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class="vector-toc-numb">2</span> <span>Ideal simple machine</span> </div> </a> <ul id="toc-Ideal_simple_machine-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Friction_and_efficiency" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Friction_and_efficiency"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Friction and efficiency</span> </div> </a> <ul id="toc-Friction_and_efficiency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compound_machines" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Compound_machines"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Compound machines</span> </div> </a> <ul id="toc-Compound_machines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Self-locking_machines" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Self-locking_machines"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Self-locking machines</span> </div> </a> <button aria-controls="toc-Self-locking_machines-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 Self-locking machines subsection</span> </button> <ul id="toc-Self-locking_machines-sublist" class="vector-toc-list"> <li id="toc-Proof" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Proof</span> </div> </a> <ul id="toc-Proof-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Modern_machine_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Modern_machine_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Modern machine theory</span> </div> </a> <button aria-controls="toc-Modern_machine_theory-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 Modern machine theory subsection</span> </button> <ul id="toc-Modern_machine_theory-sublist" class="vector-toc-list"> <li id="toc-Kinematic_chains" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kinematic_chains"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Kinematic chains</span> </div> </a> <ul id="toc-Kinematic_chains-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classification_of_machines" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classification_of_machines"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Classification of machines</span> </div> </a> <ul id="toc-Classification_of_machines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kinematic_synthesis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kinematic_synthesis"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Kinematic synthesis</span> </div> </a> <ul id="toc-Kinematic_synthesis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-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">8</span> <span>References</span> </div> </a> <ul id="toc-References-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">Simple machine</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 67 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-67" 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">67 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Einfache_Maschine" title="Einfache Maschine – Alemannic" lang="gsw" hreflang="gsw" data-title="Einfache Maschine" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A2%D9%84%D8%A9_%D8%A8%D8%B3%D9%8A%D8%B7%D8%A9" title="آلة بسيطة – Arabic" lang="ar" hreflang="ar" data-title="آلة بسيطة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Maquina_simpla" title="Maquina simpla – Aragonese" lang="an" hreflang="an" data-title="Maquina simpla" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%B8%E0%A7%B0%E0%A6%B2_%E0%A6%AF%E0%A6%A8%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A7%B0" title="সৰল যন্ত্ৰ – Assamese" lang="as" hreflang="as" data-title="সৰল যন্ত্ৰ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/M%C3%A1quina_cenciella" title="Máquina cenciella – Asturian" lang="ast" hreflang="ast" data-title="Máquina cenciella" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A6%B0%E0%A6%B2_%E0%A6%AF%E0%A6%A8%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="সরল যন্ত্র – Bangla" lang="bn" hreflang="bn" data-title="সরল যন্ত্র" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%B7%D1%8A%D0%BC" title="Прост механизъм – Bulgarian" lang="bg" hreflang="bg" data-title="Прост механизъм" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Jednostavne_ma%C5%A1ine" title="Jednostavne mašine – Bosnian" lang="bs" hreflang="bs" data-title="Jednostavne mašine" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/M%C3%A0quina_simple" title="Màquina simple – Catalan" lang="ca" hreflang="ca" data-title="Màquina simple" 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%90%D0%BD%D1%81%D0%B0%D1%82_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%B7%D0%BC" title="Ансат механизм – Chuvash" lang="cv" hreflang="cv" data-title="Ансат механизм" 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/Jednoduch%C3%BD_stroj" title="Jednoduchý stroj – Czech" lang="cs" hreflang="cs" data-title="Jednoduchý stroj" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Enkel_maskine" title="Enkel maskine – Danish" lang="da" hreflang="da" data-title="Enkel maskine" 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/Einfache_Maschine" title="Einfache Maschine – German" lang="de" hreflang="de" data-title="Einfache Maschine" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/M%C3%A1quina_simple" title="Máquina simple – Spanish" lang="es" hreflang="es" data-title="Máquina simple" 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/Simpla_ma%C5%9Dino" title="Simpla maŝino – Esperanto" lang="eo" hreflang="eo" data-title="Simpla maŝino" 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/Makina_sinple" title="Makina sinple – Basque" lang="eu" hreflang="eu" data-title="Makina sinple" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A7%D8%B4%DB%8C%D9%86%E2%80%8C%D9%87%D8%A7%DB%8C_%D8%B3%D8%A7%D8%AF%D9%87" title="ماشین‌های ساده – Persian" lang="fa" hreflang="fa" data-title="ماشین‌های ساده" 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/Machine_simple" title="Machine simple – French" lang="fr" hreflang="fr" data-title="Machine simple" 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/M%C3%A1quina_simple" title="Máquina simple – Galician" lang="gl" hreflang="gl" data-title="Máquina simple" 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/%EB%8B%A8%EC%88%9C_%EA%B8%B0%EA%B3%84" title="단순 기계 – Korean" lang="ko" hreflang="ko" data-title="단순 기계" 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/%D5%8A%D5%A1%D6%80%D5%A6_%D5%B4%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%A6%D5%B4" title="Պարզ մեխանիզմ – Armenian" lang="hy" hreflang="hy" data-title="Պարզ մեխանիզմ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%B0%E0%A4%B2_%E0%A4%AF%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="सरल यंत्र – Hindi" lang="hi" hreflang="hi" data-title="सरल यंत्र" 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/Jednostavni_stroj" title="Jednostavni stroj – Croatian" lang="hr" hreflang="hr" data-title="Jednostavni stroj" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Simpla_mashino" title="Simpla mashino – Ido" lang="io" hreflang="io" data-title="Simpla mashino" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Pesawat_sederhana" title="Pesawat sederhana – Indonesian" lang="id" hreflang="id" data-title="Pesawat sederhana" 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-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/INguxa_elula" title="INguxa elula – Zulu" lang="zu" hreflang="zu" data-title="INguxa elula" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Grunnv%C3%A9l" title="Grunnvél – Icelandic" lang="is" hreflang="is" data-title="Grunnvél" 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/Macchina_semplice" title="Macchina semplice – Italian" lang="it" hreflang="it" data-title="Macchina semplice" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%9B%D7%95%D7%A0%D7%94_%D7%A4%D7%A9%D7%95%D7%98%D7%94" title="מכונה פשוטה – Hebrew" lang="he" hreflang="he" data-title="מכונה פשוטה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%96%D0%B0%D0%B9_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%B7%D0%BC" title="Жай механизм – Kazakh" lang="kk" hreflang="kk" data-title="Жай механизм" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Machin_senp" title="Machin senp – Haitian Creole" lang="ht" hreflang="ht" data-title="Machin senp" 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-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Machin_senp" title="Machin senp – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Machin senp" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Machina_simplex" title="Machina simplex – Latin" lang="la" hreflang="la" data-title="Machina simplex" 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/Vienk%C4%81r%C5%A1s_meh%C4%81nisms" title="Vienkāršs mehānisms – Latvian" lang="lv" hreflang="lv" data-title="Vienkāršs mehānisms" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Egyszer%C5%B1_g%C3%A9p" title="Egyszerű gép – Hungarian" lang="hu" hreflang="hu" data-title="Egyszerű gép" 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%9F%D1%80%D0%BE%D1%81%D1%82%D0%B0_%D0%BC%D0%B0%D1%88%D0%B8%D0%BD%D0%B0" title="Проста машина – Macedonian" lang="mk" hreflang="mk" data-title="Проста машина" 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%B2%E0%B4%98%E0%B5%81%E0%B4%AF%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82" title="ലഘുയന്ത്രം – Malayalam" lang="ml" hreflang="ml" data-title="ലഘുയന്ത്രം" 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/Mesin_ringkas" title="Mesin ringkas – Malay" lang="ms" hreflang="ms" data-title="Mesin ringkas" 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-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Eenvoudige_machine" title="Eenvoudige machine – Dutch" lang="nl" hreflang="nl" data-title="Eenvoudige machine" 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/%E5%8D%98%E7%B4%94%E6%A9%9F%E6%A2%B0" title="単純機械 – Japanese" lang="ja" hreflang="ja" data-title="単純機械" 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/Enkel_maskin" title="Enkel maskin – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Enkel maskin" 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/Enkel_maskin" title="Enkel maskin – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Enkel maskin" 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/Maquina_simpla" title="Maquina simpla – Occitan" lang="oc" hreflang="oc" data-title="Maquina simpla" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Maashinoota_sasalphoo" title="Maashinoota sasalphoo – Oromo" lang="om" hreflang="om" data-title="Maashinoota sasalphoo" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Maszyny_proste" title="Maszyny proste – Polish" lang="pl" hreflang="pl" data-title="Maszyny proste" 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/M%C3%A1quina_simples" title="Máquina simples – Portuguese" lang="pt" hreflang="pt" data-title="Máquina simples" 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/Ma%C8%99ini_simple" title="Mașini simple – Romanian" lang="ro" hreflang="ro" data-title="Mașini simple" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Sikllalla_kuyuna" title="Sikllalla kuyuna – Quechua" lang="qu" hreflang="qu" data-title="Sikllalla kuyuna" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%B5%D0%B9%D1%88%D0%B8%D0%B9_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%B7%D0%BC" title="Простейший механизм – Russian" lang="ru" hreflang="ru" data-title="Простейший механизм" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%A1%D1%83%D0%B4%D1%83%D1%80%D0%B3%D1%83_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%B7%D0%BC" title="Судургу механизм – Yakut" lang="sah" hreflang="sah" data-title="Судургу механизм" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Simple_machine" title="Simple machine – Simple English" lang="en-simple" hreflang="en-simple" data-title="Simple machine" 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/Jednoduch%C3%BD_stroj" title="Jednoduchý stroj – Slovak" lang="sk" hreflang="sk" data-title="Jednoduchý stroj" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%DB%95%DA%A9%DB%8C%D9%86%DB%95%DB%8C_%D8%B3%D8%A7%DA%A9%D8%A7%D8%B1" title="مەکینەی ساکار – Central Kurdish" lang="ckb" hreflang="ckb" data-title="مەکینەی ساکار" 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/Jednostavna_ma%C5%A1ina" title="Jednostavna mašina – Serbian" lang="sr" hreflang="sr" data-title="Jednostavna mašina" 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/Jednostavni_stroj" title="Jednostavni stroj – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Jednostavni stroj" 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/Yksinkertainen_kone" title="Yksinkertainen kone – Finnish" lang="fi" hreflang="fi" data-title="Yksinkertainen kone" 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/Enkel_maskin" title="Enkel maskin – Swedish" lang="sv" hreflang="sv" data-title="Enkel maskin" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Makinang_payak" title="Makinang payak – Tagalog" lang="tl" hreflang="tl" data-title="Makinang payak" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8E%E0%AE%B3%E0%AE%BF%E0%AE%AF_%E0%AE%87%E0%AE%AF%E0%AE%A8%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%B0%E0%AE%AE%E0%AF%8D" title="எளிய இயந்திரம் – Tamil" lang="ta" hreflang="ta" data-title="எளிய இயந்திரம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B8%E0%B0%B0%E0%B0%B3_%E0%B0%AF%E0%B0%82%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%82" title="సరళ యంత్రం – Telugu" lang="te" hreflang="te" data-title="సరళ యంత్రం" 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%80%E0%B8%84%E0%B8%A3%E0%B8%B7%E0%B9%88%E0%B8%AD%E0%B8%87%E0%B8%81%E0%B8%A5%E0%B8%AD%E0%B8%A2%E0%B9%88%E0%B8%B2%E0%B8%87%E0%B8%87%E0%B9%88%E0%B8%B2%E0%B8%A2" title="เครื่องกลอย่างง่าย – Thai" lang="th" hreflang="th" data-title="เครื่องกลอย่างง่าย" 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/Basit_makine" title="Basit makine – Turkish" lang="tr" hreflang="tr" data-title="Basit makine" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li 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srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/30px-Semi-protection-shackle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/40px-Semi-protection-shackle.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mechanical device that changes the direction or magnitude of a force</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the concept in physics. For independent record label, see <a href="/wiki/Simple_Machines" title="Simple Machines">Simple Machines</a>. For the Internet forum software, see <a href="/wiki/Simple_Machines_Forum" title="Simple Machines Forum">Simple Machines Forum</a>. For broader coverage of this topic, see <a href="/wiki/Mechanism_(engineering)" title="Mechanism (engineering)">Mechanism (engineering)</a>.</div> <p class="mw-empty-elt"> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Six_Mechanical_Powers.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Six_Mechanical_Powers.png/250px-Six_Mechanical_Powers.png" decoding="async" width="250" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Six_Mechanical_Powers.png/375px-Six_Mechanical_Powers.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Six_Mechanical_Powers.png/500px-Six_Mechanical_Powers.png 2x" data-file-width="1005" data-file-height="915" /></a><figcaption>The six classical simple machines</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="background:#ccccff;"><a href="/wiki/History_of_technology" title="History of technology">History of technology</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">By technological eras</div><div class="sidebar-list-content mw-collapsible-content"> <dl><dt>Premodern / <a href="/wiki/Pre-industrial_society" title="Pre-industrial society">Pre-industrial</a></dt></dl> <ul><li><a href="/wiki/Prehistoric_technology" title="Prehistoric technology">Prehistoric</a></li> <li><a href="/wiki/Stone_Age" title="Stone Age">Stone Age</a> (<a href="/wiki/Lithic_technology" title="Lithic technology">lithic</a>)</li> <li><a href="/wiki/Neolithic_Revolution" title="Neolithic Revolution">Neolithic Revolution</a></li> <li><a href="/wiki/Chalcolithic" title="Chalcolithic">Copper Age</a></li> <li><a href="/wiki/Bronze_Age" title="Bronze Age">Bronze Age</a></li> <li><a href="/wiki/Iron_Age" title="Iron Age">Iron Age</a></li> <li><a href="/wiki/Ancient_technology" title="Ancient technology">Ancient</a></li></ul> <dl><dt>Modern</dt></dl> <ul><li><i><a href="/wiki/Proto-industrialization" title="Proto-industrialization">Proto-industrialization</a></i></li> <li><a href="/wiki/Industrial_Revolution" title="Industrial Revolution">First Industrial Revolution</a></li> <li><a href="/wiki/Standardization" title="Standardization">Standardization</a></li> <li><a href="/wiki/Second_Industrial_Revolution" title="Second Industrial Revolution">Second Industrial Revolution</a></li> <li><a href="/wiki/Machine_Age" title="Machine Age">Machine Age</a></li> <li><a href="/wiki/Atomic_Age" title="Atomic Age">Atomic Age</a></li> <li><a href="/wiki/Jet_Age" title="Jet Age">Jet Age</a></li> <li><a href="/wiki/Space_Age" title="Space Age">Space Age</a></li> <li><a href="/wiki/Digital_Revolution" class="mw-redirect" title="Digital Revolution">Third Industrial Revolution</a></li> <li><a href="/wiki/Digital_transformation" title="Digital transformation">Digital transformation</a></li> <li><a href="/wiki/Information_Age" title="Information Age">Information Age</a></li> <li><a href="/wiki/Fourth_Industrial_Revolution" title="Fourth Industrial Revolution">Fourth Industrial Revolution</a></li> <li><a href="/wiki/Imagination_age" class="mw-redirect" title="Imagination age">Imagination Age</a></li></ul> <dl><dt>Future</dt></dl> <ul><li><a href="/wiki/Post-industrial_society" title="Post-industrial society">Post-industrial</a></li> <li><a href="/wiki/Emerging_technologies" title="Emerging technologies">Emerging technologies</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="color: var(--color-base)">By historical regions</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/History_of_science_and_technology_in_Africa" title="History of science and technology in Africa">Ancient Africa</a></li> <li><a href="/wiki/Ancient_Egyptian_technology" title="Ancient Egyptian technology">Ancient Egypt</a></li> <li><a href="/wiki/History_of_science_and_technology_in_the_Indian_subcontinent" class="mw-redirect" title="History of science and technology in the Indian subcontinent">Indian subcontinent</a></li> <li><a href="/wiki/History_of_science_and_technology_in_China" title="History of science and technology in China">Ancient China</a></li> <li><a href="/wiki/Maya_civilization" title="Maya civilization">Maya civilization</a></li> <li><a href="/wiki/Ancient_Greek_technology" title="Ancient Greek technology">Hellenistic world</a></li> <li><a href="/wiki/Roman_technology" class="mw-redirect" title="Roman technology">Roman Empire</a></li> <li><a href="/wiki/List_of_Byzantine_inventions" title="List of Byzantine inventions">Byzantine Empire</a></li> <li><a href="/wiki/List_of_inventions_in_the_medieval_Islamic_world" title="List of inventions in the medieval Islamic world">Medieval Islamic world</a></li> <li><a href="/wiki/Arab_Agricultural_Revolution" title="Arab Agricultural Revolution">Arab Agricultural Revolution</a></li> <li><a href="/wiki/Medieval_technology" title="Medieval technology">Medieval Europe</a></li> <li><a href="/wiki/Renaissance_technology" title="Renaissance technology">Renaissance Europe</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="color: var(--color-base)">By type of technology</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/History_of_agriculture" title="History of agriculture">History of agriculture</a></li> <li><a href="/wiki/History_of_biotechnology" title="History of biotechnology">History of biotechnology</a></li> <li><a href="/wiki/History_of_communication" title="History of communication">History of communication</a></li> <li><a href="/wiki/History_of_computer_hardware" class="mw-redirect" title="History of computer hardware">History of computer hardware</a></li> <li><a href="/wiki/Electrical_engineering#History" title="Electrical engineering">History of electrical engineering</a></li> <li><a href="/wiki/History_of_manufacturing" class="mw-redirect" title="History of manufacturing">History of manufacturing</a></li> <li><a href="/wiki/Maritime_history" title="Maritime history">History of maritime</a></li> <li><a href="/wiki/History_of_materials_science" title="History of materials science">History of materials science</a></li> <li><a href="/wiki/History_of_measurement" title="History of measurement">History of measurement</a></li> <li><a href="/wiki/History_of_medicine" title="History of medicine">History of medicine</a></li> <li><a class="mw-selflink selflink">History of simple machine</a></li> <li><a href="/wiki/Nuclear_technology#History" title="Nuclear technology">History of nuclear technology</a></li> <li><a href="/wiki/History_of_transport" title="History of transport">History of transport</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="color: var(--color-base)">Technology timelines</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Timeline_of_historic_inventions" title="Timeline of historic inventions">Timeline of historic inventions</a></li> <li><a href="/wiki/Technological_revolution" title="Technological revolution">Technological revolution</a></li> <li><a href="/wiki/Category:Technology_timelines" title="Category:Technology timelines">Complete list by category</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="color: var(--color-base)"><a href="/wiki/Portal:Technology" title="Portal:Technology">Article indices</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Outline_of_technology" title="Outline of technology">Outline of technology</a></li> <li><a href="/wiki/Outline_of_prehistoric_technology" title="Outline of prehistoric technology">Outline of prehistoric technology</a></li></ul></div></div></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:History_of_technology_sidebar" title="Template:History of technology sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:History_of_technology_sidebar" title="Template talk:History of technology sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:History_of_technology_sidebar" title="Special:EditPage/Template:History of technology sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>A <b>simple machine</b> is a <a href="/wiki/Machine" title="Machine">mechanical device</a> that changes the direction or magnitude of a <a href="/wiki/Force" title="Force">force</a>.