the spring, acting upward. Since the object is stationary and not accelerating, there is no net force acting on the object and the force from the spring is equal in magnitude to the force of gravity on the object. In this case, the balance shows the value of the force of gravity on the object. </p><p>When the same object is weighed on the <a href="/wiki/Equator" title="Equator">equator</a>, the same two real forces act upon the object. However, the object is moving in a circular path as the Earth rotates and therefore experiencing a centripetal acceleration. When considered in an inertial frame (that is to say, one that is not rotating with the Earth), the non-zero acceleration means that force of gravity will not balance with the force from the spring. In order to have a net centripetal force, the magnitude of the restoring force of the spring must be less than the magnitude of force of gravity. This reduced restoring force in the spring is reflected on the scale as less weight — about 0.3% less at the equator than at the poles.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In the Earth reference frame (in which the object being weighed is at rest), the object does not appear to be accelerating; however, the two real forces, gravity and the force from the spring, are the same magnitude and do not balance. The centrifugal force must be included to make the sum of the forces be zero to match the apparent lack of acceleration. </p><p><small> <b>Note:</b> <i>In fact, the observed weight difference is more — about 0.53%. Earth's gravity is a bit stronger at the poles than at the equator, because the Earth is <a href="/wiki/Oblate_spheroid" class="mw-redirect" title="Oblate spheroid">not a perfect sphere</a>, so an object at the poles is slightly closer to the center of the Earth than one at the equator; this effect combines with the centrifugal force to produce the observed weight difference.</i><sup id="cite_ref-Boynton_21-0" class="reference"><a href="#cite_note-Boynton-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </small> </p> <div class="mw-heading mw-heading2"><h2 id="Derivation">Derivation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=8" title="Edit section: Derivation"><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/Rotating_reference_frame" title="Rotating reference frame">Rotating reference frame</a></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/Fictitious_force" title="Fictitious force">Fictitious force</a> and <a href="/w/index.php?title=Mechanics_of_planar_particle_motion&action=edit&redlink=1" class="new" title="Mechanics of planar particle motion (page does not exist)">Mechanics of planar particle motion</a></div> <p>For the following formalism, the <a href="/wiki/Rotating_frame_of_reference" class="mw-redirect" title="Rotating frame of reference">rotating frame of reference</a> is regarded as a special case of a <a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">non-inertial reference frame</a> that is rotating relative to an inertial reference frame denoted the stationary frame. </p> <div class="mw-heading mw-heading3"><h3 id="Time_derivatives_in_a_rotating_frame">Time derivatives in a rotating frame</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=9" title="Edit section: Time derivatives in a rotating frame"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a rotating frame of reference, the time derivatives of any vector function <span class="texhtml"><i><b>P</b></i></span> of time—such as the velocity and acceleration vectors of an object—will differ from its time derivatives in the stationary frame. If <span class="texhtml"><i>P</i><sub>1</sub> <i>P</i><sub>2</sub>, <i>P</i><sub>3</sub></span> are the components of <span class="texhtml"><i><b>P</b></i></span> with respect to unit vectors <span class="texhtml"><i><b>i</b></i>, <i><b>j</b></i>, <i><b>k</b></i></span> directed along the axes of the rotating frame (i.e. <span class="texhtml"><i><b>P</b></i> = <i>P</i><sub>1</sub> <i><b>i</b></i> + <i>P</i><sub>2</sub> <i><b>j</b></i> +<i>P</i><sub>3</sub> <i><b>k</b></i></span>), then the first time derivative <span class="texhtml">[d<i><b>P</b></i>/d<i>t</i>]</span> of <span class="texhtml"><i><b>P</b></i></span> with respect to the rotating frame is, by definition, <span class="texhtml">d<i>P</i><sub>1</sub>/d<i>t</i> <i><b>i</b></i> + d<i>P</i><sub>2</sub>/d<i>t</i> <i><b>j</b></i> + d<i>P</i><sub>3</sub>/d<i>t</i> <i><b>k</b></i></span>. If the absolute <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> of the rotating frame is <span class="texhtml mvar" style="font-style:italic;"><b>ω</b></span> then the derivative <span class="texhtml">d<i><b>P</b></i>/d<i>t</i></span> of <span class="texhtml"><i><b>P</b></i></span> with respect to the stationary frame is related to <span class="texhtml">[d<i><b>P</b></i>/d<i>t</i>]</span> by the equation:<sup id="cite_ref-Synge_22-0" class="reference"><a href="#cite_note-Synge-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> <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 {\frac {\mathrm {d} {\boldsymbol {P}}}{\mathrm {d} t}}=\left[{\frac {\mathrm {d} {\boldsymbol {P}}}{\mathrm {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {P}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">P</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">P</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">P</mi> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} {\boldsymbol {P}}}{\mathrm {d} t}}=\left[{\frac {\mathrm {d} {\boldsymbol {P}}}{\mathrm {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {P}}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f92943ec6cb61027f18fed127eda785a533db0e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.293ex; height:6.176ex;" alt="{\displaystyle {\frac {\mathrm {d} {\boldsymbol {P}}}{\mathrm {d} t}}=\left[{\frac {\mathrm {d} {\boldsymbol {P}}}{\mathrm {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {P}}\ ,}"></span> where <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 \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"></span> denotes the <a href="/wiki/Vector_cross_product" class="mw-redirect" title="Vector cross product">vector cross product</a>. In other words, the rate of change of <span class="texhtml mvar" style="font-style:italic;"><b>P</b></span> in the stationary frame is the sum of its apparent rate of change in the rotating frame and a rate of rotation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}\times {\boldsymbol {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}\times {\boldsymbol {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0fea7593cb3180075bd91446e43abc5ba41d58a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.478ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\omega }}\times {\boldsymbol {P}}}"></span> attributable to the motion of the rotating frame. The vector <span class="texhtml mvar" style="font-style:italic;"><b>ω</b></span> has magnitude <span class="texhtml mvar" style="font-style:italic;">ω</span> equal to the rate of rotation and is directed along the axis of rotation according to the <a href="/wiki/Right-hand_rule" title="Right-hand rule">right-hand rule</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Acceleration">Acceleration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=10" title="Edit section: Acceleration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Newton's law of motion for a particle of mass <span class="texhtml mvar" style="font-style:italic;">m</span> written in vector form 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 {\boldsymbol {F}}=m{\boldsymbol {a}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">a</mi> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {F}}=m{\boldsymbol {a}}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce27e6aaf51704332676aa6eba83d02454cae488" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.718ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {F}}=m{\boldsymbol {a}}\ ,}"></span> where <span class="texhtml mvar" style="font-style:italic;"><b>F</b></span> is the vector sum of the physical forces applied to the particle and <span class="texhtml mvar" style="font-style:italic;"><b>a</b></span> is the absolute <a href="/wiki/Acceleration" title="Acceleration">acceleration</a> (that is, acceleration in an inertial frame) of the particle, 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 {\boldsymbol {a}}={\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {a}}={\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cb1afc5cddd325d6e6b73996d9294fcb12e610" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.21ex; height:6.009ex;" alt="{\displaystyle {\boldsymbol {a}}={\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\ ,}"></span> where <span class="texhtml mvar" style="font-style:italic;"><b>r</b></span> is the position vector of the particle (not to be confused with radius, as used above.) </p><p>By applying the transformation above from the stationary to the rotating frame three times (twice to <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 {\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3170a7a4b57d55d47a52c9d6856264d43e7b1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.62ex; height:3.843ex;" alt="{\textstyle {\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}}"></span> and once to <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 {\frac {\mathrm {d} }{\mathrm {d} t}}\left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e674e00661addabf4cfc4492e2efa3ed432549b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.545ex; height:4.843ex;" alt="{\textstyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]}"></span>), the absolute acceleration of the particle can be written 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 {\begin{aligned}{\boldsymbol {a}}&={\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}\left(\left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {r}}\ \right)\\&=\left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]+{\boldsymbol {\omega }}\times \left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r}}+{\boldsymbol {\omega }}\times {\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\\&=\left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]+{\boldsymbol {\omega }}\times \left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r}}+{\boldsymbol {\omega }}\times \left(\left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {r}}\ \right)\\&=\left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]+{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r}}+2{\boldsymbol {\omega }}\times \left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})\ .