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class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Sadržaj</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">premjesti</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sakrij</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Početak</div> </a> </li> <li id="toc-Definicija:_rad_sile" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definicija:_rad_sile"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definicija: rad sile</span> </div> </a> <ul id="toc-Definicija:_rad_sile-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Specijalni_slučaj_"sila_puta_put"" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Specijalni_slučaj_"sila_puta_put""> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Specijalni slučaj "sila puta put"</span> </div> </a> <ul id="toc-Specijalni_slučaj_"sila_puta_put"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zakon_o_promjeni_kinetičke_energije" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zakon_o_promjeni_kinetičke_energije"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Zakon o promjeni kinetičke energije</span> </div> </a> <ul id="toc-Zakon_o_promjeni_kinetičke_energije-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Objašnjenje_definicije_rada_sile" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Objašnjenje_definicije_rada_sile"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Objašnjenje definicije rada sile</span> </div> </a> <button aria-controls="toc-Objašnjenje_definicije_rada_sile-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>Sadržaj cjeline Objašnjenje definicije rada sile</span> </button> <ul id="toc-Objašnjenje_definicije_rada_sile-sublist" class="vector-toc-list"> <li id="toc-Značaj_tangencijalne_komponente_sile" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Značaj_tangencijalne_komponente_sile"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Značaj tangencijalne komponente sile</span> </div> </a> <ul id="toc-Značaj_tangencijalne_komponente_sile-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uloga_hvatišta_sile" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uloga_hvatišta_sile"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Uloga hvatišta sile</span> </div> </a> <ul id="toc-Uloga_hvatišta_sile-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Promjenjivu_silu_treba_integrirati" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Promjenjivu_silu_treba_integrirati"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Promjenjivu silu treba integrirati</span> </div> </a> <ul id="toc-Promjenjivu_silu_treba_integrirati-sublist" class="vector-toc-list"> <li id="toc-Tumačenje_i_primjer_integrala_rada" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Tumačenje_i_primjer_integrala_rada"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.1</span> <span>Tumačenje i primjer integrala rada</span> </div> </a> <ul id="toc-Tumačenje_i_primjer_integrala_rada-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Opis_rada_skalarnim_produktom" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Opis_rada_skalarnim_produktom"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Opis rada skalarnim produktom</span> </div> </a> <ul id="toc-Opis_rada_skalarnim_produktom-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Opaska:_pređeni_put_ili_pomak?" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Opaska:_pređeni_put_ili_pomak?"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Opaska: pređeni put ili pomak?</span> </div> </a> <ul id="toc-Opaska:_pređeni_put_ili_pomak?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rad_momenta_sile" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rad_momenta_sile"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Rad momenta sile</span> </div> </a> <ul id="toc-Rad_momenta_sile-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Izvori" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Izvori"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Izvori</span> </div> </a> <ul id="toc-Izvori-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="Sadržaj" 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="Prikaz sadržaja stranice" > <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">Prikaz sadržaja stranice</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">Rad (fizika)</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="Idi na druge jezične varijante članka. Dostupan je na 102 jezika" > <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-102" 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">102 jezika</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Arbeid" title="Arbeid – afrikaans" lang="af" hreflang="af" data-title="Arbeid" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Arbeit_(Physik)" title="Arbeit (Physik) – švicarski njemački" lang="gsw" hreflang="gsw" data-title="Arbeit (Physik)" data-language-autonym="Alemannisch" data-language-local-name="švicarski njemački" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%A5%E1%88%AB" title="ሥራ – amharski" lang="am" hreflang="am" data-title="ሥራ" data-language-autonym="አማርኛ" data-language-local-name="amharski" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Treballo_mecanico" title="Treballo mecanico – aragonski" lang="an" hreflang="an" data-title="Treballo mecanico" data-language-autonym="Aragonés" data-language-local-name="aragonski" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B4%D8%BA%D9%84_(%D9%81%D9%8A%D8%B2%D9%8A%D8%A7%D8%A1)" title="شغل (فيزياء) – arapski" lang="ar" hreflang="ar" data-title="شغل (فيزياء)" data-language-autonym="العربية" data-language-local-name="arapski" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%95%E0%A6%BE%E0%A7%B0%E0%A7%8D%E0%A6%AF%E0%A7%8D%E0%A6%AF" title="কাৰ্য্য – asamski" lang="as" hreflang="as" data-title="কাৰ্য্য" data-language-autonym="অসমীয়া" data-language-local-name="asamski" 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/Trabayu_(f%C3%ADsica)" title="Trabayu (física) – asturijski" lang="ast" hreflang="ast" data-title="Trabayu (física)" data-language-autonym="Asturianu" data-language-local-name="asturijski" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Mexaniki_i%C5%9F" title="Mexaniki iş – azerbajdžanski" lang="az" hreflang="az" data-title="Mexaniki iş" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdžanski" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D1%80%D0%B0%D0%B1%D0%BE%D1%82%D0%B0" title="Механічная работа – bjeloruski" lang="be" hreflang="be" data-title="Механічная работа" data-language-autonym="Беларуская" data-language-local-name="bjeloruski" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D1%8D%D1%85%D0%B0%D0%BD%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D0%BF%D1%80%D0%B0%D1%86%D0%B0" title="Мэханічная праца – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Мэханічная праца" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D1%80%D0%B0%D0%B1%D0%BE%D1%82%D0%B0" title="Механична работа – bugarski" lang="bg" hreflang="bg" data-title="Механична работа" data-language-autonym="Български" data-language-local-name="bugarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A6%BE%E0%A6%9C_(%E0%A6%AA%E0%A6%A6%E0%A6%BE%E0%A6%B0%E0%A7%8D%E0%A6%A5%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8)" title="কাজ (পদার্থবিজ্ঞান) – bangla" lang="bn" hreflang="bn" data-title="কাজ (পদার্থবিজ্ঞান)" data-language-autonym="বাংলা" data-language-local-name="bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Labour_un_nerzh" title="Labour un nerzh – bretonski" lang="br" hreflang="br" data-title="Labour un nerzh" data-language-autonym="Brezhoneg" data-language-local-name="bretonski" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Rad_(fizika)" title="Rad (fizika) – bosanski" lang="bs" hreflang="bs" data-title="Rad (fizika)" data-language-autonym="Bosanski" data-language-local-name="bosanski" 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/Treball_(f%C3%ADsica)" title="Treball (física) – katalonski" lang="ca" hreflang="ca" data-title="Treball (física)" data-language-autonym="Català" data-language-local-name="katalonski" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%A9%D8%A7%D8%B1_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="کار (فیزیک) – soranski kurdski" lang="ckb" hreflang="ckb" data-title="کار (فیزیک)" data-language-autonym="کوردی" data-language-local-name="soranski kurdski" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Pr%C3%A1ce_(fyzika)" title="Práce (fyzika) – češki" lang="cs" hreflang="cs" data-title="Práce (fyzika)" data-language-autonym="Čeština" data-language-local-name="češki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-csb mw-list-item"><a href="https://csb.wikipedia.org/wiki/Rob%C3%B2ta_(fizyka)" title="Robòta (fizyka) – kašupski" lang="csb" hreflang="csb" data-title="Robòta (fizyka)" data-language-autonym="Kaszëbsczi" data-language-local-name="kašupski" class="interlanguage-link-target"><span>Kaszëbsczi</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%C4%95%C3%A7" title="Механикăлла ĕç – čuvaški" lang="cv" hreflang="cv" data-title="Механикăлла ĕç" data-language-autonym="Чӑвашла" data-language-local-name="čuvaški" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Arbejde_(fysik)" title="Arbejde (fysik) – danski" lang="da" hreflang="da" data-title="Arbejde (fysik)" data-language-autonym="Dansk" data-language-local-name="danski" 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/Arbeit_(Physik)" title="Arbeit (Physik) – njemački" lang="de" hreflang="de" data-title="Arbeit (Physik)" data-language-autonym="Deutsch" data-language-local-name="njemački" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%88%CF%81%CE%B3%CE%BF_(%CF%86%CF%85%CF%83%CE%B9%CE%BA%CE%AE)" title="Έργο (φυσική) – grčki" lang="el" hreflang="el" data-title="Έργο (φυσική)" data-language-autonym="Ελληνικά" data-language-local-name="grčki" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Work_(physics)" title="Work (physics) – engleski" lang="en" hreflang="en" data-title="Work (physics)" data-language-autonym="English" data-language-local-name="engleski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Laboro_(fiziko)" title="Laboro (fiziko) – esperanto" lang="eo" hreflang="eo" data-title="Laboro (fiziko)" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Trabajo_(f%C3%ADsica)" title="Trabajo (física) – španjolski" lang="es" hreflang="es" data-title="Trabajo (física)" data-language-autonym="Español" data-language-local-name="španjolski" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Mehaaniline_t%C3%B6%C3%B6" title="Mehaaniline töö – estonski" lang="et" hreflang="et" data-title="Mehaaniline töö" data-language-autonym="Eesti" data-language-local-name="estonski" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Lan_(fisika)" title="Lan (fisika) – baskijski" lang="eu" hreflang="eu" data-title="Lan (fisika)" data-language-autonym="Euskara" data-language-local-name="baskijski" 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/%DA%A9%D8%A7%D8%B1_(%D9%81%DB%8C%D8%B2%DB%8C%DA%A9)" title="کار (فیزیک) – perzijski" lang="fa" hreflang="fa" data-title="کار (فیزیک)" data-language-autonym="فارسی" data-language-local-name="perzijski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ty%C3%B6_(fysiikka)" title="Työ (fysiikka) – finski" lang="fi" hreflang="fi" data-title="Työ (fysiikka)" data-language-autonym="Suomi" data-language-local-name="finski" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Travail_d%27une_force" title="Travail d'une force – francuski" lang="fr" hreflang="fr" data-title="Travail d'une force" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Werk" title="Werk – sjevernofrizijski" lang="frr" hreflang="frr" data-title="Werk" data-language-autonym="Nordfriisk" data-language-local-name="sjevernofrizijski" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Obair_(fisic)" title="Obair (fisic) – irski" lang="ga" hreflang="ga" data-title="Obair (fisic)" data-language-autonym="Gaeilge" data-language-local-name="irski" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Obair_(fiosaigs)" title="Obair (fiosaigs) – škotski gaelski" lang="gd" hreflang="gd" data-title="Obair (fiosaigs)" data-language-autonym="Gàidhlig" data-language-local-name="škotski gaelski" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Traballo_(f%C3%ADsica)" title="Traballo (física) – galicijski" lang="gl" hreflang="gl" data-title="Traballo (física)" data-language-autonym="Galego" data-language-local-name="galicijski" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Obbyr_(fishig)" title="Obbyr (fishig) – manski" lang="gv" hreflang="gv" data-title="Obbyr (fishig)" data-language-autonym="Gaelg" data-language-local-name="manski" class="interlanguage-link-target"><span>Gaelg</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A2%D7%91%D7%95%D7%93%D7%94_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)" title="עבודה (פיזיקה) – hebrejski" lang="he" hreflang="he" data-title="עבודה (פיזיקה)" data-language-autonym="עברית" data-language-local-name="hebrejski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80)" title="कार्य (भौतिकी) – hindski" lang="hi" hreflang="hi" data-title="कार्य (भौतिकी)" data-language-autonym="हिन्दी" data-language-local-name="hindski" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Travay_(fizik)" title="Travay (fizik) – haićanski kreolski" lang="ht" hreflang="ht" data-title="Travay (fizik)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haićanski kreolski" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Mechanikai_munka" title="Mechanikai munka – mađarski" lang="hu" hreflang="hu" data-title="Mechanikai munka" data-language-autonym="Magyar" data-language-local-name="mađarski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%B7%D5%AD%D5%A1%D5%BF%D5%A1%D5%B6%D6%84_(%D6%86%D5%AB%D5%A6%D5%AB%D5%AF%D5%A1)" title="Աշխատանք (ֆիզիկա) – armenski" lang="hy" hreflang="hy" data-title="Աշխատանք (ֆիզիկա)" data-language-autonym="Հայերեն" data-language-local-name="armenski" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Usaha_(fisika)" title="Usaha (fisika) – indonezijski" lang="id" hreflang="id" data-title="Usaha (fisika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijski" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Vinna_(e%C3%B0lisfr%C3%A6%C3%B0i)" title="Vinna (eðlisfræði) – islandski" lang="is" hreflang="is" data-title="Vinna (eðlisfræði)" data-language-autonym="Íslenska" data-language-local-name="islandski" 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/Lavoro_(fisica)" title="Lavoro (fisica) – talijanski" lang="it" hreflang="it" data-title="Lavoro (fisica)" data-language-autonym="Italiano" data-language-local-name="talijanski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BB%95%E4%BA%8B_(%E7%89%A9%E7%90%86%E5%AD%A6)" title="仕事 (物理学) – japanski" lang="ja" hreflang="ja" data-title="仕事 (物理学)" data-language-autonym="日本語" data-language-local-name="japanski" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%A3%E1%83%A8%E1%83%90%E1%83%9D%E1%83%91%E1%83%90" title="მუშაობა – gruzijski" lang="ka" hreflang="ka" data-title="მუშაობა" data-language-autonym="ქართული" data-language-local-name="gruzijski" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%B6%D2%B1%D0%BC%D1%8B%D1%81" title="Механикалық жұмыс – kazaški" lang="kk" hreflang="kk" data-title="Механикалық жұмыс" data-language-autonym="Қазақша" data-language-local-name="kazaški" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%80%E1%9E%98%E1%9F%92%E1%9E%98%E1%9E%93%E1%9F%92%E1%9E%8F_(%E1%9E%9A%E1%9E%BC%E1%9E%94%E1%9E%9C%E1%9E%B7%E1%9E%91%E1%9F%92%E1%9E%99%E1%9E%B6)" title="កម្មន្ត (រូបវិទ្យា) – kmerski" lang="km" hreflang="km" data-title="កម្មន្ត (រូបវិទ្យា)" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="kmerski" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%95%E0%B3%86%E0%B2%B2%E0%B2%B8" title="ಕೆಲಸ – karnatački" lang="kn" hreflang="kn" data-title="ಕೆಲಸ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="karnatački" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%BC_(%EB%AC%BC%EB%A6%AC%ED%95%99)" title="일 (물리학) – korejski" lang="ko" hreflang="ko" data-title="일 (물리학)" data-language-autonym="한국어" data-language-local-name="korejski" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Kar_(fiz%C3%AEk)" title="Kar (fizîk) – kurdski" lang="ku" hreflang="ku" data-title="Kar (fizîk)" data-language-autonym="Kurdî" data-language-local-name="kurdski" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D0%B6%D1%83%D0%BC%D1%83%D1%88" title="Механикалык жумуш – kirgiski" lang="ky" hreflang="ky" data-title="Механикалык жумуш" data-language-autonym="Кыргызча" data-language-local-name="kirgiski" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Labor_(physica)" title="Labor (physica) – latinski" lang="la" hreflang="la" data-title="Labor (physica)" data-language-autonym="Latina" data-language-local-name="latinski" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Labora_(fisica)" title="Labora (fisica) – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Labora (fisica)" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/La%C3%B4_(fisica)" title="Laô (fisica) – Lombard" lang="lmo" hreflang="lmo" data-title="Laô (fisica)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Mechaninis_darbas" title="Mechaninis darbas – litavski" lang="lt" hreflang="lt" data-title="Mechaninis darbas" data-language-autonym="Lietuvių" data-language-local-name="litavski" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Darbs_(fizika)" title="Darbs (fizika) – latvijski" lang="lv" hreflang="lv" data-title="Darbs (fizika)" data-language-autonym="Latviešu" data-language-local-name="latvijski" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B1%D0%BE%D1%82%D0%B0_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA%D0%B0)" title="Работа (физика) – makedonski" lang="mk" hreflang="mk" data-title="Работа (физика)" data-language-autonym="Македонски" data-language-local-name="makedonski" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%B5%E0%B5%83%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%BF" title="പ്രവൃത്തി – malajalamski" lang="ml" hreflang="ml" data-title="പ്രവൃത്തി" data-language-autonym="മലയാളം" data-language-local-name="malajalamski" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D0%B6%D0%B8%D0%BB_(%D1%84%D0%B8%D0%B7%D0%B8%D0%BA)" title="Ажил (физик) – mongolski" lang="mn" hreflang="mn" data-title="Ажил (физик)" data-language-autonym="Монгол" data-language-local-name="mongolski" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0)" title="कार्य (भौतिकशास्त्र) – marathski" lang="mr" hreflang="mr" data-title="कार्य (भौतिकशास्त्र)" data-language-autonym="मराठी" data-language-local-name="marathski" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kerja_(fizik)" title="Kerja (fizik) – malajski" lang="ms" hreflang="ms" data-title="Kerja (fizik)" data-language-autonym="Bahasa Melayu" data-language-local-name="malajski" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%9C%E1%80%AF%E1%80%95%E1%80%BA_(%E1%80%9B%E1%80%B0%E1%80%95%E1%80%97%E1%80%B1%E1%80%92)" title="အလုပ် (ရူပဗေဒ) – burmanski" lang="my" hreflang="my" data-title="အလုပ် (ရူပဗေဒ)" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="burmanski" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%95%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0)" title="कार्य (भौतिकशास्त्र) – nepalski" lang="ne" hreflang="ne" data-title="कार्य (भौतिकशास्त्र)" data-language-autonym="नेपाली" data-language-local-name="nepalski" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%B5%E0%A5%87%E0%A4%B2%E0%A5%88_(%E0%A4%B8%E0%A4%A8%E0%A5%8D_%E0%A5%A7%E0%A5%AF%E0%A5%AF%E0%A5%AE%E0%A4%AF%E0%A4%BE_%E0%A4%B8%E0%A4%82%E0%A4%95%E0%A4%BF%E0%A4%AA%E0%A4%BE)" title="वेलै (सन् १९९८या संकिपा) – newari" lang="new" hreflang="new" data-title="वेलै (सन् १९९८या संकिपा)" data-language-autonym="नेपाल भाषा" data-language-local-name="newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Arbeid_(natuurkunde)" title="Arbeid (natuurkunde) – nizozemski" lang="nl" hreflang="nl" data-title="Arbeid (natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="nizozemski" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Arbeid_i_fysikk" title="Arbeid i fysikk – norveški nynorsk" lang="nn" hreflang="nn" data-title="Arbeid i fysikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norveški nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Arbeid_(fysikk)" title="Arbeid (fysikk) – norveški bokmål" lang="nb" hreflang="nb" data-title="Arbeid (fysikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norveški bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Trabalh_(fisica)" title="Trabalh (fisica) – okcitanski" lang="oc" hreflang="oc" data-title="Trabalh (fisica)" data-language-autonym="Occitan" data-language-local-name="okcitanski" 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/Dalagaa" title="Dalagaa – oromski" lang="om" hreflang="om" data-title="Dalagaa" data-language-autonym="Oromoo" data-language-local-name="oromski" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%B0%E0%A8%AE_(%E0%A8%AD%E0%A9%8C%E0%A8%A4%E0%A8%BF%E0%A8%95_%E0%A8%B5%E0%A8%BF%E0%A8%97%E0%A8%BF%E0%A8%86%E0%A8%A8)" title="ਕੰਮ (ਭੌਤਿਕ ਵਿਗਿਆਨ) – pandžapski" lang="pa" hreflang="pa" data-title="ਕੰਮ (ਭੌਤਿਕ ਵਿਗਿਆਨ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pandžapski" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Praca_(fizyka)" title="Praca (fizyka) – poljski" lang="pl" hreflang="pl" data-title="Praca (fizyka)" data-language-autonym="Polski" data-language-local-name="poljski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Travaj" title="Travaj – Piedmontese" lang="pms" hreflang="pms" data-title="Travaj" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Trabalho_(f%C3%ADsica)" title="Trabalho (física) – portugalski" lang="pt" hreflang="pt" data-title="Trabalho (física)" data-language-autonym="Português" data-language-local-name="portugalski" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Ruray" title="Ruray – kečuanski" lang="qu" hreflang="qu" data-title="Ruray" data-language-autonym="Runa