<sup id="cite_ref-Paul-Roy-Mukherjee-2005_1-0" class="reference"><a href="#cite_note-Paul-Roy-Mukherjee-2005-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In general, they can be defined as the simplest mechanisms that use <a href="/wiki/Mechanical_advantage" title="Mechanical advantage">mechanical advantage</a> (also called leverage) to multiply force.<sup id="cite_ref-Asimov1988_2-0" class="reference"><a href="#cite_note-Asimov1988-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Usually the term refers to the six classical simple machines that were defined by <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a> scientists:<sup id="cite_ref-Anderson_3-0" class="reference"><a href="#cite_note-Anderson-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Britannica1773_4-0" class="reference"><a href="#cite_note-Britannica1773-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Morris_5-0" class="reference"><a href="#cite_note-Morris-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <ul><li><a href="/wiki/Lever" title="Lever">Lever</a></li> <li><a href="/wiki/Wheel_and_axle" title="Wheel and axle">Wheel and axle</a></li> <li><a href="/wiki/Pulley" title="Pulley">Pulley</a></li> <li><a href="/wiki/Inclined_plane" title="Inclined plane">Inclined plane</a></li> <li><a href="/wiki/Wedge_(mechanical_device)" class="mw-redirect" title="Wedge (mechanical device)">Wedge</a></li> <li><a href="/wiki/Screw_(simple_machine)" class="mw-redirect" title="Screw (simple machine)">Screw</a></li></ul> <p>A simple machine uses a single applied force to do <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">work</a> against a single load force. Ignoring <a href="/wiki/Friction" title="Friction">friction</a> losses, the work done on the load is equal to the work done by the applied force. The machine can increase the amount of the output force, at the cost of a proportional decrease in the distance moved by the load. The ratio of the output to the applied force is called the <i><a href="/wiki/Mechanical_advantage" title="Mechanical advantage">mechanical advantage</a></i>. </p><p>Simple machines can be regarded as the elementary "building blocks" of which all more complicated <a href="/wiki/Machine" title="Machine">machines</a> (sometimes called "compound machines"<sup id="cite_ref-U_Virginia_elementary_curriculum_6-0" class="reference"><a href="#cite_note-U_Virginia_elementary_curriculum-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Usher_7-0" class="reference"><a href="#cite_note-Usher-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>) are composed.<sup id="cite_ref-Asimov1988_2-1" class="reference"><a href="#cite_note-Asimov1988-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> For example, wheels, levers, and pulleys are all used in the mechanism of a <a href="/wiki/Bicycle" title="Bicycle">bicycle</a>.<sup id="cite_ref-Prater1994_9-0" class="reference"><a href="#cite_note-Prater1994-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-USBureauNavalPersonnel1971_10-0" class="reference"><a href="#cite_note-USBureauNavalPersonnel1971-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The mechanical advantage of a compound machine is just the product of the mechanical advantages of the simple machines of which it is composed. </p><p>Although they continue to be of great importance in mechanics and applied science, modern mechanics has moved beyond the view of the simple machines as the ultimate building blocks of which all <a href="/wiki/Machine_(mechanical)" class="mw-redirect" title="Machine (mechanical)">machines</a> are composed, which arose in the Renaissance as a <a href="/wiki/Neoclassicism" title="Neoclassicism">neoclassical</a> amplification of <a href="/wiki/Ancient_Greece" title="Ancient Greece">ancient Greek</a> texts. The great variety and sophistication of modern machine linkages, which arose during the <a href="/wiki/Industrial_Revolution" title="Industrial Revolution">Industrial Revolution</a>, is inadequately described by these six simple categories. Various post-Renaissance authors have compiled expanded lists of "simple machines", often using terms like <i>basic machines</i>,<sup id="cite_ref-Prater1994_9-1" class="reference"><a href="#cite_note-Prater1994-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <i>compound machines</i>,<sup id="cite_ref-U_Virginia_elementary_curriculum_6-1" class="reference"><a href="#cite_note-U_Virginia_elementary_curriculum-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> or <i>machine elements</i> to distinguish them from the classical simple machines above. By the late 1800s, <a href="/wiki/Franz_Reuleaux" title="Franz Reuleaux">Franz Reuleaux</a><sup id="cite_ref-Reuleaux1876_11-0" class="reference"><a href="#cite_note-Reuleaux1876-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> had identified hundreds of machine elements, calling them <i>simple machines</i>.<sup id="cite_ref-KMODDL_12-0" class="reference"><a href="#cite_note-KMODDL-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Modern machine theory analyzes machines as <a href="/wiki/Kinematic_chain" title="Kinematic chain">kinematic chains</a> composed of elementary linkages called <a href="/wiki/Kinematic_pair" title="Kinematic pair">kinematic pairs</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> <p>The idea of a simple machine originated with the Greek philosopher <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> around the 3rd century BC, who studied the <a href="/wiki/Aristotelian_physics" title="Aristotelian physics">Archimedean</a> simple machines: lever, pulley, and <a href="/wiki/Screw_(simple_machine)" class="mw-redirect" title="Screw (simple machine)">screw</a>.<sup id="cite_ref-Asimov1988_2-2" class="reference"><a href="#cite_note-Asimov1988-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Chiu_13-0" class="reference"><a href="#cite_note-Chiu-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> He discovered the principle of <a href="/wiki/Mechanical_advantage" title="Mechanical advantage">mechanical advantage</a> in the lever.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Archimedes' famous remark with regard to the lever: "Give me a place to stand on, and I will move the Earth," (<a href="/wiki/Greek_language" title="Greek language">Greek</a>: <span lang="el">δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω</span>)<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> expresses his realization that there was no limit to the amount of force amplification that could be achieved by using mechanical advantage. Later Greek philosophers defined the classic five simple machines (excluding the <a href="/wiki/Inclined_plane" title="Inclined plane">inclined plane</a>) and were able to calculate their (ideal) mechanical advantage.<sup id="cite_ref-Usher_7-1" class="reference"><a href="#cite_note-Usher-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> For example, <a href="/wiki/Heron_of_Alexandria" class="mw-redirect" title="Heron of Alexandria">Heron of Alexandria</a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 10</span>–75 AD) in his work <i>Mechanics</i> lists five mechanisms that can "set a load in motion": <a href="/wiki/Lever" title="Lever">lever</a>, <a href="/wiki/Windlass" title="Windlass">windlass</a>, <a href="/wiki/Pulley" title="Pulley">pulley</a>, <a href="/wiki/Wedge_(mechanical_device)" class="mw-redirect" title="Wedge (mechanical device)">wedge</a>, and <a href="/wiki/Screw" title="Screw">screw</a>,<sup id="cite_ref-Chiu_13-1" class="reference"><a href="#cite_note-Chiu-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> and describes their fabrication and uses.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> However the Greeks' understanding was limited to the <a href="/wiki/Statics" title="Statics">statics</a> of simple machines (the balance of forces), and did not include <a href="/wiki/Dynamics_(mechanics)" class="mw-redirect" title="Dynamics (mechanics)">dynamics</a>, the tradeoff between force and distance, or the concept of <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>. </p><p>During the <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a> the dynamics of the <i>mechanical powers</i>, as the simple machines were called, began to be studied from the standpoint of how far they could lift a load, in addition to the force they could apply, leading eventually to the new concept of mechanical work. In 1586 Flemish engineer <a href="/wiki/Simon_Stevin" title="Simon Stevin">Simon Stevin</a> derived the mechanical advantage of the inclined plane, and it was included with the other simple machines. The complete dynamic theory of simple machines was worked out by Italian scientist <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a> in 1600 in <span title="Italian-language text"><i lang="it">Le Meccaniche</i></span> (<i>On Mechanics</i>), in which he showed the underlying mathematical similarity of the machines as force amplifiers.<sup id="cite_ref-Krebs_17-0" class="reference"><a href="#cite_note-Krebs-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Stephen_18-0" class="reference"><a href="#cite_note-Stephen-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> He was the first to explain that simple machines do not create <a href="/wiki/Energy" title="Energy">energy</a>, only transform it.<sup id="cite_ref-Krebs_17-1" class="reference"><a href="#cite_note-Krebs-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>The classic rules of sliding <a href="/wiki/Friction" title="Friction">friction</a> in machines were discovered by <a href="/wiki/Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a> (1452–1519), but were unpublished and merely documented in his notebooks, and were based on pre-Newtonian science such as believing friction was an <a href="/wiki/Aether_(classical_element)" title="Aether (classical element)">ethereal</a> fluid. They were rediscovered by <a href="/wiki/Guillaume_Amontons" title="Guillaume Amontons">Guillaume Amontons</a> (1699) and were further developed by <a href="/wiki/Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Charles-Augustin de Coulomb</a> (1785).<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Ideal_simple_machine">Ideal simple machine</h2></div> <p>If a simple machine does not dissipate energy through friction, wear or deformation, then energy is conserved and it is called an ideal simple machine. In this case, the power into the machine equals the power out, and the mechanical advantage can be calculated from its geometric dimensions. </p><p>Although each machine works differently mechanically, the way they function is similar mathematically.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> In each machine, a 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 F_{\text{in}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\text{in}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef2fc1fb77ff93181d8961f7aabd00962890b4a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.509ex;" alt="{\displaystyle F_{\text{in}}}"></span> is applied to the device at one point, and it does <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">work</a> moving a load <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\text{out}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\text{out}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68e9ff5a0b3dd2de8188c5fc3845fdb20acab0cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.102ex; height:2.509ex;" alt="{\displaystyle F_{\text{out}}}"></span> at another point.