\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"> <mi mathvariant="bold-italic">a</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\boldsymbol {a}}&={\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}\left(\left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {r}}\ \right)\\&=\left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]+{\boldsymbol {\omega }}\times \left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r}}+{\boldsymbol {\omega }}\times {\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\\&=\left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]+{\boldsymbol {\omega }}\times \left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r}}+{\boldsymbol {\omega }}\times \left(\left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {r}}\ \right)\\&=\left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]+{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r}}+2{\boldsymbol {\omega }}\times \left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})\ .\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19fd4a3a60a88be77f1f054ab87f1d1450dfc7da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.505ex; width:61.804ex; height:26.176ex;" alt="{\displaystyle {\begin{aligned}{\boldsymbol {a}}&={\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}={\frac {\mathrm {d} }{\mathrm {d} t}}\left(\left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {r}}\ \right)\\&=\left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]+{\boldsymbol {\omega }}\times \left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r}}+{\boldsymbol {\omega }}\times {\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\\&=\left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]+{\boldsymbol {\omega }}\times \left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r}}+{\boldsymbol {\omega }}\times \left(\left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\boldsymbol {\omega }}\times {\boldsymbol {r}}\ \right)\\&=\left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]+{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r}}+2{\boldsymbol {\omega }}\times \left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})\ .\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Force">Force</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=11" title="Edit section: Force"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The apparent acceleration in the rotating frame is <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[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b73a610718a6d3c9ac1bd34c23a31fcbd7346b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.868ex; height:6.343ex;" alt="{\displaystyle \left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]}"></span>. An observer unaware of the rotation would expect this to be zero in the absence of outside forces. However, Newton's laws of motion apply only in the inertial frame and describe dynamics in terms of the absolute acceleration <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff12959d68078a0036c33afac491f1cea0994d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:4.413ex; height:6.009ex;" alt="{\displaystyle {\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}}"></span>. Therefore, the observer perceives the extra terms as contributions due to fictitious forces. These terms in the apparent acceleration are independent of mass; so it appears that each of these fictitious forces, like gravity, pulls on an object in proportion to its mass. When these forces are added, the equation of motion has the form:<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-L&L_A_24-0" class="reference"><a href="#cite_note-L&L_A-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hand_A_25-0" class="reference"><a href="#cite_note-Hand_A-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> <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 {\boldsymbol {F}}+\underbrace {\left(-m{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r}}\right)} _{\text{Euler}}+\underbrace {\left(-2m{\boldsymbol {\omega }}\times \left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]\right)} _{\text{Coriolis}}+\underbrace {\left(-m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})\right)} _{\text{centrifugal}}=m\left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">F</mi> </mrow> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Euler</mtext> </mrow> </munder> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Coriolis</mtext> </mrow> </munder> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>centrifugal</mtext> </mrow> </munder> <mo>=</mo> <mi>m</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {F}}+\underbrace {\left(-m{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r}}\right)} _{\text{Euler}}+\underbrace {\left(-2m{\boldsymbol {\omega }}\times \left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]\right)} _{\text{Coriolis}}+\underbrace {\left(-m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})\right)} _{\text{centrifugal}}=m\left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3644f21c2c1ba71770b36caf7dcf969b6a7edea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:76.017ex; height:9.843ex;" alt="{\displaystyle {\boldsymbol {F}}+\underbrace {\left(-m{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\times {\boldsymbol {r}}\right)} _{\text{Euler}}+\underbrace {\left(-2m{\boldsymbol {\omega }}\times \left[{\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}\right]\right)} _{\text{Coriolis}}+\underbrace {\left(-m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})\right)} _{\text{centrifugal}}=m\left[{\frac {\mathrm {d} ^{2}{\boldsymbol {r}}}{\mathrm {d} t^{2}}}\right]\ .}"></span> </p><p>From the perspective of the rotating frame, the additional force terms are experienced just like the real external forces and contribute to the apparent acceleration.<sup id="cite_ref-Silverman_26-0" class="reference"><a href="#cite_note-Silverman-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> The additional terms on the force side of the equation can be recognized as, reading from left to right, the <a href="/wiki/Euler_force" title="Euler force">Euler force</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -m\mathrm {d} {\boldsymbol {\omega }}/\mathrm {d} t\times {\boldsymbol {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -m\mathrm {d} {\boldsymbol {\omega }}/\mathrm {d} t\times {\boldsymbol {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1499b9f7722f5ddb2c68b4867823173069a833a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.175ex; height:2.843ex;" alt="{\displaystyle -m\mathrm {d} {\boldsymbol {\omega }}/\mathrm {d} t\times {\boldsymbol {r}}}"></span>, the <a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2m{\boldsymbol {\omega }}\times \left[\mathrm {d} {\boldsymbol {r}}/\mathrm {d} t\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2m{\boldsymbol {\omega }}\times \left[\mathrm {d} {\boldsymbol {r}}/\mathrm {d} t\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37e97f28f344facce88c5f4c0087d69942a2a5ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.631ex; height:2.843ex;" alt="{\displaystyle -2m{\boldsymbol {\omega }}\times \left[\mathrm {d} {\boldsymbol {r}}/\mathrm {d} t\right]}"></span>, and the centrifugal force <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d6de0cd7870d607aeba039d4f22d9971c89b30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.906ex; height:2.843ex;" alt="{\displaystyle -m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})}"></span>, respectively.<sup id="cite_ref-Lanczos_A_28-0" class="reference"><a href="#cite_note-Lanczos_A-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> Unlike the other two fictitious forces, the centrifugal force always points radially outward from the axis of rotation of the rotating frame, with magnitude <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\omega ^{2}r_{\perp }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\omega ^{2}r_{\perp }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4042b3a3bed98f61cac8da6c052ae57023808ee9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.1ex; height:3.009ex;" alt="{\displaystyle m\omega ^{2}r_{\perp }}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\perp }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\perp }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20b303385f918a4189371be2f509566cab54ab99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.559ex; height:2.009ex;" alt="{\displaystyle r_{\perp }}"></span> is the component of the position vector perpendicular to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb8af7a2f64af348e559652b6b1f0d2415ba444" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\omega }}}"></span>, and unlike the Coriolis force in particular, it is independent of the motion of the particle in the rotating frame. As expected, for a non-rotating inertial frame of reference <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\boldsymbol {\omega }}=0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\boldsymbol {\omega }}=0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb8df1f7882b5b5609b9932d4a00d583ac33ad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.739ex; height:2.843ex;" alt="{\displaystyle ({\boldsymbol {\omega }}=0)}"></span> the centrifugal force and all other fictitious forces disappear.<sup id="cite_ref-Tavel_29-0" class="reference"><a href="#cite_note-Tavel-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Similarly, as the centrifugal force is proportional to the distance from object to the axis of rotation of the frame, the centrifugal force vanishes for objects that lie upon the axis. </p> <div class="mw-heading mw-heading2"><h2 id="Absolute_rotation">Absolute rotation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=12" title="Edit section: Absolute rotation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Parabola_shape_in_rotating_layers_of_fluid.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Parabola_shape_in_rotating_layers_of_fluid.jpg/220px-Parabola_shape_in_rotating_layers_of_fluid.jpg" decoding="async" width="220" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/ca/Parabola_shape_in_rotating_layers_of_fluid.jpg 1.5x" data-file-width="300" data-file-height="209" /></a><figcaption>The interface of two <a href="/wiki/Miscibility" title="Miscibility">immiscible</a> liquids rotating around a vertical axis is an upward-opening circular paraboloid.