Simi" data-language-local-name="kečuanski" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Lucru_mecanic" title="Lucru mecanic – rumunjski" lang="ro" hreflang="ro" data-title="Lucru mecanic" data-language-autonym="Română" data-language-local-name="rumunjski" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%80%D0%B0%D0%B1%D0%BE%D1%82%D0%B0" title="Механическая работа – ruski" lang="ru" hreflang="ru" data-title="Механическая работа" data-language-autonym="Русский" data-language-local-name="ruski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%A0%E1%B1%9F%E1%B1%B9%E1%B1%A2%E1%B1%A4" title="ᱠᱟᱹᱢᱤ – santalski" lang="sat" hreflang="sat" data-title="ᱠᱟᱹᱢᱤ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="santalski" class="interlanguage-link-target"><span>ᱥᱟᱱᱛᱟᱲᱤ</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%DA%AA%D9%85_(%D8%B7%D8%A8%D8%B9%D9%8A%D8%A7%D8%AA)" title="ڪم (طبعيات) – sindski" lang="sd" hreflang="sd" data-title="ڪم (طبعيات)" data-language-autonym="سنڌي" data-language-local-name="sindski" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Rad_(fizika)" title="Rad (fizika) – srpsko-hrvatski" lang="sh" hreflang="sh" data-title="Rad (fizika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srpsko-hrvatski" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9A%E0%B7%8F%E0%B6%BB%E0%B7%8A%E0%B6%BA%E0%B6%BA_(%E0%B6%B7%E0%B7%9E%E0%B6%AD%E0%B7%92%E0%B6%9A_%E0%B7%80%E0%B7%92%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B7%80)" title="කාර්යය (භෞතික විද්‍යාව) – sinhaleški" lang="si" hreflang="si" data-title="කාර්යය (භෞතික විද්‍යාව)" data-language-autonym="සිංහල" data-language-local-name="sinhaleški" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Work_(physics)" title="Work (physics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Work (physics)" 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/Pr%C3%A1ca_(fyzika)" title="Práca (fyzika) – slovački" lang="sk" hreflang="sk" data-title="Práca (fyzika)" data-language-autonym="Slovenčina" data-language-local-name="slovački" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Delo_(fizika)" title="Delo (fizika) – slovenski" lang="sl" hreflang="sl" data-title="Delo (fizika)" data-language-autonym="Slovenščina" data-language-local-name="slovenski" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Basa_(fundoyetsimba)" title="Basa (fundoyetsimba) – shona" lang="sn" hreflang="sn" data-title="Basa (fundoyetsimba)" data-language-autonym="ChiShona" data-language-local-name="shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Puna_(fizik%C3%AB)" title="Puna (fizikë) – albanski" lang="sq" hreflang="sq" data-title="Puna (fizikë)" data-language-autonym="Shqip" data-language-local-name="albanski" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D1%87%D0%BA%D0%B8_%D1%80%D0%B0%D0%B4" title="Механички рад – srpski" lang="sr" hreflang="sr" data-title="Механички рад" data-language-autonym="Српски / srpski" data-language-local-name="srpski" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Usaha_m%C3%A9kanik" title="Usaha mékanik – sundanski" lang="su" hreflang="su" data-title="Usaha mékanik" data-language-autonym="Sunda" data-language-local-name="sundanski" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Arbete_(fysik)" title="Arbete (fysik) – švedski" lang="sv" hreflang="sv" data-title="Arbete (fysik)" data-language-autonym="Svenska" data-language-local-name="švedski" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AF%87%E0%AE%B2%E0%AF%88_(%E0%AE%87%E0%AE%AF%E0%AE%B1%E0%AF%8D%E0%AE%AA%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D)" title="வேலை (இயற்பியல்) – tamilski" lang="ta" hreflang="ta" data-title="வேலை (இயற்பியல்)" data-language-autonym="தமிழ்" data-language-local-name="tamilski" 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%AA%E0%B0%A8%E0%B0%BF" title="పని – teluški" lang="te" hreflang="te" data-title="పని" data-language-autonym="తెలుగు" data-language-local-name="teluški" 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%B8%87%E0%B8%B2%E0%B8%99_(%E0%B8%9F%E0%B8%B4%E0%B8%AA%E0%B8%B4%E0%B8%81%E0%B8%AA%E0%B9%8C)" title="งาน (ฟิสิกส์) – tajlandski" lang="th" hreflang="th" data-title="งาน (ฟิสิกส์)" data-language-autonym="ไทย" data-language-local-name="tajlandski" 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/%C4%B0%C5%9F_(fizik)" title="İş (fizik) – turski" lang="tr" hreflang="tr" data-title="İş (fizik)" data-language-autonym="Türkçe" data-language-local-name="turski" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B1%D0%BE%D1%82%D0%B0_(%D1%84%D1%96%D0%B7%D0%B8%D0%BA%D0%B0)" title="Робота (фізика) – ukrajinski" lang="uk" hreflang="uk" data-title="Робота (фізика)" data-language-autonym="Українська" data-language-local-name="ukrajinski" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%A9%D8%A7%D9%85_(%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C%D8%A7%D8%AA)" title="کام (طبیعیات) – urdski" lang="ur" hreflang="ur" data-title="کام (طبیعیات)" data-language-autonym="اردو" data-language-local-name="urdski" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/C%C3%B4ng_(v%E1%BA%ADt_l%C3%BD_h%E1%BB%8Dc)" title="Công (vật lý học) – vijetnamski" lang="vi" hreflang="vi" data-title="Công (vật lý học)" data-language-autonym="Tiếng Việt" data-language-local-name="vijetnamski" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/Ligg%C3%A9ey_(j%C3%ABmm)" title="Liggéey (jëmm) – volof" lang="wo" hreflang="wo" data-title="Liggéey (jëmm)" data-language-autonym="Wolof" data-language-local-name="volof" class="interlanguage-link-target"><span>Wolof</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8A%9F" title="功 – wu kineski" lang="wuu" hreflang="wuu" data-title="功" data-language-autonym="吴语" data-language-local-name="wu kineski" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%90%D7%A8%D7%91%D7%A2%D7%98_(%D7%A4%D7%99%D7%96%D7%99%D7%A7)" title="ארבעט (פיזיק) – jidiš" lang="yi" hreflang="yi" data-title="ארבעט (פיזיק)" data-language-autonym="ייִדיש" data-language-local-name="jidiš" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8A%9F" title="功 – kineski" lang="zh" hreflang="zh" data-title="功" data-language-autonym="中文" data-language-local-name="kineski" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%8A%9F" title="功 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="功" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Kang_(bu%CC%8Dt-l%C3%AD-ha%CC%8Dk)" title="Kang (bu̍t-lí-ha̍k) – min nan kineski" lang="nan" hreflang="nan" data-title="Kang (bu̍t-lí-ha̍k)" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan kineski" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8A%9F" title="功 – kantonski" lang="yue" hreflang="yue" data-title="功" data-language-autonym="粵語" data-language-local-name="kantonski" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q42213#sitelinks-wikipedia" title="Poveznice na druge jezike" class="wbc-editpage">Uredi poveznice</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Imenski prostori"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Rad_(fizika)" title="Pogledaj sadržaj [c]" accesskey="c"><span>Stranica</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Razgovor:Rad_(fizika)" rel="discussion" title="Razgovorna stranica [t]" accesskey="t"><span>Razgovor</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Promijeni jezičnu varijantu" > <label 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href="/w/index.php?title=Rad_(fizika)&veaction=edit" title="Uredite ovu stranicu [v]" accesskey="v"><span>Uredi</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Rad_(fizika)&action=edit" title="Uredite izvorni kôd ove stranice [e]" accesskey="e"><span>Uredi kôd</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Rad_(fizika)&action=history" title="Ranije izmjene na ovoj stranici [h]" accesskey="h"><span>Vidi povijest</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Pomagala"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Pomagala" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Pomagala</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Pomagala</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">premjesti</button> <button 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<div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Izvor: Wikipedija</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="hr" dir="ltr"><table class="infobox" style="width:300px; float: right; clear: right; text-align:center; font-size:90%;"> <tbody><tr style="background:#DCF0FF"> <td><b><a href="/wiki/Klasi%C4%8Dna_mehanika" title="Klasična mehanika">Klasična mehanika</a></b> </td></tr> <tr> <td><div style="padding-top: 7px; padding-bottom: 4px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} ={\frac {\mathrm {d} }{\mathrm {d} t}}(m\mathbf {v} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} ={\frac {\mathrm {d} }{\mathrm {d} t}}(m\mathbf {v} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f9db5d305b630d9a9299ad0e7557114a7e3be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.01ex; height:5.509ex;" alt="{\displaystyle \mathbf {F} ={\frac {\mathrm {d} }{\mathrm {d} t}}(m\mathbf {v} )}"></span><br /><small><a href="/wiki/Newtonovi_zakoni_gibanja" title="Newtonovi zakoni gibanja">drugi Newtonov zakon</a></small></div> </td></tr> <tr style="line-height: 150%;"> <td><a href="/wiki/Povijest_klasi%C4%8Dne_mehanike" title="Povijest klasične mehanike">povijest klasične mehanike</a><br /><a href="/wiki/Kronologija_klasi%C4%8Dne_mehanike" title="Kronologija klasične mehanike">kronologija klasične mehanike</a> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF;">Grane </th></tr> <tr style="line-height: 150%;"> <td><a href="/wiki/Statika" title="Statika">statika</a> • <a href="/wiki/Dinamika" title="Dinamika">dinamika</a>/<a href="/w/index.php?title=Kinetika&action=edit&redlink=1" class="new" title="Kinetika (stranica ne postoji)">kinetika</a> • <a href="/wiki/Kinematika" title="Kinematika">kinematika</a> • <a href="/w/index.php?title=Primjenjena_mehanika&action=edit&redlink=1" class="new" title="Primjenjena mehanika (stranica ne postoji)">primjenjena mehanika</a> • <a href="/wiki/Nebeska_mehanika" title="Nebeska mehanika">nebeska mehanika</a> • <a href="/wiki/Mehanika_kontinuuma" title="Mehanika kontinuuma">mehanika kontinuuma</a> • <a href="/wiki/Statisti%C4%8Dka_mehanika" title="Statistička mehanika">statistička mehanika</a> </td></tr></tbody></table> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF;">Formulacije </th></tr> <tr style="line-height: 150%;"> <td><div style="text-align:left;"> <ul><li><a href="/wiki/Newtonovi_zakoni_gibanja" title="Newtonovi zakoni gibanja">Newtonova mehanika</a> (vektorska mehanika)</li> <li><a href="/wiki/Analiti%C4%8Dka_mehanika" title="Analitička mehanika">analitička mehanika</a>: <ul><li><a href="/w/index.php?title=Lagrangeova_mehanika&action=edit&redlink=1" class="new" title="Lagrangeova mehanika (stranica ne postoji)">Lagrangeova mehanika</a></li> <li><a href="/w/index.php?title=Hamiltonova_mehanika&action=edit&redlink=1" class="new" title="Hamiltonova mehanika (stranica ne postoji)">Hamiltonova