<sup id="cite_ref-Bhatnagar_21-0" class="reference"><a href="#cite_note-Bhatnagar-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Although some machines only change the direction of the force, such as a stationary pulley, most machines multiply the magnitude of the force by a factor, the <a href="/wiki/Mechanical_advantage" title="Mechanical advantage">mechanical advantage</a> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {MA} ={F_{\text{out}} \over F_{\text{in}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MA} ={F_{\text{out}} \over F_{\text{in}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4636c7f4a0ef771b56157edb38d5f9d75fc72322" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.911ex; height:5.509ex;" alt="{\displaystyle \mathrm {MA} ={F_{\text{out}} \over F_{\text{in}}}}"></span> </p><p>that can be calculated from the machine's geometry and friction. </p><p>Simple machines do not contain a source of <a href="/wiki/Energy" title="Energy">energy</a>,<sup id="cite_ref-Simmons_22-0" class="reference"><a href="#cite_note-Simmons-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> so they cannot do more <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">work</a> than they receive from the input force.<sup id="cite_ref-Bhatnagar_21-1" class="reference"><a href="#cite_note-Bhatnagar-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> A simple machine with no <a href="/wiki/Friction" title="Friction">friction</a> or <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elasticity</a> is called an <i>ideal machine</i>.<sup id="cite_ref-Gujral_23-0" class="reference"><a href="#cite_note-Gujral-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Uicker2003_24-0" class="reference"><a href="#cite_note-Uicker2003-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> Due to <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a>, in an ideal simple machine, the power output (rate of energy output) at any time <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 P_{\text{out}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{out}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48a231adb6407576ee85ecb542e45cc78a928979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.1ex; height:2.509ex;" alt="{\displaystyle P_{\text{out}}}"></span> is equal to the power input <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 P_{\text{in}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{in}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a879bbb16141edc8b2b0f0a16025342d1afdffb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.096ex; height:2.509ex;" alt="{\displaystyle P_{\text{in}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\text{out}}=P_{\text{in}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{out}}=P_{\text{in}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/298be40f65a434c35f060c5a9a5bc078831eb8be" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:10.294ex; height:2.509ex;" alt="{\displaystyle P_{\text{out}}=P_{\text{in}}\!}"></span> </p><p>The power output equals the velocity of the load <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\text{out}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\text{out}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd5718aee1ce8a100753f2506e0ad9c0a685d3cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.123ex; height:2.009ex;" alt="{\displaystyle v_{\text{out}}\,}"></span> multiplied by the load 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 P_{\text{out}}=F_{\text{out}}v_{\text{out}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{out}}=F_{\text{out}}v_{\text{out}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f3198f4bb5fb7e4ccf9392c7933e2b85ec6e28c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.424ex; height:2.509ex;" alt="{\displaystyle P_{\text{out}}=F_{\text{out}}v_{\text{out}}\,}"></span>. Similarly the power input from the applied force is equal to the velocity of the input point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\text{in}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\text{in}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5688a61934ceb1b8bd1e235b857a5fb153a6cded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.118ex; height:2.009ex;" alt="{\displaystyle v_{\text{in}}\,}"></span> multiplied by the applied 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 P_{\text{in}}=F_{\text{in}}v_{\text{in}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{in}}=F_{\text{in}}v_{\text{in}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce447ee5aeb7a0cfde44a3177c5f9026c8725dbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:12.024ex; height:2.509ex;" alt="{\displaystyle P_{\text{in}}=F_{\text{in}}v_{\text{in}}\!}"></span>. Therefore, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\text{out}}v_{\text{out}}=F_{\text{in}}v_{\text{in}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\text{out}}v_{\text{out}}=F_{\text{in}}v_{\text{in}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80be0fdf825d0bc1e1299b0ef8d9d84df2f16ef6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.153ex; height:2.509ex;" alt="{\displaystyle F_{\text{out}}v_{\text{out}}=F_{\text{in}}v_{\text{in}}\,}"></span> </p><p>So the mechanical advantage of an ideal machine <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 \mathrm {MA} _{\text{ideal}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>ideal</mtext> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MA} _{\text{ideal}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fefc839d44c08f546b0e6b96f91f44f40ae2f656" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.874ex; height:2.509ex;" alt="{\displaystyle \mathrm {MA} _{\text{ideal}}\,}"></span> is equal to the <i>velocity ratio</i>, the ratio of input velocity to output velocity </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {MA} _{\text{ideal}}={F_{\text{out}} \over F_{\text{in}}}={v_{\text{in}} \over v_{\text{out}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>ideal</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MA} _{\text{ideal}}={F_{\text{out}} \over F_{\text{in}}}={v_{\text{in}} \over v_{\text{out}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e0d1453bb2752858bbb608c4e18eac37d1fe38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.581ex; height:5.509ex;" alt="{\displaystyle \mathrm {MA} _{\text{ideal}}={F_{\text{out}} \over F_{\text{in}}}={v_{\text{in}} \over v_{\text{out}}}\,}"></span> </p><p>The <i>velocity ratio</i> is also equal to the ratio of the distances covered in any given period of time<sup id="cite_ref-Rao1_26-0" class="reference"><a href="#cite_note-Rao1-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Goyal1_27-0" class="reference"><a href="#cite_note-Goyal1-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Avison_28-0" class="reference"><a href="#cite_note-Avison-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {v_{\text{out}} \over v_{\text{in}}}={d_{\text{out}} \over d_{\text{in}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {v_{\text{out}} \over v_{\text{in}}}={d_{\text{out}} \over d_{\text{in}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e01bbb0e39ae701cf3e225e882d8211d2439c49f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.323ex; height:5.843ex;" alt="{\displaystyle {v_{\text{out}} \over v_{\text{in}}}={d_{\text{out}} \over d_{\text{in}}}}"></span> </p><p>Therefore, the mechanical advantage of an ideal machine is also equal to the <i>distance ratio</i>, the ratio of input distance moved to output distance moved </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 0px; border-width:1px; border-style: solid; border-color: black; color: inherit;text-align: center; display: table"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {MA} _{\text{ideal}}={F_{\text{out}} \over F_{\text{in}}}={d_{\text{in}} \over d_{\text{out}}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>ideal</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MA} _{\text{ideal}}={F_{\text{out}} \over F_{\text{in}}}={d_{\text{in}} \over d_{\text{out}}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84b868394415ef686c5aea8e6ae4e203af2335e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.663ex; height:5.843ex;" alt="{\displaystyle \mathrm {MA} _{\text{ideal}}={F_{\text{out}} \over F_{\text{in}}}={d_{\text{in}} \over d_{\text{out}}}\,}"></span> </p> </div> <p>This can be calculated from the geometry of the machine. For example, the mechanical advantage and distance ratio of the <a href="/wiki/Lever" title="Lever">lever</a> is equal to the ratio of its <a href="/wiki/Lever_arm" class="mw-redirect" title="Lever arm">lever arms</a>. </p><p>The mechanical advantage can be greater or less than one: </p> <ul><li>If <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 \mathrm {MA} >1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mo>></mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MA} >1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efb37baada2af4e582ef97d610393670b3fecd74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.522ex; height:2.176ex;" alt="{\displaystyle \mathrm {MA} >1\,}"></span>, the output force is greater than the input, the machine acts as a force amplifier, but the distance moved by the load <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\text{out}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\text{out}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efbc283dc7d9627770a66c8b5f0408606dbaf5dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.817ex; height:2.509ex;" alt="{\displaystyle d_{\text{out}}}"></span> is less than the distance moved by the input 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 d_{\text{in}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\text{in}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70110963d25131f8268feccb00569555ab8ea5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.2ex; height:2.509ex;" alt="{\displaystyle d_{\text{in}}\,}"></span>.</li> <li>If <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 \mathrm {MA} <1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mo><</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MA} <1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d80ab24612013b977ce7b5c6c592cd74dc7f457" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.522ex; height:2.176ex;" alt="{\displaystyle \mathrm {MA} <1\,}"></span>, the output force is less than the input, but the distance moved by the load is greater than the distance moved by the input force.