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Elipsoid_zplostely.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Elipsoid_zplostely.png/220px-Elipsoid_zplostely.png" decoding="async" width="220" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Elipsoid_zplostely.png/330px-Elipsoid_zplostely.png 1.5x, //upload.wikimedia.org/wikipedia/commons/8/88/Elipsoid_zplostely.png 2x" data-file-width="371" data-file-height="345" /></a><figcaption>When analysed in a rotating reference frame of the planet, centrifugal force causes rotating planets to assume the shape of an oblate spheroid.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Absolute_rotation" title="Absolute rotation">Absolute rotation</a></div> <p>Three scenarios were suggested by Newton to answer the question of whether the absolute rotation of a local frame can be detected; that is, if an observer can decide whether an observed object is rotating or if the observer is rotating.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <ul><li>The shape of the surface of water <a href="/wiki/Rotating_bucket" class="mw-redirect" title="Rotating bucket">rotating in a bucket</a>. The shape of the surface becomes concave to balance the centrifugal force against the other forces upon the liquid.</li> <li>The tension in a string joining two <a href="/wiki/Rotating_spheres" title="Rotating spheres">spheres rotating</a> about their center of mass. The tension in the string will be proportional to the centrifugal force on each sphere as it rotates around the common center of mass.</li></ul> <p>In these scenarios, the effects attributed to centrifugal force are only observed in the local frame (the frame in which the object is stationary) if the object is undergoing absolute rotation relative to an inertial frame. By contrast, in an inertial frame, the observed effects arise as a consequence of the inertia and the known forces without the need to introduce a centrifugal force. Based on this argument, the privileged frame, wherein the laws of physics take on the simplest form, is a stationary frame in which no fictitious forces need to be invoked. </p><p>Within this view of physics, any other phenomenon that is usually attributed to centrifugal force can be used to identify absolute rotation. For example, the oblateness of a sphere of freely flowing material is often explained in terms of centrifugal force. The <a href="/wiki/Oblate_spheroid" class="mw-redirect" title="Oblate spheroid">oblate spheroid</a> shape reflects, following <a href="/wiki/Clairaut%27s_theorem" class="mw-redirect" title="Clairaut's theorem">Clairaut's theorem</a>, the balance between containment by gravitational attraction and dispersal by centrifugal force. That the Earth is itself an oblate spheroid, bulging at the equator where the radial distance and hence the centrifugal force is larger, is taken as one of the evidences for its absolute rotation.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </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=Centrifugal_force&action=edit&section=13" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The operations of numerous common rotating mechanical systems are most easily conceptualized in terms of centrifugal force. For example: </p> <ul><li>A <a href="/wiki/Centrifugal_governor" title="Centrifugal governor">centrifugal governor</a> regulates the speed of an engine by using spinning masses that move radially, adjusting the <a href="/wiki/Throttle" title="Throttle">throttle</a>, as the engine changes speed. In the reference frame of the spinning masses, centrifugal force causes the radial movement.</li> <li>A <a href="/wiki/Centrifugal_clutch" title="Centrifugal clutch">centrifugal clutch</a> is used in small engine-powered devices such as chain saws, go-karts and model helicopters. It allows the engine to start and idle without driving the device but automatically and smoothly engages the drive as the engine speed rises. <a href="/wiki/Self-locking_device#Inertial_drum_brake_ascender" title="Self-locking device">Inertial drum brake ascenders</a> used in <a href="/wiki/Rock_climbing" title="Rock climbing">rock climbing</a> and the <a href="/wiki/Seat_belt#Technology" title="Seat belt">inertia reels</a> used in many automobile seat belts operate on the same principle.</li> <li>Centrifugal forces can be used to generate <a href="/wiki/Artificial_gravity" title="Artificial gravity">artificial gravity</a>, as in proposed designs for rotating space stations. The <a href="/wiki/Mars_Gravity_Biosatellite" title="Mars Gravity Biosatellite">Mars Gravity Biosatellite</a> would have studied the effects of <a href="/wiki/Mars" title="Mars">Mars</a>-level gravity on mice with gravity simulated in this way.</li> <li><a href="/wiki/Spin_casting" title="Spin casting">Spin casting</a> and <a href="/wiki/Centrifugal_casting_(industrial)" title="Centrifugal casting (industrial)">centrifugal casting</a> are production methods that use centrifugal force to disperse liquid metal or plastic throughout the negative space of a mold.</li> <li><a href="/wiki/Centrifuge" title="Centrifuge">Centrifuges</a> are used in science and industry to separate substances. In the reference frame spinning with the centrifuge, the centrifugal force induces a hydrostatic pressure gradient in fluid-filled tubes oriented perpendicular to the axis of rotation, giving rise to large <a href="/wiki/Buoyant_force" class="mw-redirect" title="Buoyant force">buoyant forces</a> which push low-density particles inward. Elements or particles denser than the fluid move outward under the influence of the centrifugal force. This is effectively <a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes' principle</a> as generated by centrifugal force as opposed to being generated by gravity.</li> <li>Some <a href="/wiki/Amusement_ride" class="mw-redirect" title="Amusement ride">amusement rides</a> make use of centrifugal forces. For instance, a <a href="/wiki/Gravitron" title="Gravitron">Gravitron</a>'s spin forces riders against a wall and allows riders to be elevated above the machine's floor in defiance of Earth's gravity.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></li></ul> <p>Nevertheless, all of these systems can also be described without requiring the concept of centrifugal force, in terms of motions and forces in a stationary frame, at the cost of taking somewhat more care in the consideration of forces and motions within the system. </p> <div class="mw-heading mw-heading2"><h2 id="Other_uses_of_the_term">Other uses of the term</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=14" title="Edit section: Other uses of the term"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While the majority of the scientific literature uses the term <i>centrifugal force</i> to refer to the particular fictitious force that arises in rotating frames, there are a few limited instances in the literature of the term applied to other distinct physical concepts. </p> <div class="mw-heading mw-heading3"><h3 id="In_Lagrangian_mechanics">In Lagrangian mechanics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=15" title="Edit section: In Lagrangian mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One of these instances occurs in <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a>. Lagrangian mechanics formulates mechanics in terms of <a href="/wiki/Generalized_coordinates" title="Generalized coordinates">generalized coordinates</a> {<i>q<sub>k</sub></i>}, which can be as simple as the usual polar coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\ \theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mtext> </mtext> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\ \theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6292b4608eeb2c1c6ada486c782f2d2887a8577a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle (r,\ \theta )}"></span> or a much more extensive list of variables.<sup id="cite_ref-Lanczos_34-0" class="reference"><a href="#cite_note-Lanczos-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Shabana1_35-0" class="reference"><a href="#cite_note-Shabana1-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> Within this formulation the motion is described in terms of <i><a href="/wiki/Generalized_forces" title="Generalized forces">generalized forces</a></i>, using in place of <a href="/wiki/Newton%27s_laws" class="mw-redirect" title="Newton's laws">Newton's laws</a> the <a href="/wiki/Euler%E2%80%93Lagrange_equations" class="mw-redirect" title="Euler–Lagrange equations">Euler–Lagrange equations</a>. Among the generalized forces, those involving the square of the time derivatives {(d<i>q<sub>k</sub></i> ⁄ d<i>t</i> )<sup>2</sup>} are sometimes called centrifugal forces.<sup id="cite_ref-Ott_36-0" class="reference"><a href="#cite_note-Ott-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Ge_37-0" class="reference"><a href="#cite_note-Ge-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Nagrath_38-0" class="reference"><a href="#cite_note-Nagrath-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Toda_39-0" class="reference"><a href="#cite_note-Toda-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> In the case of motion in a central potential the Lagrangian centrifugal force has the same form as the fictitious centrifugal force derived in a co-rotating frame.<sup id="cite_ref-Bini1997_40-0" class="reference"><a href="#cite_note-Bini1997-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> However, the Lagrangian use of "centrifugal force" in other, more general cases has only a limited connection to the Newtonian definition. </p> <div class="mw-heading mw-heading3"><h3 id="As_a_reactive_force">As a reactive force</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=16" title="Edit section: As a reactive force"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In another instance the term refers to the <a href="/wiki/Reaction_(physics)" title="Reaction (physics)">reaction</a> <a href="/wiki/Force" title="Force">force</a> to a centripetal force, or <a href="/wiki/Reactive_centrifugal_force" title="Reactive centrifugal force">reactive centrifugal force</a>. A body undergoing curved motion, such as <a href="/wiki/Circular_motion" title="Circular motion">circular motion</a>, is accelerating toward a center at any particular point in time. This <a href="/wiki/Centripetal_acceleration" class="mw-redirect" title="Centripetal acceleration">centripetal acceleration</a> is provided by a centripetal force, which is exerted on the body in curved motion by some other body. In accordance with <a href="/wiki/Newton%27s_laws_of_motion#Newton's_third_law" title="Newton's laws of motion">Newton's third law of motion</a>, the body in curved motion exerts an equal and opposite force on the other body. This <a href="/wiki/Reaction_(physics)" title="Reaction (physics)">reactive</a> force is exerted <i>by</i> the body in curved motion <i>on</i> the other body that provides the centripetal force and its direction is from that other body toward the body in curved motion.