mehanika</a></li></ul></li></ul></div> </td></tr></tbody></table> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF;">Osnovni koncepti </th></tr> <tr style="line-height: 150%;"> <td><a href="/wiki/Prostor" title="Prostor">prostor</a> • <a href="/wiki/Vrijeme_(fizika)" title="Vrijeme (fizika)">vrijeme</a> • <a href="/wiki/Brzina" title="Brzina">brzina</a> • <a href="/wiki/Masa" title="Masa">masa</a> • <a href="/wiki/Ubrzanje" title="Ubrzanje">ubrzanje</a> • <a href="/wiki/Gravitacija" title="Gravitacija">gravitacija</a> • <a href="/wiki/Sila" title="Sila">sila</a> • <a href="/wiki/Impuls_sile" title="Impuls sile">impuls sile</a> • <a href="/wiki/Spreg_sila" title="Spreg sila">spreg sila</a>/<a href="/wiki/Moment_sile" title="Moment sile">moment sile</a> • <a href="/wiki/Koli%C4%8Dina_gibanja" title="Količina gibanja">količina gibanja</a> • <a href="/wiki/Kutna_koli%C4%8Dina_gibanja" title="Kutna količina gibanja">kutna količina gibanja</a> • <a href="/wiki/Tromost" title="Tromost">tromost</a> • <a href="/wiki/Moment_tromosti" class="mw-redirect" title="Moment tromosti">moment tromosti</a> • <a href="/wiki/Referentni_okvir" class="mw-redirect" title="Referentni okvir">referentni okvir</a> • <a href="/wiki/Energija" title="Energija">energija</a> • <a href="/wiki/Kineti%C4%8Dka_energija" title="Kinetička energija">kinetička energija</a> • <a href="/wiki/Potencijalna_energija" title="Potencijalna energija">potencijalna energija</a> • <a class="mw-selflink selflink">rad</a> • <a href="/w/index.php?title=Virtualni_rad&action=edit&redlink=1" class="new" title="Virtualni rad (stranica ne postoji)">virtualni rad</a> • <a href="/w/index.php?title=D%27Alembertovo_na%C4%8Delo&action=edit&redlink=1" class="new" title="D'Alembertovo načelo (stranica ne postoji)">D'Alembertovo načelo</a> • <a href="/wiki/Princip_stacionarnog_djelovanja" title="Princip stacionarnog djelovanja">princip stacionarnog djelovanja</a> </td></tr></tbody></table> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF;">Ključne teme </th></tr> <tr style="line-height: 150%;"> <td><a href="/w/index.php?title=Kruto_tijelo&action=edit&redlink=1" class="new" title="Kruto tijelo (stranica ne postoji)">kruto tijelo</a> • <a href="/w/index.php?title=Dinamika_krutog_tijela&action=edit&redlink=1" class="new" title="Dinamika krutog tijela (stranica ne postoji)">dinamika krutog tijela</a> • <a href="/w/index.php?title=Eulerove_jednad%C5%BEbe_gibanja&action=edit&redlink=1" class="new" title="Eulerove jednadžbe gibanja (stranica ne postoji)">Eulerove jednadžbe gibanja</a> • <a href="/wiki/Gibanje" title="Gibanje">gibanje</a> • <a href="/wiki/Newtonovi_zakoni_gibanja" title="Newtonovi zakoni gibanja">Newtonovi zakoni gibanja</a> • <a href="/wiki/Newtonov_zakon_gravitacije" title="Newtonov zakon gravitacije">Newtonov zakon gravitacije</a> • <a href="/w/index.php?title=Jednad%C5%BEbe_gibanja&action=edit&redlink=1" class="new" title="Jednadžbe gibanja (stranica ne postoji)">jednadžbe gibanja</a> • <a href="/wiki/Inercijski_referentni_okvir" title="Inercijski referentni okvir">inercijski referentni okvir</a> • <a href="/w/index.php?title=Neinercijski_referentni_okvir&action=edit&redlink=1" class="new" title="Neinercijski referentni okvir (stranica ne postoji)">neinercijski referentni okvir</a> • <a href="/w/index.php?title=Rotiraju%C4%87i_referentni_okvir&action=edit&redlink=1" class="new" title="Rotirajući referentni okvir (stranica ne postoji)">rotirajući referentni okvir</a> • <a href="/w/index.php?title=Fiktivna_sila&action=edit&redlink=1" class="new" title="Fiktivna sila (stranica ne postoji)">fiktivna sila</a> • <a href="/w/index.php?title=Mehanika_ravninskog_gibanja_krutog_tijela&action=edit&redlink=1" class="new" title="Mehanika ravninskog gibanja krutog tijela (stranica ne postoji)">mehanika ravninskog gibanja krutog tijela</a> • <a href="/w/index.php?title=Pomak_(vektor)&action=edit&redlink=1" class="new" title="Pomak (vektor) (stranica ne postoji)">pomak (vektor)</a> • <a href="/w/index.php?title=Relativna_brzina&action=edit&redlink=1" class="new" title="Relativna brzina (stranica ne postoji)">relativna brzina</a> • <a href="/wiki/Trenje" title="Trenje">trenje</a> • <a href="/w/index.php?title=Jednostavno_harmonijsko_gibanje&action=edit&redlink=1" class="new" title="Jednostavno harmonijsko gibanje (stranica ne postoji)">jednostavno harmonijsko gibanje</a> • <a href="/wiki/Harmonijski_oscilator" class="mw-redirect" title="Harmonijski oscilator">harmonijski oscilator</a> • <a href="/wiki/Vibracije" title="Vibracije">vibracije</a> • <a href="/wiki/Prigu%C5%A1enje" title="Prigušenje">prigušenje</a> • <a href="/w/index.php?title=Koeficijent_prigu%C5%A1enja&action=edit&redlink=1" class="new" title="Koeficijent prigušenja (stranica ne postoji)">koeficijent prigušenja</a> • <a href="/w/index.php?title=Rotacija_tijela_oko_nepomi%C4%8Dne_osi&action=edit&redlink=1" class="new" title="Rotacija tijela oko nepomične osi (stranica ne postoji)">Rotacijsko gibanje</a> • <a href="/wiki/Kru%C5%BEno_gibanje" title="Kružno gibanje">Kružno gibanje</a> • <a href="/wiki/Jednoliko_gibanje_po_kru%C5%BEnici" title="Jednoliko gibanje po kružnici">jednoliko kružno gibanje</a> • <a href="/w/index.php?title=Nejednoliko_kru%C5%BEno_gibanje&action=edit&redlink=1" class="new" title="Nejednoliko kružno gibanje (stranica ne postoji)">nejednoliko kružno gibanje</a> • <a href="/wiki/Centripetalna_sila" title="Centripetalna sila">centripetalna sila</a> • <a href="/wiki/Centrifugalna_sila" title="Centrifugalna sila">centrifugalna sila</a> • <a href="/w/index.php?title=Centrifugalna_sila_(rotacijski_referentni_okvir)&action=edit&redlink=1" class="new" title="Centrifugalna sila (rotacijski referentni okvir) (stranica ne postoji)">centrifugalna sila (rotacijski referentni okvir)</a> • <a href="/w/index.php?title=Reaktivna_centrifugalna_sila&action=edit&redlink=1" class="new" title="Reaktivna centrifugalna sila (stranica ne postoji)">reaktivna centrifugalna sila</a> • <a href="/wiki/Coriolisov_u%C4%8Dinak" title="Coriolisov učinak">Coriolisov učinak</a> • <a href="/w/index.php?title=Fizi%C4%8Dko_njihalo&action=edit&redlink=1" class="new" title="Fizičko njihalo (stranica ne postoji)">fizičko njihalo</a> • <a href="/w/index.php?title=Rotacijska_brzina&action=edit&redlink=1" class="new" title="Rotacijska brzina (stranica ne postoji)">rotacijska brzina</a> • <a href="/wiki/Kutno_ubrzanje" title="Kutno ubrzanje">kutno ubrzanje</a> • <a href="/wiki/Kutna_brzina" title="Kutna brzina">kutna brzina</a> • <a href="/wiki/Kutna_frekvencija" title="Kutna frekvencija">kutna frekvencija</a> • <a href="/w/index.php?title=Kutni_pomak&action=edit&redlink=1" class="new" title="Kutni pomak (stranica ne postoji)">kutni pomak</a> </td></tr></tbody></table> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF;">Znanstvenici </th></tr> <tr style="line-height: 150%;"> <td><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> • <a href="/w/index.php?title=Jeremiah_Horrocks&action=edit&redlink=1" class="new" title="Jeremiah Horrocks (stranica ne postoji)">Jeremiah Horrocks</a> • <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> • <a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">Jean le Rond d'Alembert</a> • <a href="/w/index.php?title=Alexis_Clairaut&action=edit&redlink=1" class="new" title="Alexis Clairaut (stranica ne postoji)">Alexis Clairaut</a> • <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a> • <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> • <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> • <a href="/w/index.php?title=Sim%C3%A9on-Denis_Poisson&action=edit&redlink=1" class="new" title="Siméon-Denis Poisson (stranica ne postoji)">Siméon-Denis Poisson</a> </td></tr></tbody></table> </td></tr> </tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Gravity_gravita_grave.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/7d/Gravity_gravita_grave.gif" decoding="async" width="289" height="400" class="mw-file-element" data-file-width="289" data-file-height="400" /></a><figcaption>Kod <a href="/wiki/Slobodni_pad" title="Slobodni pad">slobodnog pada</a> rad je jednak <a href="/wiki/Mno%C5%BEenje" title="Množenje">umnošku</a> <a href="/wiki/Gravitacija" title="Gravitacija">sile teže</a> i puta, u ovom slučaju visine <i>h</i> ili <i>W = m ∙ g ∙ h</i>.</figcaption></figure> <p><b>Rad</b> ili <b>mehanički rad</b> (oznaka <i>W</i>) je <a href="/wiki/Fizikalna_veli%C4%8Dina" class="mw-redirect" title="Fizikalna veličina">fizikalna veličina</a> koja opisuje djelovanje <a href="/wiki/Sila" title="Sila">sile</a>, određena kao <a href="/wiki/Mno%C5%BEenje" title="Množenje">umnožak</a> sile i <a href="/wiki/Duljina" title="Duljina">prijeđenog puta</a> u smjeru duž kojega se obavlja rad. U općem slučaju rad se određuje ili <a href="/wiki/Skalarni_umno%C5%BEak" title="Skalarni umnožak">skalarnim produktom</a> vektora sile <i>F</i> i puta <i>s</i>:  </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W={\vec {F}}\cdot {\vec {s}}=F\cdot s\cdot cos\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>s</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>F</mi> <mo>⋅</mo> <mi>s</mi> <mo>⋅</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W={\vec {F}}\cdot {\vec {s}}=F\cdot s\cdot cos\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6030f5ba46f5648cda3d61f9ee2fd07c507ee959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.207ex; height:2.843ex;" alt="{\displaystyle W={\vec {F}}\cdot {\vec {s}}=F\cdot s\cdot cos\alpha }"></span></dd></dl> <p>ako je iznos sile <a href="/wiki/Konstanta" title="Konstanta">konstantan</a> (<i>α</i> je <a href="/wiki/Kut" title="Kut">kut</a> između vektora sile i puta), ili <a href="/wiki/Integral" title="Integral">integralom</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{A}^{B}F\cdot \mathrm {d} s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mi>F</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{A}^{B}F\cdot \mathrm {d} s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e1ab64f7d6645ad43ab543284222749789e28bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.667ex; height:6.176ex;" alt="{\displaystyle W=\int _{A}^{B}F\cdot \mathrm {d} s}"></span></dd></dl> <p>ako se iznos sile mijenja duž puta od točke <i>A</i> do <i>B</i>. <a href="/wiki/Skalarni_produkt" class="mw-redirect" title="Skalarni produkt">Skalarni produkt</a> je jednak nuli ako je sila koja djeluje na neko tijelo okomita na put <i>cos α = 0</i>, to jest sila koja djeluje okomito na smjer <a href="/wiki/Gibanje" title="Gibanje">gibanja</a> ne obavlja nikakav rad. Izvršeni rad tada ovisi samo o krajnjim točkama puta i potpuno je neovisan o njegovu obliku. </p><p>Rad se razlikuje po vrstama: rad <a href="/wiki/Elektri%C4%8Dni_naboj" title="Električni naboj">električki nabijene</a> čestice u <a href="/wiki/Elektromagnetsko_polje" title="Elektromagnetsko polje">elektromagnetskom polju</a>, rad <a href="/wiki/Materijalna_to%C4%8Dka" title="Materijalna točka">materijalne točke</a> u <a href="/wiki/Gravitacija" title="Gravitacija">gravitacijskom polju</a>, rad pri kružnim <a href="/wiki/Termodinamika" title="Termodinamika">termodinamičkim</a> procesima i drugo. Također se pri primjeni razlikuju strojni rad, tehnički rad otvorenoga ili zatvorenoga sustava, proizvodni rad, i tako dalje. Mjerna je jedinica rada <a href="/wiki/D%C5%BEul" title="Džul">džul</a> (J). Mogu se upotrebljavati i svi umnošci mjernih jedinica za silu i duljinu, na primjer <a href="/wiki/Njutn" title="Njutn">njutn</a><a href="/wiki/Metar" title="Metar">metar</a> (Nm), ili jedinica za snagu i vrijeme, na primjer <a href="/wiki/Vat" title="Vat">vat</a><a href="/wiki/Sekunda" title="Sekunda">sekunda</a> (Ws) ili kilovatsat (kWh).