</li></ul> <p>In the <a href="/wiki/Screw_(simple_machine)" class="mw-redirect" title="Screw (simple machine)">screw</a>, which uses rotational motion, the input force should be replaced by the <a href="/wiki/Torque" title="Torque">torque</a>, and the velocity by the <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> the shaft is turned. </p> <div class="mw-heading mw-heading2"><h2 id="Friction_and_efficiency">Friction and efficiency</h2></div> <p>All real machines have friction, which causes some of the input power to be dissipated as heat. If <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 P_{\text{fric}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fric</mtext> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{fric}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b641696ee15062b7f505447df87cfa11499f4751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.447ex; height:2.509ex;" alt="{\displaystyle P_{\text{fric}}\,}"></span> is the power lost to friction, from conservation of energy </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\text{in}}=P_{\text{out}}+P_{\text{fric}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fric</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{in}}=P_{\text{out}}+P_{\text{fric}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/438ba239643f61835661681c0b1ea23665f85573" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.195ex; height:2.509ex;" alt="{\displaystyle P_{\text{in}}=P_{\text{out}}+P_{\text{fric}}}"></span> </p><p>The <a href="/wiki/Mechanical_efficiency" title="Mechanical efficiency">mechanical efficiency</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 \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> of a machine (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 0<\eta \ <1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>η</mi> <mtext> </mtext> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\eta \ <1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/663a92134791c4d7440662cbb4839eb36a9f93bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.272ex; height:2.676ex;" alt="{\displaystyle 0<\eta \ <1}"></span>) is defined as the ratio of power out to the power in, and is a measure of the frictional energy losses </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\eta &\equiv {P_{\text{out}} \over P_{\text{in}}}\\P_{\text{out}}&=\eta P_{\text{in}}\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> <mi>η</mi> </mtd> <mtd> <mi></mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>η</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\eta &\equiv {P_{\text{out}} \over P_{\text{in}}}\\P_{\text{out}}&=\eta P_{\text{in}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e360715efb25dd561c4a25b437544acc96dbd29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:12.886ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}\eta &\equiv {P_{\text{out}} \over P_{\text{in}}}\\P_{\text{out}}&=\eta P_{\text{in}}\end{aligned}}}"></span> </p><p>As above, the power is equal to the product of force and velocity, so </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\text{out}}v_{\text{out}}=\eta F_{\text{in}}v_{\text{in}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mo>=</mo> <mi>η</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\text{out}}v_{\text{out}}=\eta F_{\text{in}}v_{\text{in}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04d66fc4880736c308a4d0cc9105b5da503fa495" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.935ex; height:2.676ex;" alt="{\displaystyle F_{\text{out}}v_{\text{out}}=\eta F_{\text{in}}v_{\text{in}}}"></span> </p><p>Therefore, </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 0px; border-width:1px; border-style: solid; border-color: black; color: inherit;text-align: center; display: table"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {MA} ={F_{\text{out}} \over F_{\text{in}}}=\eta {v_{\text{in}} \over v_{\text{out}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MA} ={F_{\text{out}} \over F_{\text{in}}}=\eta {v_{\text{in}} \over v_{\text{out}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2613801bf42a2578617d0d9624df5884b3b26e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.751ex; height:5.509ex;" alt="{\displaystyle \mathrm {MA} ={F_{\text{out}} \over F_{\text{in}}}=\eta {v_{\text{in}} \over v_{\text{out}}}}"></span> </p> </div> <p>So in non-ideal machines, the mechanical advantage is always less than the velocity ratio by the product with the efficiency <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 \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span>. So a machine that includes friction will not be able to move as large a load as a corresponding ideal machine using the same input force. </p> <div class="mw-heading mw-heading2"><h2 id="Compound_machines">Compound machines</h2></div> <p>A <i>compound machine</i> is a <a href="/wiki/Machine_(mechanical)" class="mw-redirect" title="Machine (mechanical)">machine</a> formed from a set of simple machines connected in series with the output force of one providing the input force to the next. For example, a <a href="/wiki/Vise" title="Vise">bench vise</a> consists of a lever (the vise's handle) in series with a screw, and a simple <a href="/wiki/Gear_train" title="Gear train">gear train</a> consists of a number of <a href="/wiki/Gear" title="Gear">gears</a> (<a href="/wiki/Wheel_and_axle" title="Wheel and axle">wheels and axles</a>) connected in series. </p><p>The mechanical advantage of a compound machine is the ratio of the output force exerted by the last machine in the series divided by the input force applied to the first machine, that is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {MA} _{\text{compound}}={F_{{\text{out}}N} \over F_{\text{in1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>compound</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> <mi>N</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in1</mtext> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MA} _{\text{compound}}={F_{{\text{out}}N} \over F_{\text{in1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/318d4ae4f0c9290175ca3f8eb65c7579033e6cee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.001ex; height:5.509ex;" alt="{\displaystyle \mathrm {MA} _{\text{compound}}={F_{{\text{out}}N} \over F_{\text{in1}}}}"></span> </p><p>Because the output force of each machine is the input of the next, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\text{out1}}=F_{\text{in2}},\;F_{\text{out2}}=F_{\text{in3}},\,\ldots \;F_{{\text{out}}K}=F_{{\text{in}}K+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out1</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in2</mtext> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out2</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in3</mtext> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mspace width="thickmathspace" /> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> <mi>K</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> <mi>K</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\text{out1}}=F_{\text{in2}},\;F_{\text{out2}}=F_{\text{in3}},\,\ldots \;F_{{\text{out}}K}=F_{{\text{in}}K+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad60d567da84ca73c558ba91faa2fafe94a7d16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:46.063ex; height:2.509ex;" alt="{\displaystyle F_{\text{out1}}=F_{\text{in2}},\;F_{\text{out2}}=F_{\text{in3}},\,\ldots \;F_{{\text{out}}K}=F_{{\text{in}}K+1}}"></span>, this mechanical advantage is also given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {MA} _{\text{compound}}={F_{\text{out1}} \over F_{\text{in1}}}{F_{\text{out2}} \over F_{\text{in2}}}{F_{\text{out3}} \over F_{\text{in3}}}\ldots {F_{{\text{out}}N} \over F_{{\text{in}}N}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>compound</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out1</mtext> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in1</mtext> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out2</mtext> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in2</mtext> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out3</mtext> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in3</mtext> </mrow> </msub> </mfrac> </mrow> <mo>…</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> <mi>N</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> <mi>N</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MA} _{\text{compound}}={F_{\text{out1}} \over F_{\text{in1}}}{F_{\text{out2}} \over F_{\text{in2}}}{F_{\text{out3}} \over F_{\text{in3}}}\ldots {F_{{\text{out}}N} \over F_{{\text{in}}N}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d560295dd6548ef48c2f58a29e1b5e0f238d32e6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.167ex; height:5.843ex;" alt="{\displaystyle \mathrm {MA} _{\text{compound}}={F_{\text{out1}} \over F_{\text{in1}}}{F_{\text{out2}} \over F_{\text{in2}}}{F_{\text{out3}} \over F_{\text{in3}}}\ldots {F_{{\text{out}}N} \over F_{{\text{in}}N}}\,}"></span> </p><p>Thus, the mechanical advantage of the compound machine is equal to the product of the mechanical advantages of the series of simple machines that form it </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {MA} _{\text{compound}}=\mathrm {MA} _{1}\mathrm {MA} _{2}\ldots \mathrm {MA} _{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>compound</mtext> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {MA} _{\text{compound}}=\mathrm {MA} _{1}\mathrm {MA} _{2}\ldots \mathrm {MA} _{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/befaa52e1f751d428137d2ea979349ac4a9e37bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.523ex; height:2.843ex;" alt="{\displaystyle \mathrm {MA} _{\text{compound}}=\mathrm {MA} _{1}\mathrm {MA} _{2}\ldots \mathrm {MA} _{N}}"></span> </p><p>Similarly, the efficiency of a compound machine is also the product of the efficiencies of the series of simple machines that form it </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\text{compound}}=\eta _{1}\eta _{2}\ldots \;\eta _{N}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>compound</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <mspace width="thickmathspace" /> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\text{compound}}=\eta _{1}\eta _{2}\ldots \;\eta _{N}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30c9de0b28b703dac27589ea4a7f93a8c75ddcee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.941ex; height:2.343ex;" alt="{\displaystyle \eta _{\text{compound}}=\eta _{1}\eta _{2}\ldots \;\eta _{N}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Self-locking_machines">Self-locking machines</h2></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:BOLT_SCREW_UBT_199.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/BOLT_SCREW_UBT_199.JPG/220px-BOLT_SCREW_UBT_199.JPG" decoding="async" width="220" height="130" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/BOLT_SCREW_UBT_199.JPG/330px-BOLT_SCREW_UBT_199.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/BOLT_SCREW_UBT_199.JPG/440px-BOLT_SCREW_UBT_199.JPG 2x" data-file-width="2172" data-file-height="1281" /></a><figcaption>The <a href="/wiki/Screw_(simple_machine)" class="mw-redirect" title="Screw (simple machine)">screw</a>'s self-locking property is the reason for its wide use in <a href="/wiki/Threaded_fastener" class="mw-redirect" title="Threaded fastener">threaded fasteners</a> like <a href="/wiki/Bolt_(fastener)" title="Bolt (fastener)">bolts</a> and <a href="/wiki/Wood_screw" class="mw-redirect" title="Wood screw">wood screws</a> </figcaption></figure> <p>In many simple machines, if the load 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 F_{\textrm {out}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\textrm {out}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abf3c6202f8b6e178e0f681510406a62e68ba7be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.102ex; height:2.509ex;" alt="{\displaystyle F_{\textrm {out}}}"></span> on the machine is high enough in relation to the input 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 F_{\textrm {in}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\textrm {in}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc3a6428e692e9cdd948a4ea8e4b2ad69733c265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.509ex;" alt="{\displaystyle F_{\textrm {in}}}"></span>, the machine will move backwards, with the load force doing work on the input force.