<sup id="cite_ref-Mook_41-0" class="reference"><a href="#cite_note-Mook-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Scott_42-0" class="reference"><a href="#cite_note-Scott-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-Signell_43-0" class="reference"><a href="#cite_note-Signell-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> </p><p>This reaction force is sometimes described as a <i>centrifugal inertial reaction</i>,<sup id="cite_ref-Roche_45-0" class="reference"><a href="#cite_note-Roche-45"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> that is, a force that is centrifugally directed, which is a reactive force equal and opposite to the centripetal force that is curving the path of the mass. </p><p>The concept of the reactive centrifugal force is sometimes used in mechanics and engineering. It is sometimes referred to as just <i>centrifugal force</i> rather than as <i>reactive</i> centrifugal force<sup id="cite_ref-Bowser_47-0" class="reference"><a href="#cite_note-Bowser-47"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Angelo_48-0" class="reference"><a href="#cite_note-Angelo-48"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> although this usage is deprecated in elementary mechanics.<sup id="cite_ref-Rogers_49-0" class="reference"><a href="#cite_note-Rogers-49"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </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=Centrifugal_force&action=edit&section=17" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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title="Portal:Physics">Physics portal</a></span></li></ul> <ul><li><a href="/wiki/Balancing_of_rotating_masses" title="Balancing of rotating masses">Balancing of rotating masses</a></li> <li><a href="/wiki/Centrifugal_mechanism_of_acceleration" class="mw-redirect" title="Centrifugal mechanism of acceleration">Centrifugal mechanism of acceleration</a></li> <li><a href="/wiki/Equivalence_principle" title="Equivalence principle">Equivalence principle</a></li> <li><a href="/wiki/Folk_physics" class="mw-redirect" title="Folk physics">Folk physics</a></li> <li><a href="/wiki/Lagrangian_point" class="mw-redirect" title="Lagrangian point">Lagrangian point</a></li> <li><a href="/wiki/Lamm_equation" title="Lamm equation">Lamm equation</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=18" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">In Latin: <i>vim centrifugam</i>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Centrifugal_force&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-yoeder-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-yoeder_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-yoeder_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px 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 id="CITEREFYoder1991" class="citation journal cs1"><a href="/wiki/Joella_Yoder" title="Joella Yoder">Yoder, Joella</a> (1991). <a rel="nofollow" class="external text" href="http://www.gewina.nl/journals/tractrix/yoder91.pdf">"Christiaan Huygens' Great Treasure"</a> <span class="cs1-format">(PDF)</span>. <i>Tractrix</i>. <b>3</b>: 1–13. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180413044740/http://www.gewina.nl/journals/tractrix/yoder91.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 13 April 2018<span class="reference-accessdate">. Retrieved <span class="nowrap">12 April</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Tractrix&rft.atitle=Christiaan+Huygens%27+Great+Treasure&rft.volume=3&rft.pages=1-13&rft.date=1991&rft.aulast=Yoder&rft.aufirst=Joella&rft_id=http%3A%2F%2Fwww.gewina.nl%2Fjournals%2Ftractrix%2Fyoder91.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Yoder2013-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Yoder2013_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYoder2013" class="citation book cs1">Yoder, Joella (17 May 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XGZlIvCOtFsC"><i>A Catalogue of the Manuscripts of Christiaan Huygens including a concordance with his Oeuvres Complètes</i></a>. BRILL. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789004235656" title="Special:BookSources/9789004235656"><bdi>9789004235656</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200316011539/https://books.google.com/books?id=XGZlIvCOtFsC">Archived</a> from the original on 16 March 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">12 April</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Catalogue+of+the+Manuscripts+of+Christiaan+Huygens+including+a+concordance+with+his+Oeuvres+Compl%C3%A8tes&rft.pub=BRILL&rft.date=2013-05-17&rft.isbn=9789004235656&rft.aulast=Yoder&rft.aufirst=Joella&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXGZlIvCOtFsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlackwell1986" class="citation book cs1">Blackwell, Richard J. (1986). <a rel="nofollow" class="external text" href="https://archive.org/details/christiaanhuygen0000huyg"><i>Christiaan Huygens' the pendulum clock, or, Geometrical demonstrations concerning the motion of pendula as applied to clocks</i></a>. Ames: Iowa State University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/christiaanhuygen0000huyg/page/173">173</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8138-0933-5" title="Special:BookSources/978-0-8138-0933-5"><bdi>978-0-8138-0933-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Christiaan+Huygens%27+the+pendulum+clock%2C+or%2C+Geometrical+demonstrations+concerning+the+motion+of+pendula+as+applied+to+clocks&rft.place=Ames&rft.pages=173&rft.pub=Iowa+State+University+Press&rft.date=1986&rft.isbn=978-0-8138-0933-5&rft.aulast=Blackwell&rft.aufirst=Richard+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fchristiaanhuygen0000huyg&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1 cs1-prop-foreign-lang-source"><a class="external text" href="https://commons.wikimedia.org/w/index.php?title=File:Huygens_-_%C5%92uvres_compl%C3%A8tes,_Tome_7,_1897.djvu"><i>Œuvres complètes de Christiaan Huygens</i></a> (in French). Vol. 7. The Hague: M. Nijhoff. 1897. p. <a class="external text" href="https://commons.wikimedia.org/w/index.php?title=File:Huygens_-_%C5%92uvres_compl%C3%A8tes,_Tome_7,_1897.djvu&page=353">325</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20231106055244/https://commons.wikimedia.org/w/index.php?title=File:Huygens_-_%C5%92uvres_compl%C3%A8tes,_Tome_7,_1897.djvu">Archived</a> from the original on 2023-11-06<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-01-14</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C5%92uvres+compl%C3%A8tes+de+Christiaan+Huygens&rft.place=The+Hague&rft.pages=325&rft.pub=M.+Nijhoff&rft.date=1897&rft_id=https%3A%2F%2Fcommons.wikimedia.org%2Fw%2Findex.php%3Ftitle%3DFile%3AHuygens_-_%25C5%2592uvres_compl%25C3%25A8tes%2C_Tome_7%2C_1897.djvu&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Newton-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Newton_6-0">^</a></b></span> <span class="reference-text">An English translation is found at <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIsaac_Newton1934" class="citation book cs1">Isaac Newton (1934). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ySYULc7VEwsC&pg=PA10"><i>Philosophiae naturalis principia mathematica</i></a> (Andrew Motte translation of 1729, revised by Florian Cajori ed.). University of California Press. pp. 10–12. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780520009271" title="Special:BookSources/9780520009271"><bdi>9780520009271</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Philosophiae+naturalis+principia+mathematica&rft.pages=10-12&rft.edition=Andrew+Motte+translation+of+1729%2C+revised+by+Florian+Cajori&rft.pub=University+of+California+Press&rft.date=1934&rft.isbn=9780520009271&rft.au=Isaac+Newton&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DySYULc7VEwsC%26pg%3DPA10&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Barbour-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Barbour_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJulian_B._BarbourHerbert_Pfister1995" class="citation book cs1">Julian B. Barbour; Herbert Pfister, eds. (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fKgQ9YpAcwMC&pg=PA69"><i>Mach's principle : from Newton's bucket to quantum gravity</i></a>. Boston: Birkhäuser. p. 69. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8176-3823-7" title="Special:BookSources/0-8176-3823-7"><bdi>0-8176-3823-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/32664808">32664808</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mach%27s+principle+%3A+from+Newton%27s+bucket+to+quantum+gravity&rft.place=Boston&rft.pages=69&rft.pub=Birkh%C3%A4user&rft.date=1995&rft_id=info%3Aoclcnum%2F32664808&rft.isbn=0-8176-3823-7&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfKgQ9YpAcwMC%26pg%3DPA69&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Eriksson-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Eriksson_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=rYW8tKzrFd4C&pg=PA194"><i>Science education in the 21st century</i></a>. Ingrid V. Eriksson. New York: Nova Science Publishers. 2008. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-60021-951-1" title="Special:BookSources/978-1-60021-951-1"><bdi>978-1-60021-951-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/165958146">165958146</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Science+education+in+the+21st+century&rft.place=New+York&rft.pub=Nova+Science+Publishers&rft.date=2008&rft_id=info%3Aoclcnum%2F165958146&rft.isbn=978-1-60021-951-1&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrYW8tKzrFd4C%26pg%3DPA194&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: others (<a href="/wiki/Category:CS1_maint:_others" title="Category:CS1 maint: others">link</a>)</span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_T._Weidner_and_Robert_L._Sells1973" class="citation book cs1">Richard T. Weidner and Robert L. Sells (1973). <i>Mechanics, mechanical waves, kinetic theory, thermodynamics</i> (2 ed.). Allyn and Bacon. p. 123.