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Rad je <a href="/wiki/Skalar" title="Skalar">skalarna</a> <a href="/wiki/Fizikalna_veli%C4%8Dina" class="mw-redirect" title="Fizikalna veličina">fizikalna veličina</a> koja je blisko povezana s <a href="/wiki/Energija" title="Energija">energijom</a>, te bi se mogao definirati kao prenošenje energije s jednog tijela na drugo ili iz jednog sustava u drugi. No, takva je definicija neprikladna ako se pojam rada koristi prilikom definiranja pojma energije, što je teško izbjeći barem za pojašnjavanje apstraktnijih definicija energije (a u <a href="/wiki/Klasi%C4%8Dna_mehanika" title="Klasična mehanika">klasičnoj mehanici</a> najjednostavnije je definirati energiju tijela upravo kao sposobnost tijela da izvrši rad). Umjesto toga, moguće je (a često prikladnije i iz drugih praktičnih razloga) definirati rad kao rad <a href="/wiki/Sila" title="Sila">sile</a>, jer se i prenošenje energije može opisivati kao proces koji posreduju sile kojima tijela djeluju jedno na drugo. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definicija:_rad_sile">Definicija: rad sile</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rad_(fizika)&veaction=edit&section=1" title="Uredi odlomak: Definicija: rad sile" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rad_(fizika)&action=edit&section=1" title="Uredi kôd odjeljka Definicija: rad sile"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Frame"><a href="/wiki/Datoteka:Rad_po_krivulji.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/hr/5/53/Rad_po_krivulji.JPG" decoding="async" width="252" height="181" class="mw-file-element" data-file-width="252" data-file-height="181" /></a><figcaption>Ilustracija i oznake uz definiciju rada sile.</figcaption></figure> <p><b>Rad sile</b> je <a href="/wiki/Integral_(matematika)" class="mw-redirect" title="Integral (matematika)">integral</a> tangencijalne skalarne komponente sile duž putanje njezinog hvatišta: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{A}^{B}F\cos \alpha \,\mathrm {d} s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mi>F</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{A}^{B}F\cos \alpha \,\mathrm {d} s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94934a049aff7158d8dd5ad1a2b7a5a99765deee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.748ex; height:6.176ex;" alt="{\displaystyle W=\int _{A}^{B}F\cos \alpha \,\mathrm {d} s}"></span></dd></dl> <p>gdje je <i>F</i> iznos sile, α je kut između smjera sile <sup><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 \scriptstyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02da86fc900a1585e3a01a08d897a7f8a715e064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.309ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {\vec {F}}}"></span></sup> i smjera gibanja (to jest brzine <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 \scriptstyle {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a22ea722fa1095eb63f41c95385e74b4eae216b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.85ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {\vec {v}}}"></span>) hvatišta sile (pa je "<i>F</i> cos α" tangencijalna skalarna komponenta sile), dok je <i>s</i> put hvatišta sile od točke A do točke B.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>To je najopćenitija definicija za rad proizvoljne sile koja može bilo kako mijenjati iznos i smjer duž putanje proizvoljnoga oblika, ako se djelovanje sile može reducirati na jednu točku (koja se zove hvatište sile); u protivnom, treba posebno promatrati komponente sile i računati njihove radove za sve točke na koje sila djeluje. Pripadajuća formula (integral) može se pisati i drugačije (kako se pokazuje kasnije u tekstu). </p> <div class="mw-heading mw-heading2"><h2 id="Specijalni_slučaj_"sila_puta_put""><span id="Specijalni_slu.C4.8Daj_.22sila_puta_put.22"></span>Specijalni slučaj "sila puta put"</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rad_(fizika)&veaction=edit&section=2" title="Uredi odlomak: Specijalni slučaj "sila puta put"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rad_(fizika)&action=edit&section=2" title="Uredi kôd odjeljka Specijalni slučaj "sila puta put""><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Najjednostavnija formula za rad sile</b>, koja je najbolje polazište za razumijevanje pojma rada, vrijedi npr. u slučaju kada konstantna sila djeluje na tijelo koje se translacijski (tj. bez rotacije) giba u smjeru njezina djelovanja. Tada je (kao što proizlazi i iz gornjeg integrala): </p> <figure class="mw-halign-left" typeof="mw:File/Frame"><a href="/wiki/Datoteka:Rad_konstantne_sile_u_smjeru_gibanja.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/hr/b/b3/Rad_konstantne_sile_u_smjeru_gibanja.JPG" decoding="async" width="261" height="76" class="mw-file-element" data-file-width="261" data-file-height="76" /></a><figcaption>Rad konstantne sile u smjeru gibanja.</figcaption></figure> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=F\cdot s\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>F</mi> <mo>⋅</mo> <mi>s</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=F\cdot s\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8672cdc1dd3404cd0a286d0b1ae84702cdb6a041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.431ex; height:2.176ex;" alt="{\displaystyle W=F\cdot s\,}"></span></dd></dl> <p>gdje je <i>F</i> iznos sile, dok je <i>s</i> pređeni put. Na taj se slučaj odnosi definicija iz osnovne škole "rad je sila puta put", koja ne uzima u obzir da je sila <a href="/wiki/Vektor" title="Vektor">vektor</a> (što je ovdje irelevantno zato što su i sila i gibanje u istom smjeru koji se ne mijenja), niti precizira da treba promatrati put hvatišta sile (jer se sve točke tijela jednako gibaju, pa može biti i "put tijela"). Npr. ako sila od 5 <a href="/wiki/Njutn" title="Njutn">N</a> vuče tijelo na putu od 3 <a href="/wiki/Metar" title="Metar">m</a>, ona izvrši rad <i>W</i> = <i>F ∙ s</i> = 5 N ∙ 3 m = 15 J. Odatle se vidi da je <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> <a href="/wiki/Mjerna_jedinica" title="Mjerna jedinica">mjerna jedinica</a> za rad, <a href="/wiki/D%C5%BEul" title="Džul">džul</a> (J), skraćenica za umnožak "Nm". </p><p>Ipak, ova jednostavna <a href="/wiki/Jednad%C5%BEba" title="Jednadžba">jednadžba</a> nije ograničena samo na <a href="/wiki/Jednoliko_pravocrtno_gibanje" title="Jednoliko pravocrtno gibanje">pravocrtno gibanje</a>. Ona vrijedi uvijek kada se iznos sile ne mijenja, a hvatište sile se giba točno u smjeru djelovanja sile (koji se može po volji mijenjati). </p> <div class="mw-heading mw-heading2"><h2 id="Zakon_o_promjeni_kinetičke_energije"><span id="Zakon_o_promjeni_kineti.C4.8Dke_energije"></span>Zakon o promjeni kinetičke energije</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rad_(fizika)&veaction=edit&section=3" title="Uredi odlomak: Zakon o promjeni kinetičke energije" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rad_(fizika)&action=edit&section=3" title="Uredi kôd odjeljka Zakon o promjeni kinetičke energije"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Odnos rada i promjene energije ključna je odrednica za razumijevanje definicije rada sile. Ako je u prethodnom jednostavnom primjeru sila <i>F</i> jedina sila koja djeluje na tijelo mase <i>m</i> koje je do tada mirovalo (nije imalo <a href="/wiki/Kineti%C4%8Dka_energija" title="Kinetička energija">kinetičke energije</a>), tijelo na putu <i>s</i> ima stalnu <a href="/wiki/Akceleracija" class="mw-redirect" title="Akceleracija">akceleraciju</a> <i>a</i> = <i>F</i>/<i>m</i> i giba se jednoliko ubrzano, te na kraju puta <i>s</i> = <i>a ∙ t</i><sup>2</sup>/2 postiže <a href="/wiki/Brzina" title="Brzina">brzinu</a> <i>v</i> = <i>a ∙ t</i>. Odatle se lako vidi da je rad sile jednak kinetičkoj energiji koju tijelo dobije na tom putu: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=F\cdot s=m\cdot a\cdot {\frac {a\cdot t^{2}}{2}}=m\cdot {\frac {(a\cdot t)^{2}}{2}}={\frac {m\cdot v^{2}}{2}}=E_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>F</mi> <mo>⋅</mo> <mi>s</mi> <mo>=</mo> <mi>m</mi> <mo>⋅</mo> <mi>a</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>⋅</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo>⋅</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=F\cdot s=m\cdot a\cdot {\frac {a\cdot t^{2}}{2}}=m\cdot {\frac {(a\cdot t)^{2}}{2}}={\frac {m\cdot v^{2}}{2}}=E_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b057cae0abca92b5a4e4c2b3b3c24986c2b668" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.414ex; height:5.843ex;" alt="{\displaystyle W=F\cdot s=m\cdot a\cdot {\frac {a\cdot t^{2}}{2}}=m\cdot {\frac {(a\cdot t)^{2}}{2}}={\frac {m\cdot v^{2}}{2}}=E_{k}}"></span></dd></dl> <p>Poopćenje tog rezultata (moglo bi se dobiti iz opće formule za rad sile, uz malo više računa) zove se zakon o promjeni kinetičke energije: rad svih sila koje djeluju na kruto tijelo jednak je promjeni njegove kinetičke energije. Pritom treba imati na umu da kinetička energija tijela ne mora biti samo translacijska (kako je opisana u gornjem jednostavnom primjeru), nego može imati i rotacijski dio (osim u slučaju čestice, tj. tijela zanemarivih dimenzija). </p> <div class="mw-heading mw-heading2"><h2 id="Objašnjenje_definicije_rada_sile"><span id="Obja.C5.A1njenje_definicije_rada_sile"></span>Objašnjenje definicije rada sile</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rad_(fizika)&veaction=edit&section=4" title="Uredi odlomak: Objašnjenje definicije rada sile" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rad_(fizika)&action=edit&section=4" title="Uredi kôd odjeljka Objašnjenje definicije rada sile"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Objašnjenje opće formule za rad sile polazi od opisane veze s energijom: formula je konstruirana upravo tako da promjena kinetičke energije bude jednaka ukupnom radu svih sila. </p> <div class="mw-heading mw-heading3"><h3 id="Značaj_tangencijalne_komponente_sile"><span id="Zna.C4.8Daj_tangencijalne_komponente_sile"></span>Značaj tangencijalne komponente sile</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rad_(fizika)&veaction=edit&section=5" title="Uredi odlomak: Značaj tangencijalne komponente sile" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rad_(fizika)&action=edit&section=5" title="Uredi kôd odjeljka Značaj tangencijalne komponente sile"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Frame"><a href="/wiki/Datoteka:Rastav_sile_na_tangencijalnu_i_normalnu_komponentu.