<sup id="cite_ref-Gujral2_29-0" class="reference"><a href="#cite_note-Gujral2-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> So these machines can be used in either direction, with the driving force applied to either input point. For example, if the load force on a lever is high enough, the lever will move backwards, moving the input arm backwards against the input force. These are called <i>reversible</i>, <i>non-locking</i> or <i>overhauling</i> machines, and the backward motion is called <i>overhauling</i>. </p><p>However, in some machines, if the frictional forces are high enough, no amount of load force can move it backwards, even if the input force is zero. This is called a <i>self-locking</i>, <i>nonreversible</i>, or <i>non-overhauling</i> machine.<sup id="cite_ref-Gujral2_29-1" class="reference"><a href="#cite_note-Gujral2-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> These machines can only be set in motion by a force at the input, and when the input force is removed will remain motionless, "locked" by friction at whatever position they were left. </p><p>Self-locking occurs mainly in those machines with large areas of sliding contact between moving parts: the <a href="/wiki/Screw_(simple_machine)" class="mw-redirect" title="Screw (simple machine)">screw</a>, <a href="/wiki/Inclined_plane" title="Inclined plane">inclined plane</a>, and <a href="/wiki/Wedge_(mechanical_device)" class="mw-redirect" title="Wedge (mechanical device)">wedge</a>: </p> <ul><li>The most common example is a screw. In most screws, one can move the screw forward or backward by turning it, and one can move the nut along the shaft by turning it, but no amount of pushing the screw or the nut will cause either of them to turn.</li> <li>On an inclined plane, a load can be pulled up the plane by a sideways input force, but if the plane is not too steep and there is enough friction between load and plane, when the input force is removed the load will remain motionless and will not slide down the plane, regardless of its weight.</li> <li>A wedge can be driven into a block of wood by force on the end, such as from hitting it with a sledge hammer, forcing the sides apart, but no amount of <a href="/wiki/Compression_(physics)" title="Compression (physics)">compression</a> force from the wood walls will cause it to pop back out of the block.</li></ul> <p>A machine will be self-locking if and only if its efficiency <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 \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> is below 50%:<sup id="cite_ref-Gujral2_29-2" class="reference"><a href="#cite_note-Gujral2-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta \equiv {\frac {F_{\text{out}}/F_{\text{in}}}{d_{\text{in}}/d_{\text{out}}}}<0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> </mrow> <mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo><</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta \equiv {\frac {F_{\text{out}}/F_{\text{in}}}{d_{\text{in}}/d_{\text{out}}}}<0.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39a552849eada2f32eb054f4234c9158dc3cfb36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.537ex; height:6.509ex;" alt="{\displaystyle \eta \equiv {\frac {F_{\text{out}}/F_{\text{in}}}{d_{\text{in}}/d_{\text{out}}}}<0.5}"></span> </p><p>Whether a machine is self-locking depends on both the friction forces (<a href="/wiki/Coefficient_of_friction" class="mw-redirect" title="Coefficient of friction">coefficient of static friction</a>) between its parts, and the distance ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\textrm {in}}/d_{\textrm {out}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>out</mtext> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\textrm {in}}/d_{\textrm {out}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85c621b3796fe07530f23a0ee7421260103e9f41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.792ex; height:2.843ex;" alt="{\displaystyle d_{\textrm {in}}/d_{\textrm {out}}}"></span> (ideal mechanical advantage). If both the friction and ideal mechanical advantage are high enough, it will self-lock. </p> <div class="mw-heading mw-heading3"><h3 id="Proof">Proof</h3></div> <p>When a machine moves in the forward direction from point 1 to point 2, with the input force doing work on a load force, from conservation of energy<sup id="cite_ref-Rao_30-0" class="reference"><a href="#cite_note-Rao-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Goyal_31-0" class="reference"><a href="#cite_note-Goyal-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> the input work <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 W_{\text{1,2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>1,2</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\text{1,2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0b9ad0cf19f74d90303ae347b5bf5ae600806f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.527ex; height:2.843ex;" alt="{\displaystyle W_{\text{1,2}}}"></span> is equal to the sum of the work done on the load 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 W_{\text{load}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>load</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\text{load}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f81b2bb93b780698a814b0c44dd4c08a7b8cfbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.441ex; height:2.509ex;" alt="{\displaystyle W_{\text{load}}}"></span> and the work lost to friction <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 W_{\text{fric}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fric</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\text{fric}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54a9adff8b56498ff7cb76f08fa635b46d22581" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.761ex; height:2.509ex;" alt="{\displaystyle W_{\text{fric}}}"></span> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><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 W_{\text{1,2}}=W_{\text{load}}+W_{\text{fric}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>1,2</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>load</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fric</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\text{1,2}}=W_{\text{load}}+W_{\text{fric}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da9d215c9ca5298adf34eb034a50935b26b0bbda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.669ex; height:2.843ex;" alt="{\displaystyle W_{\text{1,2}}=W_{\text{load}}+W_{\text{fric}}}"></span></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_Eq._1" class="reference nourlexpansion" style="font-weight:bold;">Eq. 1</span>)</b></td></tr></tbody></table> <p>If the efficiency is below 50% <span class="nowrap">(<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 \eta =W_{\text{load}}/W_{\text{1,2}}<0.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>load</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>1,2</mtext> </mrow> </msub> <mo><</mo> <mn>0.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =W_{\text{load}}/W_{\text{1,2}}<0.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81bd87dd6820d6b5682660c2b3348e868947aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.469ex; height:3.009ex;" alt="{\displaystyle \eta =W_{\text{load}}/W_{\text{1,2}}<0.5}"></span>):</span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2W_{\text{load}}<W_{\text{1,2}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>load</mtext> </mrow> </msub> <mo><</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>1,2</mtext> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2W_{\text{load}}<W_{\text{1,2}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc02bae3e7a406831269804ca115f2279f4724d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.617ex; height:2.843ex;" alt="{\displaystyle 2W_{\text{load}}<W_{\text{1,2}}\,}"></span> </p><p>From <b><a href="#math_Eq._1">Eq. 1</a></b> <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}2W_{\text{load}}&<W_{\text{load}}+W_{\text{fric}}\\W_{\text{load}}&<W_{\text{fric}}\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> <mn>2</mn> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>load</mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo><</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>load</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fric</mtext> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>load</mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo><</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fric</mtext> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}2W_{\text{load}}&<W_{\text{load}}+W_{\text{fric}}\\W_{\text{load}}&<W_{\text{fric}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d8fe71d4f19ce7fbffe77a09a598c31cee68436" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.497ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}2W_{\text{load}}&<W_{\text{load}}+W_{\text{fric}}\\W_{\text{load}}&<W_{\text{fric}}\end{aligned}}}"></span> </p><p>When the machine moves backward from point 2 to point 1 with the load force doing work on the input force, the work lost to friction <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 W_{\text{fric}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fric</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\text{fric}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54a9adff8b56498ff7cb76f08fa635b46d22581" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.761ex; height:2.509ex;" alt="{\displaystyle W_{\text{fric}}}"></span> is the same </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{\text{load}}=W_{\text{2,1}}+W_{\text{fric}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>load</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>2,1</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fric</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\text{load}}=W_{\text{2,1}}+W_{\text{fric}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc7b906ec074f8a55fef89b4ab1127ae41274d98" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.669ex; height:2.843ex;" alt="{\displaystyle W_{\text{load}}=W_{\text{2,1}}+W_{\text{fric}}}"></span> </p><p>So the output work 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 W_{\text{2,1}}=W_{\text{load}}-W_{\text{fric}}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>2,1</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>load</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>fric</mtext> </mrow> </msub> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{\text{2,1}}=W_{\text{load}}-W_{\text{fric}}<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded8762bd1a1c4fb166397e7694f5e40cdafeb2a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.93ex; height:2.843ex;" alt="{\displaystyle W_{\text{2,1}}=W_{\text{load}}-W_{\text{fric}}<0}"></span> </p><p>Thus the machine self-locks, because the work dissipated in friction is greater than the work done by the load force moving it backwards even with no input force. </p> <div class="mw-heading mw-heading2"><h2 id="Modern_machine_theory">Modern machine theory</h2></div> <p>Machines are studied as mechanical systems consisting of <a href="/wiki/Actuator" title="Actuator">actuators</a> and <a href="/wiki/Mechanism_(engineering)" title="Mechanism (engineering)">mechanisms</a> that transmit forces and movement, monitored by sensors and controllers. The components of actuators and mechanisms consist of links and joints that form kinematic chains. </p> <div class="mw-heading mw-heading3"><h3 id="Kinematic_chains">Kinematic chains</h3></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Kinematics_of_Machinery_-_Figure_21.jpg" class="mw-file-description"><img alt="Illustration of a Four-bar linkage from Kinematics of Machinery, 1876" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Kinematics_of_Machinery_-_Figure_21.jpg/200px-Kinematics_of_Machinery_-_Figure_21.jpg" decoding="async" width="200" height="241" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Kinematics_of_Machinery_-_Figure_21.jpg/300px-Kinematics_of_Machinery_-_Figure_21.