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics%2C+mechanical+waves%2C+kinetic+theory%2C+thermodynamics&rft.pages=123&rft.edition=2&rft.pub=Allyn+and+Bacon&rft.date=1973&rft.au=Richard+T.+Weidner+and+Robert+L.+Sells&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRestucciaTorošGibsonUlbricht2019" class="citation journal cs1">Restuccia, S.; Toroš, M.; Gibson, G. M.; Ulbricht, H.; Faccio, D.; Padgett, M. J. (2019). <a rel="nofollow" class="external text" href="https://doi.org/10.1103/physrevlett.123.110401">"Photon Bunching in a Rotating Reference Frame"</a>. <i>Physical Review Letters</i>. <b>123</b> (11): 110401. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1906.03400">1906.03400</a></span>. <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/2019PhRvL.123k0401R">2019PhRvL.123k0401R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevlett.123.110401">10.1103/physrevlett.123.110401</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/31573252">31573252</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:182952610">182952610</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Photon+Bunching+in+a+Rotating+Reference+Frame&rft.volume=123&rft.issue=11&rft.pages=110401&rft.date=2019&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A182952610%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2019PhRvL.123k0401R&rft_id=info%3Aarxiv%2F1906.03400&rft_id=info%3Apmid%2F31573252&rft_id=info%3Adoi%2F10.1103%2Fphysrevlett.123.110401&rft.aulast=Restuccia&rft.aufirst=S.&rft.au=Toro%C5%A1%2C+M.&rft.au=Gibson%2C+G.+M.&rft.au=Ulbricht%2C+H.&rft.au=Faccio%2C+D.&rft.au=Padgett%2C+M.+J.&rft_id=https%3A%2F%2Fdoi.org%2F10.1103%2Fphysrevlett.123.110401&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Taylor1-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Taylor1_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Robert_Taylor2004" class="citation book cs1">John Robert Taylor (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1"><i>Classical Mechanics</i></a>. Sausalito CA: University Science Books. Chapter 9, pp. 344 ff. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-891389-22-1" title="Special:BookSources/978-1-891389-22-1"><bdi>978-1-891389-22-1</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007141548/https://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics&rft.place=Sausalito+CA&rft.pages=Chapter+9%2C+pp.+344+ff&rft.pub=University+Science+Books&rft.date=2004&rft.isbn=978-1-891389-22-1&rft.au=John+Robert+Taylor&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DP1kCtNr-pJsC%26pg%3DPP1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKobayashi2008" class="citation journal cs1">Kobayashi, Yukio (2008). "Remarks on viewing situation in a rotating frame". <i>European Journal of Physics</i>. <b>29</b> (3): 599–606. <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/2008EJPh...29..599K">2008EJPh...29..599K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0143-0807%2F29%2F3%2F019">10.1088/0143-0807/29/3/019</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:120947179">120947179</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=European+Journal+of+Physics&rft.atitle=Remarks+on+viewing+situation+in+a+rotating+frame&rft.volume=29&rft.issue=3&rft.pages=599-606&rft.date=2008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120947179%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0143-0807%2F29%2F3%2F019&rft_id=info%3Abibcode%2F2008EJPh...29..599K&rft.aulast=Kobayashi&rft.aufirst=Yukio&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_P._Stern2006" class="citation web cs1">David P. Stern (2006). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200406211413/https://www-spof.gsfc.nasa.gov/stargaze/Sframes1.htm">"Frames of Reference: The Basics"</a>. <i>From Stargazers to Starships</i>. Goddard Space Flight Center Space Physics Data Facility. Archived from <a rel="nofollow" class="external text" href="https://www-spof.gsfc.nasa.gov/stargaze/Sframes1.htm">the original</a> on 6 April 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">20 April</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=From+Stargazers+to+Starships&rft.atitle=Frames+of+Reference%3A+The+Basics&rft.date=2006&rft.au=David+P.+Stern&rft_id=http%3A%2F%2Fwww-spof.gsfc.nasa.gov%2Fstargaze%2FSframes1.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/EBchecked/topic/102850/centrifuge">"Centrifuge"</a>. <i>Encyclopædia Britannica</i>. April 30, 2015. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007141550/https://www.britannica.com/technology/centrifuge">Archived</a> from the original on October 7, 2024<span class="reference-accessdate">. Retrieved <span class="nowrap">June 2,</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Centrifuge&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.date=2015-04-30&rft_id=https%3A%2F%2Fwww.britannica.com%2FEBchecked%2Ftopic%2F102850%2Fcentrifuge&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</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://feynmanlectures.caltech.edu/I_12.html#Ch12-S5-p2">"The Feynman Lectures on Physics Vol. I Ch. 12: Characteristics of Force"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007141549/https://www.feynmanlectures.caltech.edu/I_12.html#Ch12-S5-p2">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-05-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Feynman+Lectures+on+Physics+Vol.+I+Ch.+12%3A+Characteristics+of+Force&rft_id=https%3A%2F%2Ffeynmanlectures.caltech.edu%2FI_12.html%23Ch12-S5-p2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Fetter-16"><span class="mw-cite-backlink">^ <a href="#cite_ref-Fetter_16-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Fetter_16-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexander_L._FetterJohn_Dirk_Walecka2003" class="citation book cs1"><a href="/wiki/Alexander_L._Fetter" class="mw-redirect" title="Alexander L. Fetter">Alexander L. Fetter</a>; <a href="/wiki/John_Dirk_Walecka" title="John Dirk Walecka">John Dirk Walecka</a> (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=olMpStYOlnoC&pg=PA39"><i>Theoretical Mechanics of Particles and Continua</i></a>. Courier Dover Publications. pp. 38–39. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43261-8" title="Special:BookSources/978-0-486-43261-8"><bdi>978-0-486-43261-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theoretical+Mechanics+of+Particles+and+Continua&rft.pages=38-39&rft.pub=Courier+Dover+Publications&rft.date=2003&rft.isbn=978-0-486-43261-8&rft.au=Alexander+L.+Fetter&rft.au=John+Dirk+Walecka&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DolMpStYOlnoC%26pg%3DPA39&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Marsden-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Marsden_17-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJerrold_E._MarsdenTudor_S._Ratiu1999" class="citation book cs1">Jerrold E. Marsden; Tudor S. Ratiu (1999). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=I2gH9ZIs-3AC&pg=PA251"><i>Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems</i></a>. Springer. p. 251. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98643-2" title="Special:BookSources/978-0-387-98643-2"><bdi>978-0-387-98643-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007141657/https://books.google.com/books?id=I2gH9ZIs-3AC&pg=PA251#v=onepage&q&f=false">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mechanics+and+Symmetry%3A+A+Basic+Exposition+of+Classical+Mechanical+Systems&rft.pages=251&rft.pub=Springer&rft.date=1999&rft.isbn=978-0-387-98643-2&rft.au=Jerrold+E.+Marsden&rft.au=Tudor+S.+Ratiu&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DI2gH9ZIs-3AC%26pg%3DPA251&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-EB-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-EB_18-0">^</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.britannica.com/science/centrifugal-force">"Centrifugal force"</a>. Encyclopædia Britannica. 17 August 2016. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170421011514/https://www.britannica.com/science/centrifugal-force">Archived</a> from the original on 21 April 2017<span class="reference-accessdate">. Retrieved <span class="nowrap">20 April</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Centrifugal+force&rft.pub=Encyclop%C3%A6dia+Britannica&rft.date=2016-08-17&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2Fcentrifugal-force&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Science_of_Everyday_Things-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-Science_of_Everyday_Things_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnight2016" class="citation book cs1">Knight, Judson (2016). Schlager, Neil (ed.). <a rel="nofollow" class="external text" href="https://archive.org/stream/ScienceOfEverydayThingsVol2-RealLifePhysics/ScienceOfEverydayThingsVol.2-Physics365s-o#page/n49/mode/2up/search/Centrifugal+force"><i>Centripetal Force</i></a>. Thomson Learning. p. 47<span class="reference-accessdate">. Retrieved <span class="nowrap">19 April</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Centripetal+Force&rft.pages=47&rft.pub=Thomson+Learning&rft.date=2016&rft.aulast=Knight&rft.aufirst=Judson&rft_id=https%3A%2F%2Farchive.org%2Fstream%2FScienceOfEverydayThingsVol2-RealLifePhysics%2FScienceOfEverydayThingsVol.2-Physics365s-o%23page%2Fn49%2Fmode%2F2up%2Fsearch%2FCentrifugal%2Bforce&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: </span><span class="cs1-visible-error citation-comment"><code class="cs1-code">|work=</code> ignored (<a href="/wiki/Help:CS1_errors#periodical_ignored" title="Help:CS1 errors">help</a>)</span></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"><a rel="nofollow" class="external text" href="http://curious.astro.cornell.edu/question.php?number=310">"Curious About Astronomy?"