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/hr/1/1d/Rastav_sile_na_tangencijalnu_i_normalnu_komponentu.JPG" decoding="async" width="301" height="151" class="mw-file-element" data-file-width="301" data-file-height="151" /></a><figcaption>Rastav sile na tangencijalnu i normalnu vektorsku komponentu</figcaption></figure> <p>Prva posljedica te veze jest da sila koja djeluje na česticu okomito na smjer njezinoga gibanja (kaže se: normalna sila; primjer: <a href="/wiki/Centripetalna_sila" title="Centripetalna sila">centripetalna sila</a>) ne vrši rad - jer ne mijenja iznos brzine (nego samo njezin smjer) pa ne utječe na kinetičku energiju čestice. Sila koja leži na pravcu gibanja čestice (kaže se: tangencijalna sila) vrši pozitivan rad ako je u smjeru gibanja jer povećava brzinu a time i kinetičku energiju čestice, odnosno negativan rad ako je u suprotnom smjeru od gibanja jer umanjuje kinetičku energiju. Zato se proizvoljni vektor sile <sup><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 \scriptstyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02da86fc900a1585e3a01a08d897a7f8a715e064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.309ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {\vec {F}}}"></span></sup> prikaže kao zbroj tangencijalne i normalne sile (formalnije rečeno: rastavi na tangencijalnu i normalnu vektorsku komponentu), od čega se za izračun rada koristi samo tangencijalna. Pritom je <i>F</i> cos α tangencijalna skalarna komponenta vektora sile <sup><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 \scriptstyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02da86fc900a1585e3a01a08d897a7f8a715e064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.309ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {\vec {F}}}"></span></sup>, to jest to je broj koji je jednak iznosu tangencijalne vektorske komponente sile ako je ona u smjeru gibanja, odnosno njezinom negativnom iznosu ako je u suprotnom smjeru (što daje i odgovarajući predznak rada). </p> <figure typeof="mw:File/Frame"><a href="/wiki/Datoteka:Sila_daje_tijelu_translacijsko_i_kutno_ubrzanje.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/hr/7/7e/Sila_daje_tijelu_translacijsko_i_kutno_ubrzanje.JPG" decoding="async" width="161" height="255" class="mw-file-element" data-file-width="161" data-file-height="255" /></a><figcaption>Sila daje tijelu <a href="/wiki/Translacija" title="Translacija">translacijsko</a> i <a href="/wiki/Kutno_ubrzanje" title="Kutno ubrzanje">kutno ubrzanje</a>.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Uloga_hvatišta_sile"><span id="Uloga_hvati.C5.A1ta_sile"></span>Uloga hvatišta sile</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rad_(fizika)&veaction=edit&section=6" title="Uredi odlomak: Uloga hvatišta sile" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rad_(fizika)&action=edit&section=6" title="Uredi kôd odjeljka Uloga hvatišta sile"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Računanje rada pomoću puta koji prelazi hvatište sile (dok druge točke tijela mogu prelaziti različite puteve) također je posljedica opisane veze s energijom. Sila koja djeluje na tijelo daje akceleraciju <i>a</i><sub>CM</sub> = <i>F</i>/<i>m</i> njegovom centru masa (točki koja se na Zemlji izvrsno podudara s <a href="/wiki/Te%C5%BEi%C5%A1te" title="Težište">težištem</a> tijela); ta akceleracija opisuje kako se mijenja brzina centra masa, a pomoću te brzine računa se translacijska kinetička energija tijela. Ako pravac djelovanja sile ne prolazi kroz centar masa, osim opisanoga učinka sila daje tijelu i kutnu akceleraciju α, pa mu mijenja i rotacijsku kinetičku energiju. Tada sila mora vršiti veći rad nego kad djeluje na centar masa, a to se dešava zato što njezino hvatište prelazi veći put nego što je put centar masa (tj. "put tijela"). Zakon poluge još zornije dokazuje da se rad sile računa pomoću puta hvatišta sile: na većem kraku dovoljna je manja sila za isti rad zato što njezino hvatište prelazi veći put. </p> <div class="mw-heading mw-heading3"><h3 id="Promjenjivu_silu_treba_integrirati">Promjenjivu silu treba integrirati</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rad_(fizika)&veaction=edit&section=7" title="Uredi odlomak: Promjenjivu silu treba integrirati" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rad_(fizika)&action=edit&section=7" title="Uredi kôd odjeljka Promjenjivu silu treba integrirati"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rad sile možemo izračunati kao umnožak dvaju brojeva (komponente <i>F</i> cos α i puta njezinoga hvatišta <i>s</i>) samo ako znamo koliko ti brojevi iznose, tj. ako se na odabranom putu tangencijalna skalarna komponenta sile ne mijenja. No, u općem slučaju sila može proizvoljno mijenjati iznos i smjer: tada se rad mora računati pomoću integrala, jer ne postoji jednostavniji postupak da se odredi prosječna vrijednost <i>F</i> cos α za računanje rada na nekom putu. </p> <div class="mw-heading mw-heading4"><h4 id="Tumačenje_i_primjer_integrala_rada"><span id="Tuma.C4.8Denje_i_primjer_integrala_rada"></span>Tumačenje i primjer integrala rada</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rad_(fizika)&veaction=edit&section=8" title="Uredi odlomak: Tumačenje i primjer integrala rada" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rad_(fizika)&action=edit&section=8" title="Uredi kôd odjeljka Tumačenje i primjer integrala rada"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Postupak integriraranja može se najlakše razumjeti kao zamisao da se zbroje radovi promatrane sile po vrlo malim komadićima ukupnoga puta, tako malima da se sila na pojedinom komadiću "ne stigne" promijeniti. Naravno, sve dok je broj komadića konačan, sila će se na svakome bar malo promijeniti (ako se stalno mijenja), ali ta promjena može biti u tako dalekoj decimali da nas to u konačnom rezultatu uopće ne zanima (pa uzimamo bilo koju vrijednost s pojedinog komadića puta). Ako nije tako, podijelit ćemo put u još sitnije komadiće prije zbrajanja radova, sve dok ne dobijemo rezultat koji je točan u željenom broju znamenki (što se provjerava usporedbom s narednom još sitnijom razdiobom puta). Takav se postupak zove numeričko integriranje. </p><p>No, u mislima možemo nastaviti proces usitnjavanja u nedogled, znajući da bismo tako dobivali uzastopne rezultate sa sve većim brojem točnih znamenki. Integral je (ako postoji) onaj broj (granična vrijednost ili limes) kojemu se ti uzastopni zbrojevi sve manjih komadića rada sve više približavaju (uz dovoljno usitnjavanje puta, zbroj radova je po volji blizu granične vrijednosti). A kako pokazuje matematička analiza, tu točnu graničnu vrijednost možemo za mnoge konkretne sile izračunati na posve drugačiji način, pomoću pravila integriranja za pojedine vrste funkcija. Na primjer, potencija se integrira tako da joj se eksponent uveća za 1, i potom se podijeli s novim eksponentom. Za rastezanje elastične opruge (učvršćene na drugom kraju) potrebna je sila <i>F</i> = <i>ks</i> promjenjljivog iznosa i u smjeru rastezanja, gdje je <i>k</i> konstanta opruge, dok je <i>s</i> produljenje (potencija <i>s</i> na prvu), tj. put što ga je prešlo hvatište sile od nerastegnutog položaja <i>s</i> = 0. Da bi rastegnula oprugu za iznos <i>A</i>, sila će izvršiti rad: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{0}^{A}k\cdot s\,\mathrm {d} s=k\cdot \int _{0}^{A}s\,\mathrm {d} s=k\left[{\frac {s^{2}}{2}}\right]_{0}^{A}={\frac {k\cdot A^{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mi>k</mi> <mo>⋅</mo> <mi>s</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> <mo>=</mo> <mi>k</mi> <mo>⋅</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mi>s</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> <mo>=</mo> <mi>k</mi> <msubsup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mo>⋅</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{0}^{A}k\cdot s\,\mathrm {d} s=k\cdot \int _{0}^{A}s\,\mathrm {d} s=k\left[{\frac {s^{2}}{2}}\right]_{0}^{A}={\frac {k\cdot A^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce66df05659b5dbbaa4d05a839dd00a4bbc49d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.599ex; height:6.843ex;" alt="{\displaystyle W=\int _{0}^{A}k\cdot s\,\mathrm {d} s=k\cdot \int _{0}^{A}s\,\mathrm {d} s=k\left[{\frac {s^{2}}{2}}\right]_{0}^{A}={\frac {k\cdot A^{2}}{2}}}"></span></dd></dl> <p>Na znaku integrala (stilizirani rastegnuti znak sume, najavljuje zbrajanje "beskonačno mnogo beskonačno malih pribrojnika") donja i gornja granica označavaju početnu i završnu točku puta. Slijedi iznos sile <i>ks</i> (kosinusa nema jer je jednak 1), što se zove podintegralna funkcija. Integral završava diferencijalom puta d<i>s</i> (kojega možemo smatrati "beskonačno malim komadićem puta"). (Standardna matematička analiza smatra ovakav "tehničarski" opis nekorektinim, ali noviji radovi pokazuju da ga je moguće i rigorozno opravdati.) U narednom koraku "vadi" se ispred integrala konstanta koja množi ostatak podintegralne funkcije, a potom se <i>s</i><sup>1</sup> integrira u <i>s</i><sup>2</sup>/2. U zadnjem koraku uvrste se, umjesto <i>s</i>, granice integriranja: najprije gornja granica <i>A</i>, od čega se oduzme isti izraz s uvršenom donjom granicom (ovdje 0, pa se ne piše). </p> <div class="mw-heading mw-heading2"><h2 id="Opis_rada_skalarnim_produktom">Opis rada skalarnim produktom</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rad_(fizika)&veaction=edit&section=9" title="Uredi odlomak: Opis rada skalarnim produktom" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rad_(fizika)&action=edit&section=9" title="Uredi kôd odjeljka Opis rada skalarnim produktom"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Skalarnim množenjem dvaju vektora dobiva se skalar koji je jednak umnošku njihovih iznosa i kosinusa kuta među njima. Ako je sila <sup><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 \scriptstyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02da86fc900a1585e3a01a08d897a7f8a715e064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.309ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {\vec {F}}}"></span></sup> konstantnog iznosa i smjera, a smjer pravocrtnog gibanja njezinog hvatišta zatvara stalni kut α sa smjerom sile, rad se može zapisati na dva načina: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=F\cdot s\cdot \cos \alpha ={\vec {F}}\cdot {\vec {d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>F</mi> <mo>⋅</mo> <mi>s</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=F\cdot s\cdot \cos \alpha ={\vec {F}}\cdot {\vec {d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a90ada1438167b53ad244c5a2afa28c83c28b5fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.805ex; height:2.843ex;" alt="{\displaystyle W=F\cdot s\cdot \cos \alpha ={\vec {F}}\cdot {\vec {d}}}"></span></dd></dl> <p>Drugi izraz označava <a href="/wiki/Skalarni_produkt" class="mw-redirect" title="Skalarni produkt">skalarni produkt</a> vektora