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Kinematics_of_Machinery_-_Figure_21.jpg/400px-Kinematics_of_Machinery_-_Figure_21.jpg 2x" data-file-width="1768" data-file-height="2128" /></a><figcaption>Illustration of a four-bar linkage from <a href="https://en.wikisource.org/wiki/The_Kinematics_of_Machinery" class="extiw" title="s:The Kinematics of Machinery">Kinematics of Machinery, 1876</a></figcaption></figure><p>Simple machines are elementary examples of <a href="/wiki/Kinematic_chain" title="Kinematic chain">kinematic chains</a> that are used to model <a href="/wiki/Mechanical_systems" class="mw-redirect" title="Mechanical systems">mechanical systems</a> ranging from the steam engine to robot manipulators. The bearings that form the fulcrum of a lever and that allow the wheel and axle and pulleys to rotate are examples of a <a href="/wiki/Kinematic_pair" title="Kinematic pair">kinematic pair</a> called a hinged joint. Similarly, the flat surface of an inclined plane and wedge are examples of the kinematic pair called a sliding joint. The screw is usually identified as its own kinematic pair called a helical joint. </p><p>Two levers, or cranks, are combined into a planar <a href="/wiki/Four-bar_linkage" title="Four-bar linkage">four-bar linkage</a> by attaching a link that connects the output of one crank to the input of another. Additional links can be attached to form a <a href="/wiki/Six-bar_linkage" title="Six-bar linkage">six-bar linkage</a> or in series to form a robot.<sup id="cite_ref-Uicker2003_24-1" class="reference"><a href="#cite_note-Uicker2003-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Classification_of_machines">Classification of machines</h3></div> <p>The identification of simple machines arises from a desire for a systematic method to invent new machines. Therefore, an important concern is how simple machines are combined to make more complex machines. One approach is to attach simple machines in series to obtain compound machines. </p><p>However, a more successful strategy was identified by <a href="/wiki/Franz_Reuleaux" title="Franz Reuleaux">Franz Reuleaux</a>, who collected and studied over 800 elementary machines. He realized that a lever, pulley, and wheel and axle are in essence the same device: a body rotating about a hinge. Similarly, an inclined plane, wedge, and screw are a block sliding on a flat surface.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p>This realization shows that it is the joints, or the connections that provide movement, that are the primary elements of a machine. Starting with four types of joints, the <a href="/wiki/Revolute_joint" title="Revolute joint">revolute joint</a>, <a href="/wiki/Prismatic_joint" title="Prismatic joint">sliding joint</a>, <a href="/wiki/Cam_(mechanism)" title="Cam (mechanism)">cam joint</a> and <a href="/wiki/Gear_train" title="Gear train">gear joint</a>, and related connections such as cables and belts, it is possible to understand a machine as an assembly of solid parts that connect these joints.<sup id="cite_ref-Uicker2003_24-2" class="reference"><a href="#cite_note-Uicker2003-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Kinematic_synthesis">Kinematic synthesis</h3></div> <p>The design of mechanisms to perform required movement and force transmission is known as <a href="/wiki/Kinematic_synthesis" title="Kinematic synthesis">kinematic synthesis</a>. This is a collection of geometric techniques for the mechanical design of <a href="/wiki/Linkage_(mechanical)" title="Linkage (mechanical)">linkages</a>, <a href="/wiki/Cam_(mechanism)" title="Cam (mechanism)">cam and follower mechanisms</a> and <a href="/wiki/Gear_train" title="Gear train">gears and gear trains</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/Linkage_(mechanical)" title="Linkage (mechanical)">Linkage (mechanical)</a></li> <li><a href="/wiki/Cam_(mechanism)" title="Cam (mechanism)">Cam and follower mechanisms</a></li> <li><a href="/wiki/Gear_train" title="Gear train">Gears and gear trains</a></li> <li><a href="/wiki/Mechanism_(engineering)" title="Mechanism (engineering)">Mechanism (engineering)</a></li> <li><a href="/wiki/Rolamite" title="Rolamite">Rolamite</a>, the only elementary machine discovered in the 20th century</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Paul-Roy-Mukherjee-2005-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Paul-Roy-Mukherjee-2005_1-0">^</a></b></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="CITEREFPaulRoyMukherjee2005" class="citation cs2">Paul, Akshoy; 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Kennedy)</i>, New York: reprinted by Dover.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+kinematics+of+machinery+%28translated+and+annotated+by+A.B.W.+Kennedy%29&rft.place=New+York&rft.pub=reprinted+by+Dover&rft.date=1963&rft.aulast=Reuleaux&rft.aufirst=F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-KMODDL-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-KMODDL_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCornell_University" class="citation cs2"><a href="/wiki/Cornell_University" title="Cornell University">Cornell University</a>, <a rel="nofollow" class="external text" href="http://kmoddl.library.cornell.edu/rx_collection.php"><i>Reuleaux Collection of Mechanisms and Machines at Cornell University</i></a>, Cornell University.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Reuleaux+Collection+of+Mechanisms+and+Machines+at+Cornell+University&rft.pub=Cornell+University&rft.au=Cornell+University&rft_id=http%3A%2F%2Fkmoddl.library.cornell.edu%2Frx_collection.php&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-Chiu-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Chiu_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Chiu_13-1"><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="CITEREFChiu2010" class="citation cs2">Chiu, Y. C. (2010), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=osNrPO3ivZoC&q=%22heron+of+alexandria%22++load+motion&pg=PA42"><i>An introduction to the History of Project Management</i></a>, Delft: Eburon Academic Publishers, p. 42, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-5972-437-2" title="Special:BookSources/978-90-5972-437-2"><bdi>978-90-5972-437-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+the+History+of+Project+Management&rft.place=Delft&rft.pages=42&rft.pub=Eburon+Academic+Publishers&rft.date=2010&rft.isbn=978-90-5972-437-2&rft.aulast=Chiu&rft.aufirst=Y.+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DosNrPO3ivZoC%26q%3D%2522heron%2Bof%2Balexandria%2522%2B%2Bload%2Bmotion%26pg%3DPA42&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" 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 id="CITEREFOstdiekBord2005" class="citation book cs1">Ostdiek, Vern; Bord, Donald (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7kz2pd14hPUC&pg=PA123"><i>Inquiry into Physics</i></a>. Thompson Brooks/Cole. p. 123. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-534-49168-0" title="Special:BookSources/978-0-534-49168-0"><bdi>978-0-534-49168-0</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">May 22,</span> 2008</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Inquiry+into+Physics&rft.pages=123&rft.pub=Thompson+Brooks%2FCole&rft.date=2005&rft.isbn=978-0-534-49168-0&rft.aulast=Ostdiek&rft.aufirst=Vern&rft.au=Bord%2C+Donald&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7kz2pd14hPUC%26pg%3DPA123&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" 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">Quoted by <a href="/wiki/Pappus_of_Alexandria" title="Pappus of Alexandria">Pappus of Alexandria</a> in <i>Synagoge</i>, Book VIII</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrizhakIgor_PenkovToivo_Pappel2004" class="citation conference cs1">Strizhak, Viktor; Igor Penkov; Toivo Pappel (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FqZvlMnjqY0C&q=%22archimedean+simple+machine%22">"Evolution of design, use, and strength calculations of screw threads and threaded joints"</a>. <i>HMM2004 International Symposium on History of Machines and Mechanisms</i>. Kluwer Academic. p. 245. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-2203-4" title="Special:BookSources/1-4020-2203-4"><bdi>1-4020-2203-4</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">May 21,</span> 2008</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Evolution+of+design%2C+use%2C+and+strength+calculations+of+screw+threads+and+threaded+joints&rft.btitle=HMM2004+International+Symposium+on+History+of+Machines+and+Mechanisms&rft.pages=245&rft.pub=Kluwer+Academic&rft.date=2004&rft.isbn=1-4020-2203-4&rft.aulast=Strizhak&rft.aufirst=Viktor&rft.au=Igor+Penkov&rft.au=Toivo+Pappel&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DFqZvlMnjqY0C%26q%3D%2522archimedean%2Bsimple%2Bmachine%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-Krebs-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-Krebs_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Krebs_17-1"><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="CITEREFKrebs2004" class="citation book cs1">Krebs, Robert E. (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MTXdplfiz-cC&q=%22mechanics+Galileo+analyzed%22&pg=PA163"><i>Groundbreaking Experiments, Inventions, and Discoveries of the Middle Ages</i></a>. Greenwood. p. 163. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-313-32433-8" title="Special:BookSources/978-0-313-32433-8"><bdi>978-0-313-32433-8</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">May 21,</span> 2008</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Groundbreaking+Experiments%2C+Inventions%2C+and+Discoveries+of+the+Middle+Ages&rft.pages=163&rft.pub=Greenwood&rft.date=2004&rft.isbn=978-0-313-32433-8&rft.aulast=Krebs&rft.aufirst=Robert+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DMTXdplfiz-cC%26q%3D%2522mechanics%2BGalileo%2Banalyzed%2522%26pg%3DPA163&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-Stephen-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Stephen_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephenLowell_Cardwell2001" class="citation book cs1">Stephen, Donald; Lowell Cardwell (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BSfpFLV1nkAC&q=%22simple+machine%22+galileo&pg=PA86"><i>Wheels, clocks, and rockets: a history of technology</i></a>. US: W. W. Norton & Company. pp. 85–87. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-32175-3" title="Special:BookSources/978-0-393-32175-3"><bdi>978-0-393-32175-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wheels%2C+clocks%2C+and+rockets%3A+a+history+of+technology&rft.place=US&rft.pages=85-87&rft.pub=W.