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150117191330/http://curious.astro.cornell.edu/question.php?number=310">Archived</a> January 17, 2015, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Cornell University, retrieved June 2007</span> </li> <li id="cite_note-Boynton-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-Boynton_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoynton2001" class="citation conference cs1">Boynton, Richard (2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070227132140/http://www.space-electronics.com/Literature/Precise_Measurement_of_Mass.PDF">"Precise Measurement of Mass"</a> <span class="cs1-format">(PDF)</span>. <i>Sawe Paper No. 3147</i>. Arlington, Texas: S.A.W.E., Inc. Archived from <a rel="nofollow" class="external text" href="http://www.space-electronics.com/Literature/Precise_Measurement_of_Mass.PDF">the original</a> <span class="cs1-format">(PDF)</span> on 2007-02-27<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-01-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Precise+Measurement+of+Mass&rft.btitle=Sawe+Paper+No.+3147&rft.place=Arlington%2C+Texas&rft.pub=S.A.W.E.%2C+Inc.&rft.date=2001&rft.aulast=Boynton&rft.aufirst=Richard&rft_id=http%3A%2F%2Fwww.space-electronics.com%2FLiterature%2FPrecise_Measurement_of_Mass.PDF&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Synge-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-Synge_22-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_L._SyngeByron_A._Griffith2007" class="citation book cs1">John L. Synge; Byron A. Griffith (2007). <a rel="nofollow" class="external text" href="https://archive.org/stream/principlesofmech031468mbp#page/n342/mode/1up"><i>Principles of Mechanics</i></a> (Reprint of Second Edition of 1942 ed.). Read Books. p. 347. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4067-4670-9" title="Special:BookSources/978-1-4067-4670-9"><bdi>978-1-4067-4670-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Mechanics&rft.pages=347&rft.edition=Reprint+of+Second+Edition+of+1942&rft.pub=Read+Books&rft.date=2007&rft.isbn=978-1-4067-4670-9&rft.au=John+L.+Synge&rft.au=Byron+A.+Griffith&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fprinciplesofmech031468mbp%23page%2Fn342%2Fmode%2F1up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Taylor (2005). p. 342.</span> </li> <li id="cite_note-L&L_A-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-L&L_A_24-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLD_LandauLM_Lifshitz1976" class="citation book cs1">LD Landau; LM Lifshitz (1976). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=e-xASAehg1sC&pg=PA40"><i>Mechanics</i></a> (Third ed.). Oxford: Butterworth-Heinemann. p. 128. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7506-2896-9" title="Special:BookSources/978-0-7506-2896-9"><bdi>978-0-7506-2896-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007141549/https://books.google.com/books?id=e-xASAehg1sC&pg=PA40#v=onepage&q&f=false">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</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&rft.place=Oxford&rft.pages=128&rft.edition=Third&rft.pub=Butterworth-Heinemann&rft.date=1976&rft.isbn=978-0-7506-2896-9&rft.au=LD+Landau&rft.au=LM+Lifshitz&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3De-xASAehg1sC%26pg%3DPA40&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Hand_A-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hand_A_25-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLouis_N._HandJanet_D._Finch1998" class="citation book cs1">Louis N. Hand; Janet D. Finch (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1J2hzvX2Xh8C&q=Hand+inauthor:Finch&pg=PA267"><i>Analytical Mechanics</i></a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 267. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-57572-0" title="Special:BookSources/978-0-521-57572-0"><bdi>978-0-521-57572-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007141658/https://books.google.com/books?id=1J2hzvX2Xh8C&q=Hand+inauthor:Finch&pg=PA267">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytical+Mechanics&rft.pages=267&rft.pub=Cambridge+University+Press&rft.date=1998&rft.isbn=978-0-521-57572-0&rft.au=Louis+N.+Hand&rft.au=Janet+D.+Finch&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1J2hzvX2Xh8C%26q%3DHand%2Binauthor%3AFinch%26pg%3DPA267&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Silverman-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-Silverman_26-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMark_P_Silverman2002" class="citation book cs1">Mark P Silverman (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-Er5pIsYe_AC&pg=PA249"><i>A universe of atoms, an atom in the universe</i></a> (2 ed.). Springer. p. 249. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95437-0" title="Special:BookSources/978-0-387-95437-0"><bdi>978-0-387-95437-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007142053/https://books.google.com/books?id=-Er5pIsYe_AC&pg=PA249#v=onepage&q&f=false">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+universe+of+atoms%2C+an+atom+in+the+universe&rft.pages=249&rft.edition=2&rft.pub=Springer&rft.date=2002&rft.isbn=978-0-387-95437-0&rft.au=Mark+P+Silverman&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-Er5pIsYe_AC%26pg%3DPA249&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">Taylor (2005). p. 329.</span> </li> <li id="cite_note-Lanczos_A-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lanczos_A_28-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCornelius_Lanczos1986" class="citation book cs1">Cornelius Lanczos (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PA103"><i>The Variational Principles of Mechanics</i></a> (Reprint of Fourth Edition of 1970 ed.). Dover Publications. Chapter 4, §5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65067-8" title="Special:BookSources/978-0-486-65067-8"><bdi>978-0-486-65067-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Variational+Principles+of+Mechanics&rft.pages=Chapter+4%2C+%C2%A75&rft.edition=Reprint+of+Fourth+Edition+of+1970&rft.pub=Dover+Publications&rft.date=1986&rft.isbn=978-0-486-65067-8&rft.au=Cornelius+Lanczos&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZWoYYr8wk2IC%26pg%3DPA103&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Tavel-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-Tavel_29-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorton_Tavel2002" class="citation book cs1">Morton Tavel (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SELS0HbIhjYC&q=Einstein+equivalence+laws+physics+frame&pg=PA95"><i>Contemporary Physics and the Limits of Knowledge</i></a>. <a href="/wiki/Rutgers_University_Press" title="Rutgers University Press">Rutgers University Press</a>. p. 93. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8135-3077-2" title="Special:BookSources/978-0-8135-3077-2"><bdi>978-0-8135-3077-2</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007142054/https://books.google.com/books?id=SELS0HbIhjYC&q=Einstein+equivalence+laws+physics+frame&pg=PA95#v=snippet&q=Einstein%20equivalence%20laws%20physics%20frame&f=false">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</span></span>. <q>Noninertial forces, like centrifugal and Coriolis forces, can be eliminated by jumping into a reference frame that moves with constant velocity, the frame that Newton called inertial.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Contemporary+Physics+and+the+Limits+of+Knowledge&rft.pages=93&rft.pub=Rutgers+University+Press&rft.date=2002&rft.isbn=978-0-8135-3077-2&rft.au=Morton+Tavel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSELS0HbIhjYC%26q%3DEinstein%2Bequivalence%2Blaws%2Bphysics%2Bframe%26pg%3DPA95&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLouis_N._HandJanet_D._Finch1998" class="citation book cs1">Louis N. Hand; Janet D. Finch (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1J2hzvX2Xh8C&pg=PA324"><i>Analytical Mechanics</i></a>. Cambridge University Press. p. 324. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-57572-0" title="Special:BookSources/978-0-521-57572-0"><bdi>978-0-521-57572-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytical+Mechanics&rft.pages=324&rft.pub=Cambridge+University+Press&rft.date=1998&rft.isbn=978-0-521-57572-0&rft.au=Louis+N.+Hand&rft.au=Janet+D.+Finch&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1J2hzvX2Xh8C%26pg%3DPA324&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFI._Bernard_CohenGeorge_Edwin_Smith2002" class="citation book cs1">I. Bernard Cohen; George Edwin Smith (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3wIzvqzfUXkC&pg=PA43"><i>The Cambridge companion to Newton</i></a>. Cambridge University Press. p. 43. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-65696-2" title="Special:BookSources/978-0-521-65696-2"><bdi>978-0-521-65696-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Cambridge+companion+to+Newton&rft.pages=43&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=978-0-521-65696-2&rft.au=I.+Bernard+Cohen&rft.au=George+Edwin+Smith&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3wIzvqzfUXkC%26pg%3DPA43&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimon_Newcomb1878" class="citation book cs1">Simon Newcomb (1878). <a rel="nofollow" class="external text" href="https://archive.org/details/popularastronomy1878newc"><i>Popular astronomy</i></a>. Harper & Brothers. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/popularastronomy1878newc/page/86">86</a>–88.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Popular+astronomy&rft.pages=86-88&rft.pub=Harper+%26+Brothers&rft.date=1878&rft.au=Simon+Newcomb&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpopularastronomy1878newc&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMyers2006" class="citation book cs1">Myers, Rusty L. (2006). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/basicsofphysics0000myer"><i>The basics of physics</i></a></span>. Greenwood Publishing Group. p. <a rel="nofollow" class="external text" href="https://archive.org/details/basicsofphysics0000myer/page/57">57</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-313-32857-2" title="Special:BookSources/978-0-313-32857-2"><bdi>978-0-313-32857-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+basics+of+physics&rft.pages=57&rft.pub=Greenwood+Publishing+Group&rft.date=2006&rft.isbn=978-0-313-32857-2&rft.aulast=Myers&rft.aufirst=Rusty+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbasicsofphysics0000myer&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Lanczos-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lanczos_34-0">^</a></b></span> <span class="reference-text">For an introduction, see for example <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCornelius_Lanczos1986" class="citation book cs1">Cornelius Lanczos (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PR4"><i>The variational principles of mechanics</i></a> (Reprint of 1970 University of Toronto ed.). Dover. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65067-8" title="Special:BookSources/978-0-486-65067-8"><bdi>978-0-486-65067-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007142120/https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PR4">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+variational+principles+of+mechanics&rft.pages=1&rft.edition=Reprint+of+1970+University+of+Toronto&rft.pub=Dover&rft.date=1986&rft.isbn=978-0-486-65067-8&rft.au=Cornelius+Lanczos&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZWoYYr8wk2IC%26pg%3DPR4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Shabana1-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-Shabana1_35-0">^</a></b></span> <span class="reference-text">For a description of generalized coordinates, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAhmed_A._Shabana2003" class="citation book cs1">Ahmed A. Shabana (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zxuG-l7J5rgC">"Generalized coordinates and kinematic constraints"</a>. <i>Dynamics of Multibody Systems</i> (2 ed.). Cambridge University Press. p. 90 <i>ff</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-54411-5" title="Special:BookSources/978-0-521-54411-5"><bdi>978-0-521-54411-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007142055/https://books.google.com/books?id=zxuG-l7J5rgC">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Generalized+coordinates+and+kinematic+constraints&rft.btitle=Dynamics+of+Multibody+Systems&rft.pages=90+%27%27ff%27%27&rft.edition=2&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-54411-5&rft.au=Ahmed+A.+Shabana&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DzxuG-l7J5rgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Ott-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ott_36-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristian_Ott2008" class="citation book cs1">Christian Ott (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wKQvUfwzqjAC&pg=PA23"><i>Cartesian Impedance Control of Redundant and Flexible-Joint Robots</i></a>. Springer. p. 23. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-69253-9" title="Special:BookSources/978-3-540-69253-9"><bdi>978-3-540-69253-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007142219/https://books.google.com/books?id=wKQvUfwzqjAC&pg=PA23#v=onepage&q&f=false">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cartesian+Impedance+Control+of+Redundant+and+Flexible-Joint+Robots&rft.pages=23&rft.pub=Springer&rft.date=2008&rft.isbn=978-3-540-69253-9&rft.au=Christian+Ott&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DwKQvUfwzqjAC%26pg%3DPA23&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Ge-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ge_37-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShuzhi_S._GeTong_Heng_LeeChristopher_John_Harris1998" class="citation book cs1">Shuzhi S. Ge; Tong Heng Lee; Christopher John Harris (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cdBENqlY_ucC&q=CHristoffel+centrifugal"><i>Adaptive Neural Network Control of Robotic Manipulators</i></a>. World Scientific. pp. 47–48. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-3452-2" title="Special:BookSources/978-981-02-3452-2"><bdi>978-981-02-3452-2</bdi></a>. <q>In the above <a href="/wiki/Euler%E2%80%93Lagrange_equations" class="mw-redirect" title="Euler–Lagrange equations">Euler–Lagrange equations</a>, there are three types of terms. The first involves the second derivative of the generalized co-ordinates. The second is quadratic in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\dot {q}}}}"> <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-italic">q</mi> <mo mathvariant="bold">˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\dot {q}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c8807ca3d57306997056e576d2c92d90f40734a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.571ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {\dot {q}}}}"></span> where the coefficients may depend on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edf74db7c59a404f691ec204e3152a01ef488b83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {q}}}"></span>. These are further classified into two types. Terms involving a product of the type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{\dot {q}}_{i}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{\dot {q}}_{i}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad7ace9f9fc4e0b3a3e8151958a83bff20548bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.231ex; height:3.176ex;" alt="{\displaystyle {{\dot {q}}_{i}}^{2}}"></span> are called <i>centrifugal forces</i> while those involving a product of the type <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {q}}_{i}{\dot {q}}_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>q</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {q}}_{i}{\dot {q}}_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89b82a47e15ae0c0285600d238dd79c98580fa17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.464ex; height:3.009ex;" alt="{\displaystyle {\dot {q}}_{i}{\dot {q}}_{j}}"></span> for <i>i ≠ j</i> are called <i>Coriolis forces</i>. The third type is functions 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 {\boldsymbol {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edf74db7c59a404f691ec204e3152a01ef488b83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {q}}}"></span> only and are called <i>gravitational forces</i>.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Adaptive+Neural+Network+Control+of+Robotic+Manipulators&rft.pages=47-48&rft.pub=World+Scientific&rft.date=1998&rft.isbn=978-981-02-3452-2&rft.au=Shuzhi+S.+Ge&rft.au=Tong+Heng+Lee&rft.au=Christopher+John+Harris&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DcdBENqlY_ucC%26q%3DCHristoffel%2Bcentrifugal&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Nagrath-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nagrath_38-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFR._K._MittalI._J._Nagrath2003" class="citation book cs1">R. K. Mittal; I. J. Nagrath (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZtwMEQzMVlMC&pg=PA202"><i>Robotics and Control</i></a>. Tata McGraw-Hill. p. 202. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-048293-7" title="Special:BookSources/978-0-07-048293-7"><bdi>978-0-07-048293-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007142204/https://books.google.com/books?id=ZtwMEQzMVlMC&pg=PA202">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robotics+and+Control&rft.pages=202&rft.pub=Tata+McGraw-Hill&rft.date=2003&rft.isbn=978-0-07-048293-7&rft.au=R.+K.+Mittal&rft.au=I.+J.+Nagrath&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZtwMEQzMVlMC%26pg%3DPA202&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Toda-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-Toda_39-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFT_YanaoK_Takatsuka2005" class="citation book cs1">T Yanao; K Takatsuka (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2M4qIUTITI0C&pg=PA98">"Effects of an intrinsic metric of molecular internal space"</a>. In Mikito Toda; Tamiki Komatsuzaki; Stuart A. Rice; Tetsuro Konishi; R. Stephen Berry (eds.). <i>Geometrical Structures Of Phase Space In Multi-dimensional Chaos: Applications to chemical reaction dynamics in complex systems</i>. Wiley. p. 98. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-71157-5" title="Special:BookSources/978-0-471-71157-5"><bdi>978-0-471-71157-5</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007142108/https://books.google.com/books?id=2M4qIUTITI0C&pg=PA98">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</span></span>. <q>As is evident from the first terms ..., which are proportional to the square 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 {\dot {\phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0446aa46e762e6b105ed6cd084731c4a37b8a3e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.467ex; height:3.009ex;" alt="{\displaystyle {\dot {\phi }}}"></span>, a kind of "centrifugal force" arises ... We call this force "democratic centrifugal force". Of course, DCF is different from the ordinary centrifugal force, and it arises even in a system of zero angular momentum.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Effects+of+an+intrinsic+metric+of+molecular+internal+space&rft.btitle=Geometrical+Structures+Of+Phase+Space+In+Multi-dimensional+Chaos%3A+Applications+to+chemical+reaction+dynamics+in+complex+systems&rft.pages=98&rft.pub=Wiley&rft.date=2005&rft.isbn=978-0-471-71157-5&rft.au=T+Yanao&rft.au=K+Takatsuka&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2M4qIUTITI0C%26pg%3DPA98&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Bini1997-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bini1997_40-0">^</a></b></span> <span class="reference-text">See p. 5 in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDonato_BiniPaolo_CariniRobert_T_Jantzen1997" class="citation journal cs1">Donato Bini; Paolo Carini; Robert T Jantzen (1997). <a rel="nofollow" class="external text" href="https://cds.cern.ch/record/503373">"The intrinsic derivative and centrifugal forces in general relativity: I. Theoretical foundations"</a>. <i>International Journal of Modern Physics D</i> (Submitted manuscript). <b>6</b> (1): 143–198. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0106014v1">gr-qc/0106014v1</a></span>. <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/1997IJMPD...6..143B">1997IJMPD...6..143B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS021827189700011X">10.1142/S021827189700011X</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:10652293">10652293</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Modern+Physics+D&rft.atitle=The+intrinsic+derivative+and+centrifugal+forces+in+general+relativity%3A+I.