sile i vektora pomaka <sup><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 \scriptstyle {\vec {d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\vec {d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d233578c10741467ef9dab0051bcfba83dca4cc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.208ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {\vec {d}}}"></span></sup> (engleski naziv množenja <i>dot product</i> potječe od točke koja se piše među vektorima). Pomak je usmjerena dužina koja "ide" od početne do završne točke puta na putanji hvatišta sile (opisuje koliko se i u kojemu smjeru ta točka "pomakla"). Jednakost navedenih izraza je očigledna iz definicije skalarnog produkta, jer je u opisanom slučaju iznos pomaka jednak putu <i>s</i>. No, može se dokazati da formula sa skalarnim produktom sile i pomaka vrijedi i za proizvoljni oblik putanje, uz uvjet da sila ne mijenja iznos i smjer. </p><p>Za proizvoljni oblik putanje hvatišta, potrebno je najprije matematički opisati krivulju duž koje se ta točka giba. Gibanje točke u cijelosti je opisano ako za svaki trenutak znademo njezine koordinate, npr. <i>x</i>(<i>t</i>), <i>y</i>(<i>t</i>) i <i>z</i>(<i>t</i>) u pravokutnom Kartezijevom sustavu gdje ih možemo smatrati skalarnim komponentama vektora položaja (radij-vektora) <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 \scriptstyle {\vec {r}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\vec {r}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a31972e56b6ea879c1f5b2e2fb2fba66426b0c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.776ex; height:2.509ex;" alt="{\displaystyle \scriptstyle {\vec {r}}(t)}"></span> te točke. Vektor položaja je usmjerena dužina kojoj je početak u ishodištu sustava a kraj (strelica) "prati" točku po putanji. Koordinate i vektor položaja često se pišu bez eksplicitne oznake ovisnosti o vremenu, jer se ona kod gibanja točke i tako podrazumijeva. </p> <figure typeof="mw:File/Frame"><a href="/wiki/Datoteka:Pomak_i_put.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/hr/6/66/Pomak_i_put.JPG" decoding="async" width="583" height="241" class="mw-file-element" data-file-width="583" data-file-height="241" /></a><figcaption>Iznos pomaka <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 \scriptstyle |\Delta {\vec {r}}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle |\Delta {\vec {r}}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1adb32cffc21b850321c4162d225a04642ffb060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.186ex; height:2.509ex;" alt="{\displaystyle \scriptstyle |\Delta {\vec {r}}|}"></span> i pripadajući komadić puta <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 \scriptstyle \Delta s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">Δ</mi> <mi>s</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Delta s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9547acc63f674957798790380c808fbc71d1b70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.14ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \Delta s}"></span> postaju jednaki za dovoljno mali vremenski interval</figcaption></figure> <p>Kod takvog opisa gibanja, prikladnije je za vektor pomaka iz neke točke 1 u točku 2 koristiti oznaku <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 \scriptstyle \Delta {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Delta {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/effc08237ac328c07fc1c30f36429974a782cb23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.272ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \Delta {\vec {r}}}"></span> (ako znamo da <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 \scriptstyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">Δ</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7428df33efb5339a7c947720caf317992ac224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.369ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \Delta }"></span> označava razliku odnosno promjenu) jer ona eksplicitno pokazuje da se pomak dobiva oduzimanjem pripadnih vektora položaja: <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 \scriptstyle \Delta {\vec {r}}={\vec {r}}_{2}-{\vec {r}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Delta {\vec {r}}={\vec {r}}_{2}-{\vec {r}}_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5621eaff7251b979f379af06658a3f8e6e6acbc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.297ex; height:2.343ex;" alt="{\displaystyle \scriptstyle \Delta {\vec {r}}={\vec {r}}_{2}-{\vec {r}}_{1}}"></span>, tj. da pomak možemo promatrati kao "promjenu položaja". Duljina putanje (pređeni put <i>s</i>) na krivulji može biti znatno veća od iznosa vektora pomaka <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 \scriptstyle |\Delta {\vec {r}}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle |\Delta {\vec {r}}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1adb32cffc21b850321c4162d225a04642ffb060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.186ex; height:2.509ex;" alt="{\displaystyle \scriptstyle |\Delta {\vec {r}}|}"></span>. No, ako se promatraju sve manji pomaci (vremenski interval <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 \scriptstyle \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3177d420bcd53822fc672fb7d85f76ba2477fe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.963ex; height:1.843ex;" alt="{\displaystyle \scriptstyle \Delta t}"></span> između promatranih položaja "teži" prema nuli; na skici je ilustriran početak graničnog procesa) iznosi puta i pomaka postaju sve više jednaki. Jednakost graničnih vrijednosti možemo zapisati pomoću diferencijala: <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 \scriptstyle |\mathrm {d} {\vec {r}}|=\mathrm {d} s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle |\mathrm {d} {\vec {r}}|=\mathrm {d} s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d735644bd9834677edcf15b03a86014e997ef3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.695ex; height:2.509ex;" alt="{\displaystyle \scriptstyle |\mathrm {d} {\vec {r}}|=\mathrm {d} s}"></span>. Stoga se integral iz opće definicije za rad proizvoljne sile na proizvoljnom putu može kraće zapisati pomoću skalarnog produkta: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{A}^{B}{\vec {F}}\cdot \mathrm {d} {\vec {r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{A}^{B}{\vec {F}}\cdot \mathrm {d} {\vec {r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e30f16ac0aa35124e72bf4b637788d16e997f896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.83ex; height:6.176ex;" alt="{\displaystyle W=\int _{A}^{B}{\vec {F}}\cdot \mathrm {d} {\vec {r}}}"></span></dd></dl> <p>Uzimajući u obzir da je <a href="/wiki/Brzina" title="Brzina">brzina</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 \scriptstyle {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a22ea722fa1095eb63f41c95385e74b4eae216b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.85ex; height:2.176ex;" alt="{\displaystyle \scriptstyle {\vec {v}}}"></span> neke točke derivacija njezinog vektora položaja po vremenu (pa je <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 \scriptstyle \mathrm {d} {\vec {r}}={\vec {v}}\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \mathrm {d} {\vec {r}}={\vec {v}}\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/286148cfba21bc86070fb0408b1202a1bf0c8e0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.453ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \mathrm {d} {\vec {r}}={\vec {v}}\mathrm {d} t}"></span>), može se taj integral prevesti u integral po vremenu: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\int _{A}^{B}{\vec {F}}\cdot {\vec {v}}\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=\int _{A}^{B}{\vec {F}}\cdot {\vec {v}}\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8828d6d8acbe09ebe70d7930c1f53b2dfea391de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.009ex; height:6.176ex;" alt="{\displaystyle W=\int _{A}^{B}{\vec {F}}\cdot {\vec {v}}\,\mathrm {d} t}"></span></dd></dl> <p>Podintegralna funkcija <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 \scriptstyle {\vec {F}}\cdot {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\vec {F}}\cdot {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b1dd9556f770dce1156737079df5ba44c9a2370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.616ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {\vec {F}}\cdot {\vec {v}}}"></span> je <a href="/wiki/Snaga" title="Snaga">snaga</a> sile. Budući da se snaga definira kao derivacija rada po vremenu, jasno je da rad mora biti jednak integralu snage po vremenu. </p> <div class="mw-heading mw-heading2"><h2 id="Opaska:_pređeni_put_ili_pomak?"><span id="Opaska:_pre.C4.91eni_put_ili_pomak.3F"></span>Opaska: pređeni put ili pomak?</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rad_(fizika)&veaction=edit&section=10" title="Uredi odlomak: Opaska: pređeni put ili pomak?" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rad_(fizika)&action=edit&section=10" title="Uredi kôd odjeljka Opaska: pređeni put ili pomak?"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Frame"><a href="/wiki/Datoteka:Rad_je_skalarni_produkt_konstantne_sile_i_pomaka.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/hr/e/e4/Rad_je_skalarni_produkt_konstantne_sile_i_pomaka.JPG" decoding="async" width="177" height="159" class="mw-file-element" data-file-width="177" data-file-height="159" /></a><figcaption>Rad težine po krivulji</figcaption></figure> <p>U hrvatskim udžbenicima obično se rad opisuje pomoću puta hvatišta sile. Nasuprot tome, u američkima<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> se obično polazi od pomaka (možda zato što nemaju jednostavne riječi za "put", nego koriste "duljinu staze"). Engleski jezik nema ni jednostavne riječi za "hvatište" sile, nego koristi "točku u kojoj sila djeluje na tijelo", pa se u površnim tekstovima o radu često previđa uloga hvatišta sile. </p><p>U općoj formuli za rad promjenjljive sile duž proizvoljne putanje svejedno je (kao što je pokazano kod gornjeg integrala) da li se promatra diferencijal puta ili vektora položaja (tzv. diferecijalni pomak), jer su im iznosi isti, a integriranje je krivuljni integral duž putanje hvatišta sile. No, u specijalnim slučajevima, ponekad može jedan pristup omogućiti upotrebu jednostavnije formule nego drugi. </p><p>Korištenje vektora pomaka <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 \scriptstyle {\vec {d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\vec {d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d233578c10741467ef9dab0051bcfba83dca4cc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.208ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {\vec {d}}}"></span> olakšava račun ako se kod gibanja po krivulji promatra rad sile konstantnog iznosa i smjera (kakva je npr. <a href="/wiki/Te%C5%BEina" title="Težina">težina</a> tijela). Na gornjoj skici desno, <a href="/wiki/Te%C5%BEi%C5%A1te" title="Težište">težište</a> tijela