+W.+Norton+%26+Company&rft.date=2001&rft.isbn=978-0-393-32175-3&rft.aulast=Stephen&rft.aufirst=Donald&rft.au=Lowell+Cardwell&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBSfpFLV1nkAC%26q%3D%2522simple%2Bmachine%2522%2Bgalileo%26pg%3DPA86&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArmstrong-Hélouvry1991" class="citation book cs1">Armstrong-Hélouvry, Brian (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0zk_zI3xACgC&q=friction+leonardo+da+vinci+amontons+coulomb&pg=PA10"><i>Control of machines with friction</i></a>. Springer. p. 10. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-9133-3" title="Special:BookSources/978-0-7923-9133-3"><bdi>978-0-7923-9133-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Control+of+machines+with+friction&rft.pages=10&rft.pub=Springer&rft.date=1991&rft.isbn=978-0-7923-9133-3&rft.aulast=Armstrong-H%C3%A9louvry&rft.aufirst=Brian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0zk_zI3xACgC%26q%3Dfriction%2Bleonardo%2Bda%2Bvinci%2Bamontons%2Bcoulomb%26pg%3DPA10&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></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">This fundamental insight was the subject of Galileo Galilei's 1600 work <span title="Italian-language text"><i lang="it">Le Meccaniche</i></span> (<i>On Mechanics</i>).</span> </li> <li id="cite_note-Bhatnagar-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bhatnagar_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bhatnagar_21-1"><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="CITEREFBhatnagar1996" class="citation book cs1">Bhatnagar, V. P. (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pb45yhuNltEC&q=%22simple+machine%22+%22&pg=PA28"><i>A Complete Course in Certificate Physics</i></a>. India: Pitambar. pp. 28–30. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-209-0868-0" title="Special:BookSources/978-81-209-0868-0"><bdi>978-81-209-0868-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Complete+Course+in+Certificate+Physics&rft.place=India&rft.pages=28-30&rft.pub=Pitambar&rft.date=1996&rft.isbn=978-81-209-0868-0&rft.aulast=Bhatnagar&rft.aufirst=V.+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dpb45yhuNltEC%26q%3D%2522simple%2Bmachine%2522%2B%2522%26pg%3DPA28&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-Simmons-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-Simmons_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimmonsCindy2008" class="citation book cs1">Simmons, Ron; Cindy, Barden (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=geddKUGjI3cC&q=%22simple+machine%22+%22mechanical+advantage%22&pg=PA30"><i>Discover! Work & Machines</i></a>. US: Milliken. p. 29. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4291-0947-5" title="Special:BookSources/978-1-4291-0947-5"><bdi>978-1-4291-0947-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discover%21+Work+%26+Machines&rft.place=US&rft.pages=29&rft.pub=Milliken&rft.date=2008&rft.isbn=978-1-4291-0947-5&rft.aulast=Simmons&rft.aufirst=Ron&rft.au=Cindy%2C+Barden&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DgeddKUGjI3cC%26q%3D%2522simple%2Bmachine%2522%2B%2522mechanical%2Badvantage%2522%26pg%3DPA30&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-Gujral-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-Gujral_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGujral2005" class="citation book cs1">Gujral, I. S. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JM0OG-XUyu0C&q=%22simple+machine%22+self-locking&pg=PA382"><i>Engineering Mechanics</i></a>. Firewall Media. pp. 378–380. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-7008-636-9" title="Special:BookSources/978-81-7008-636-9"><bdi>978-81-7008-636-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics&rft.pages=378-380&rft.pub=Firewall+Media&rft.date=2005&rft.isbn=978-81-7008-636-9&rft.aulast=Gujral&rft.aufirst=I.+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJM0OG-XUyu0C%26q%3D%2522simple%2Bmachine%2522%2Bself-locking%26pg%3DPA382&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-Uicker2003-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-Uicker2003_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Uicker2003_24-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Uicker2003_24-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUickerPennockShigley2003" class="citation cs2">Uicker, John J. Jr.; Pennock, Gordon R.; Shigley, Joseph E. (2003), <i>Theory of Machines and Mechanisms</i> (third ed.), New York: Oxford University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-515598-3" title="Special:BookSources/978-0-19-515598-3"><bdi>978-0-19-515598-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Machines+and+Mechanisms&rft.place=New+York&rft.edition=third&rft.pub=Oxford+University+Press&rft.date=2003&rft.isbn=978-0-19-515598-3&rft.aulast=Uicker&rft.aufirst=John+J.+Jr.&rft.au=Pennock%2C+Gordon+R.&rft.au=Shigley%2C+Joseph+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaul1979" class="citation book cs1">Paul, Burton (1979). <i>Kinematics and Dynamics of Planar Machinery</i>. Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-516062-6" title="Special:BookSources/978-0-13-516062-6"><bdi>978-0-13-516062-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kinematics+and+Dynamics+of+Planar+Machinery&rft.pub=Prentice+Hall&rft.date=1979&rft.isbn=978-0-13-516062-6&rft.aulast=Paul&rft.aufirst=Burton&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-Rao1-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rao1_26-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRaoDurgaiah2005" class="citation book cs1">Rao, S.; Durgaiah, R. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vRR4FKAkJl4C&q=%22simple+machine%22+%22velocity+ratio%22+distance&pg=PA80"><i>Engineering Mechanics</i></a>. Universities Press. p. 80. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-7371-543-3" title="Special:BookSources/978-81-7371-543-3"><bdi>978-81-7371-543-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics&rft.pages=80&rft.pub=Universities+Press&rft.date=2005&rft.isbn=978-81-7371-543-3&rft.aulast=Rao&rft.aufirst=S.&rft.au=Durgaiah%2C+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvRR4FKAkJl4C%26q%3D%2522simple%2Bmachine%2522%2B%2522velocity%2Bratio%2522%2Bdistance%26pg%3DPA80&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-Goyal1-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-Goyal1_27-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoyalRaghuvanshee2011" class="citation book cs1">Goyal, M. C.; Raghuvanshee, G. S. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qNPE9RVkSTUC&q=%22simple+machine%22+%22velocity+ratio%22+distance&pg=PA212"><i>Engineering Mechanics</i></a>. PHI Learning. p. 212. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-203-4327-6" title="Special:BookSources/978-81-203-4327-6"><bdi>978-81-203-4327-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics&rft.pages=212&rft.pub=PHI+Learning&rft.date=2011&rft.isbn=978-81-203-4327-6&rft.aulast=Goyal&rft.aufirst=M.+C.&rft.au=Raghuvanshee%2C+G.+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqNPE9RVkSTUC%26q%3D%2522simple%2Bmachine%2522%2B%2522velocity%2Bratio%2522%2Bdistance%26pg%3DPA212&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-Avison-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-Avison_28-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAvison2014" class="citation book cs1">Avison, John (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DojwZzKAvN8C&q=machine+%22velocity+ratio%22+%22distance+ratio%22&pg=PA110"><i>The World of Physics</i></a>. Nelson Thornes. p. 110. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-17-438733-6" title="Special:BookSources/978-0-17-438733-6"><bdi>978-0-17-438733-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+World+of+Physics&rft.pages=110&rft.pub=Nelson+Thornes&rft.date=2014&rft.isbn=978-0-17-438733-6&rft.aulast=Avison&rft.aufirst=John&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDojwZzKAvN8C%26q%3Dmachine%2B%2522velocity%2Bratio%2522%2B%2522distance%2Bratio%2522%26pg%3DPA110&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-Gujral2-29"><span class="mw-cite-backlink">^ <a href="#cite_ref-Gujral2_29-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Gujral2_29-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Gujral2_29-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGujral2005" class="citation book cs1">Gujral, I. S. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JM0OG-XUyu0C&q=%22simple+machine%22+self-locking&pg=PA382"><i>Engineering Mechanics</i></a>. Firewall Media. p. 382. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-7008-636-9" title="Special:BookSources/978-81-7008-636-9"><bdi>978-81-7008-636-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics&rft.pages=382&rft.pub=Firewall+Media&rft.date=2005&rft.isbn=978-81-7008-636-9&rft.aulast=Gujral&rft.aufirst=I.+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJM0OG-XUyu0C%26q%3D%2522simple%2Bmachine%2522%2Bself-locking%26pg%3DPA382&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-Rao-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rao_30-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRaoDurgaiah2005" class="citation book cs1">Rao, S.; Durgaiah, R. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vRR4FKAkJl4C&q=%22simple+machine%22+%22&pg=PA80"><i>Engineering Mechanics</i></a>. Universities Press. p. 82. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-7371-543-3" title="Special:BookSources/978-81-7371-543-3"><bdi>978-81-7371-543-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics&rft.pages=82&rft.pub=Universities+Press&rft.date=2005&rft.isbn=978-81-7371-543-3&rft.aulast=Rao&rft.aufirst=S.&rft.au=Durgaiah%2C+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvRR4FKAkJl4C%26q%3D%2522simple%2Bmachine%2522%2B%2522%26pg%3DPA80&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" class="Z3988"></span></span> </li> <li id="cite_note-Goyal-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-Goyal_31-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoyalRaghuvanshi2009" class="citation book cs1">Goyal, M. C.; Raghuvanshi, G. S. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vRR4FKAkJl4C&pg=PA82"><i>Engineering Mechanics</i></a>. New Delhi: PHI Learning Private Ltd. p. 202. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-203-3789-3" title="Special:BookSources/978-81-203-3789-3"><bdi>978-81-203-3789-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics&rft.place=New+Delhi&rft.pages=202&rft.pub=PHI+Learning+Private+Ltd.&rft.date=2009&rft.isbn=978-81-203-3789-3&rft.aulast=Goyal&rft.aufirst=M.+C.&rft.au=Raghuvanshi%2C+G.+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvRR4FKAkJl4C%26pg%3DPA82&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASimple+machine" 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">Hartenberg, R.S. & J. Denavit (1964) <a rel="nofollow" class="external text" href="http://kmoddl.library.cornell.edu/bib.php?m=23">Kinematic synthesis of linkages</a>, New York: McGraw-Hill, online link from <a href="/wiki/Cornell_University" title="Cornell University">Cornell University</a>.</span> </li> </ol></div></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 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