+Theoretical+foundations&rft.volume=6&rft.issue=1&rft.pages=143-198&rft.date=1997&rft_id=info%3Aarxiv%2Fgr-qc%2F0106014v1&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10652293%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2FS021827189700011X&rft_id=info%3Abibcode%2F1997IJMPD...6..143B&rft.au=Donato+Bini&rft.au=Paolo+Carini&rft.au=Robert+T+Jantzen&rft_id=https%3A%2F%2Fcds.cern.ch%2Frecord%2F503373&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span>. The companion paper is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDonato_BiniPaolo_CariniRobert_T_Jantzen1997" class="citation journal cs1">Donato Bini; Paolo Carini; Robert T Jantzen (1997). <a rel="nofollow" class="external text" href="https://cds.cern.ch/record/503373">"The intrinsic derivative and centrifugal forces in general relativity: II. Applications to circular orbits in some stationary axisymmetric spacetimes"</a>. <i>International Journal of Modern Physics D</i> (Submitted manuscript). <b>6</b> (1): 143–198. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0106014v1">gr-qc/0106014v1</a></span>. <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/1997IJMPD...6..143B">1997IJMPD...6..143B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS021827189700011X">10.1142/S021827189700011X</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:10652293">10652293</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210429005245/http://cds.cern.ch/record/503373">Archived</a> from the original on 2021-04-29<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-06-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Modern+Physics+D&rft.atitle=The+intrinsic+derivative+and+centrifugal+forces+in+general+relativity%3A+II.+Applications+to+circular+orbits+in+some+stationary+axisymmetric+spacetimes&rft.volume=6&rft.issue=1&rft.pages=143-198&rft.date=1997&rft_id=info%3Aarxiv%2Fgr-qc%2F0106014v1&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10652293%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2FS021827189700011X&rft_id=info%3Abibcode%2F1997IJMPD...6..143B&rft.au=Donato+Bini&rft.au=Paolo+Carini&rft.au=Robert+T+Jantzen&rft_id=https%3A%2F%2Fcds.cern.ch%2Frecord%2F503373&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Mook-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-Mook_41-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMookThomas_Vargish1987" class="citation book cs1">Mook, Delo E.; Thomas Vargish (1987). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QnJqIyk_dzIC&pg=PA47"><i>Inside relativity</i></a>. Princeton, N.J.: Princeton University Press. p. 47. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08472-6" title="Special:BookSources/0-691-08472-6"><bdi>0-691-08472-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/16089285">16089285</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007142711/https://books.google.com/books?id=QnJqIyk_dzIC&pg=PA47#v=onepage&q&f=false">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-03-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Inside+relativity&rft.place=Princeton%2C+N.J.&rft.pages=47&rft.pub=Princeton+University+Press&rft.date=1987&rft_id=info%3Aoclcnum%2F16089285&rft.isbn=0-691-08472-6&rft.aulast=Mook&rft.aufirst=Delo+E.&rft.au=Thomas+Vargish&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQnJqIyk_dzIC%26pg%3DPA47&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Scott-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-Scott_42-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFG._David_Scott1957" class="citation news cs1">G. David Scott (1957). <a rel="nofollow" class="external text" href="http://www.deepdyve.com/lp/american-association-of-physics-teachers/centrifugal-forces-and-newton-s-laws-of-motion-0bO8fgiEUy">"Centrifugal Forces and Newton's Laws of Motion"</a>. Vol. 25. American Journal of Physics. p. 325.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Centrifugal+Forces+and+Newton%27s+Laws+of+Motion&rft.volume=25&rft.pages=325&rft.date=1957&rft.au=G.+David+Scott&rft_id=http%3A%2F%2Fwww.deepdyve.com%2Flp%2Famerican-association-of-physics-teachers%2Fcentrifugal-forces-and-newton-s-laws-of-motion-0bO8fgiEUy&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Signell-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-Signell_43-0">^</a></b></span> <span class="reference-text">Signell, Peter (2002). <a rel="nofollow" class="external text" href="http://physnet.org/modules/pdf_modules/m17.pdf">"Acceleration and force in circular motion"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007142603/http://physnet.org/modules/pdf_modules/m17.pdf">Archived</a> 2024-10-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> <i>Physnet</i>. Michigan State University, "Acceleration and force in circular motion", §5b, p. 7.</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMohanty1994" class="citation book cs1">Mohanty, A. K. (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eF-H6O11fdkC&pg=PA121"><i>Fluid mechanics</i></a> (2nd ed.). New Delhi: Prentice-Hall of India. p. 121. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/81-203-0894-8" title="Special:BookSources/81-203-0894-8"><bdi>81-203-0894-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/44020947">44020947</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007142716/https://books.google.com/books?id=eF-H6O11fdkC&pg=PA121#v=onepage&q&f=false">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-03-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+mechanics&rft.place=New+Delhi&rft.pages=121&rft.edition=2nd&rft.pub=Prentice-Hall+of+India&rft.date=1994&rft_id=info%3Aoclcnum%2F44020947&rft.isbn=81-203-0894-8&rft.aulast=Mohanty&rft.aufirst=A.+K.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeF-H6O11fdkC%26pg%3DPA121&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Roche-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-Roche_45-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoche2001" class="citation journal cs1">Roche, John (September 2001). <a rel="nofollow" class="external text" href="http://www.iop.org/EJ/article/0031-9120/36/5/305/pe1505.pdf">"Introducing motion in a circle"</a> <span class="cs1-format">(PDF)</span>. <i>Physics Education</i>. <b>43</b> (5): 399–405. <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/2001PhyEd..36..399R">2001PhyEd..36..399R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0031-9120%2F36%2F5%2F305">10.1088/0031-9120/36/5/305</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:250827660">250827660</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Education&rft.atitle=Introducing+motion+in+a+circle&rft.volume=43&rft.issue=5&rft.pages=399-405&rft.date=2001-09&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A250827660%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0031-9120%2F36%2F5%2F305&rft_id=info%3Abibcode%2F2001PhyEd..36..399R&rft.aulast=Roche&rft.aufirst=John&rft_id=http%3A%2F%2Fwww.iop.org%2FEJ%2Farticle%2F0031-9120%2F36%2F5%2F305%2Fpe1505.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLloyd_William_Taylor1959" class="citation journal cs1">Lloyd William Taylor (1959). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=fp84AAAAIAAJ&q=%22centrifugal+inertial+reaction%22">"Physics, the pioneer science"</a></span>. <i>American Journal of Physics</i>. <b>1</b> (8): 173. <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/1961AmJPh..29..563T">1961AmJPh..29..563T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.1937847">10.1119/1.1937847</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Physics%2C+the+pioneer+science&rft.volume=1&rft.issue=8&rft.pages=173&rft.date=1959&rft_id=info%3Adoi%2F10.1119%2F1.1937847&rft_id=info%3Abibcode%2F1961AmJPh..29..563T&rft.au=Lloyd+William+Taylor&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dfp84AAAAIAAJ%26q%3D%2522centrifugal%2Binertial%2Breaction%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Bowser-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bowser_47-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdward_Albert_Bowser1920" class="citation book cs1">Edward Albert Bowser (1920). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mE4GAQAAIAAJ&pg=PA357"><i>An elementary treatise on analytic mechanics: with numerous examples</i></a> (25th ed.). D. Van Nostrand Company. p. 357. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007143122/https://books.google.com/books?id=mE4GAQAAIAAJ&pg=PA357#v=onepage&q&f=false">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+elementary+treatise+on+analytic+mechanics%3A+with+numerous+examples&rft.pages=357&rft.edition=25th&rft.pub=D.+Van+Nostrand+Company&rft.date=1920&rft.au=Edward+Albert+Bowser&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmE4GAQAAIAAJ%26pg%3DPA357&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Angelo-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-Angelo_48-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoseph_A._Angelo2007" class="citation book cs1">Joseph A. Angelo (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=73kNFV4sDx8C&pg=PA267"><i>Robotics: a reference guide to the new technology</i></a>. Greenwood Press. p. 267. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-57356-337-6" title="Special:BookSources/978-1-57356-337-6"><bdi>978-1-57356-337-6</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20241007143207/https://books.google.com/books?id=73kNFV4sDx8C&pg=PA267">Archived</a> from the original on 2024-10-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-09</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robotics%3A+a+reference+guide+to+the+new+technology&rft.pages=267&rft.pub=Greenwood+Press&rft.date=2007&rft.isbn=978-1-57356-337-6&rft.au=Joseph+A.+Angelo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D73kNFV4sDx8C%26pg%3DPA267&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> <li id="cite_note-Rogers-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rogers_49-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEric_M_Rogers1960" class="citation book cs1">Eric M Rogers (1960). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/physicsforinquir00roge"><i>Physics for the Inquiring Mind</i></a></span>. Princeton University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/physicsforinquir00roge/page/302">302</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+the+Inquiring+Mind&rft.pages=302&rft.pub=Princeton+University+Press&rft.date=1960&rft.au=Eric+M+Rogers&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fphysicsforinquir00roge&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACentrifugal+force" class="Z3988"></span></span> </li> </ol></div> <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=Centrifugal_force&action=edit&section=20" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" 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