spušta se po krivudavom putu <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 \scriptstyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>s</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6873ee1af33d3eedd289d361059739e7f38bcbc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.771ex; height:1.343ex;" alt="{\displaystyle \scriptstyle s}"></span> (zbog istovremenog djelovanja drugih sila). Pritom težina <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 \scriptstyle {\vec {G}}=m{\vec {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\vec {G}}=m{\vec {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deb0bb2bc64c22c84578a9b04a049f0839a4dab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.862ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\vec {G}}=m{\vec {g}}}"></span> izvrši rad <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 \scriptstyle \ W=mgh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mtext> </mtext> <mi>W</mi> <mo>=</mo> <mi>m</mi> <mi>g</mi> <mi>h</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \ W=mgh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f61566a9dec7589c383a8725b5fb6088f13fca1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.76ex; height:2.009ex;" alt="{\displaystyle \scriptstyle \ W=mgh}"></span> (što je općenito poznati rezultat, jednak negativnoj promjeni <a href="/wiki/Potencijalna_energija" title="Potencijalna energija">potencijalne energije</a>). A taj se rezultat najlakše dobiva kao skalarni umnožak težine i pomaka <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 \scriptstyle W={\vec {G}}\cdot {\vec {d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle W={\vec {G}}\cdot {\vec {d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7528a43116066c1d9671699db009d97a9c41561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.957ex; height:2.343ex;" alt="{\displaystyle \scriptstyle W={\vec {G}}\cdot {\vec {d}}}"></span>, jer je <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 \scriptstyle d\cos \alpha =h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>d</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mi>h</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle d\cos \alpha =h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee88bbde0e2bb61143204bbf89878f1566c87c7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.111ex; height:1.676ex;" alt="{\displaystyle \scriptstyle d\cos \alpha =h}"></span> . </p> <figure typeof="mw:File/Frame"><a href="/wiki/Datoteka:Work_on_the_lever_arm.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Work_on_the_lever_arm.svg/211px-Work_on_the_lever_arm.svg.png" decoding="async" width="211" height="184" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Work_on_the_lever_arm.svg/317px-Work_on_the_lever_arm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Work_on_the_lever_arm.svg/422px-Work_on_the_lever_arm.svg.png 2x" data-file-width="211" data-file-height="184" /></a><figcaption>Sila konstantnog iznosa okomita na polugu</figcaption></figure> <p>Nasuprot tome, ako sila ima samo konstantan iznos, a njezin promjenjljivi smjer se točno podudara sa smjerom gibanja hvatišta (ili je točno u suprotnom smjeru), rad se najlakše računa kao umnožak iznosa sile i puta hvatišta (s tim da je rad negativan ako su smjerovi suprotni). Primjerice, ako tijelo kliže po podlozi po putanji proizvoljnog oblika, na njega djeluje <a href="/wiki/Trenje" title="Trenje">trenje</a> u suprotnom smjeru od smjera klizanja, a iznos trenja <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 \scriptstyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>T</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1111d99ee3dae3d02cc676db9b35388ad660af7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.157ex; height:1.676ex;" alt="{\displaystyle \scriptstyle T}"></span> je konstantan ako se na tome putu <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 \scriptstyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>s</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6873ee1af33d3eedd289d361059739e7f38bcbc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.771ex; height:1.343ex;" alt="{\displaystyle \scriptstyle s}"></span> ne mijenjaju pritisak tijela na podlogu ni svojstva podloge. Tada se rad trenja jednostavno dobiva kao <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 \scriptstyle \ W=-Ts}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mtext> </mtext> <mi>W</mi> <mo>=</mo> <mo>−</mo> <mi>T</mi> <mi>s</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \ W=-Ts}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fef91c6702752577fa5a7ea0e035f6c605eedd25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.788ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \ W=-Ts}"></span>. Drugi primjer (na donjoj skici desno) je rotacija tijela, npr. neke poluge, oko čvrste osovine; rotaciju najefikasnije ubrzava sila koja djeluje okomito na polugu (dakle, u smjeru gibanja hvatišta duž kružnice). Ako je iznos sile <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 \scriptstyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02da86fc900a1585e3a01a08d897a7f8a715e064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.309ex; height:2.343ex;" alt="{\displaystyle \scriptstyle {\vec {F}}}"></span> konstantan na prikazanom putu <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 \scriptstyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>s</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6873ee1af33d3eedd289d361059739e7f38bcbc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.771ex; height:1.343ex;" alt="{\displaystyle \scriptstyle s}"></span> (kružni luk), rad sile je <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 \scriptstyle \ W=Fs}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mtext> </mtext> <mi>W</mi> <mo>=</mo> <mi>F</mi> <mi>s</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \ W=Fs}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2336a5a09c09b790b005269a6fece4c75d256c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.583ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \ W=Fs}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Rad_momenta_sile">Rad momenta sile</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rad_(fizika)&veaction=edit&section=11" title="Uredi odlomak: Rad momenta sile" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rad_(fizika)&action=edit&section=11" title="Uredi kôd odjeljka Rad momenta sile"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Prilikom rotacije tijela oko čvrste osi, često se rad opisuje kao rad <a href="/wiki/Moment_sile" title="Moment sile">momenta sile</a>, umjesto kao rad sile. Dakako, to je isti rad, a formula se lako prevodi iz jednog oblika u drugi. U primjeru s rotiranjem poluge, kružni luk <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 \scriptstyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>s</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6873ee1af33d3eedd289d361059739e7f38bcbc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.771ex; height:1.343ex;" alt="{\displaystyle \scriptstyle s}"></span> može se opisati pomoću polumjera kružnice i kuta zakreta <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 \scriptstyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>φ</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5ec1d10f6956a0278229dcadd3362a04841f7b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.075ex; height:1.676ex;" alt="{\displaystyle \scriptstyle \varphi }"></span> (izraženog u <a href="/wiki/Radijan" title="Radijan">radijanima</a>) kao <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 \scriptstyle s=r\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>s</mi> <mo>=</mo> <mi>r</mi> <mi>φ</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle s=r\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4e0711484a34a533b79515b33a9fe6c9e629a45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.866ex; height:1.676ex;" alt="{\displaystyle \scriptstyle s=r\varphi }"></span>. Ako se to uvrsti u gornji izraz za rad sile koja gura polugu, dobiva se <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 \scriptstyle \ W=Fs=Fr\varphi =M\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mtext> </mtext> <mi>W</mi> <mo>=</mo> <mi>F</mi> <mi>s</mi> <mo>=</mo> <mi>F</mi> <mi>r</mi> <mi>φ</mi> <mo>=</mo> <mi>M</mi> <mi>φ</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \ W=Fs=Fr\varphi =M\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5cdabbfe5f0afd2bedb320ebd61b84a34caedc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.989ex; height:2.009ex;" alt="{\displaystyle \scriptstyle \ W=Fs=Fr\varphi =M\varphi }"></span>, jer je <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 \scriptstyle M=Fr}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>M</mi> <mo>=</mo> <mi>F</mi> <mi>r</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle M=Fr}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af3dfb9e509ee5bbe61c81511dfff3d1b31a5e86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.978ex; height:1.676ex;" alt="{\displaystyle \scriptstyle M=Fr}"></span> iznos momenta sile koji zakreće polugu (sila puta krak). </p><p>Odatle slijedi da se rad konstantnog momenta sile računa pomoću kuta zakreta (u radijanima) kao: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=M\cdot \varphi \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>=</mo> <mi>M</mi> <mo>⋅</mo> <mi>φ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W=M\cdot \varphi \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee405dedc8a2583d8d748da8b31b34302cc0ac50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.562ex; height:2.676ex;" alt="{\displaystyle W=M\cdot \varphi \,}"></span></dd></dl> <p>U slučaju da moment sile oko čvrste osi mijenja iznos, jasno je da navedenu formulu treba pretvoriti u integral iznosa momenta po kutu zakreta. Još općenitije, moguće je (ali nije uobičajeno u svakodnevnim tehničkim primjenama) računati rad promjenjljivog momenta sile i u slučaju da tijelo nije na čvrstoj osovini, pa mu vektor kutne brzine može mijenjati smjer. Tada se u integralu može koristiti skalarni produkt vektora momenta sile i vektorskog diferencijalnog zakreta. </p><p>Budući da je kod rotacije oko čvrste osi uobičajeno rad računati samo pomoću iznosa momenta sile, potrebno je dopisati negativni predznak radu u slučaju da moment djeluje u suprotnom smjeru od kutne brzine koju tijelo već ima. To bi, naime, u gornjem primjeru značilo da sila na polugu djeluje u suprotnom smjeru od smjera gibanja njezinog hvatišta (koji na skici nije naznačen). Tada sila, odnosno njezin moment, umanjuju kinetičku energiju tijela (koja se za rotacijsko gibanje računa po formuli <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 \scriptstyle \ E_{k}=I\omega ^{2}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mtext> </mtext> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>I</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \ E_{k}=I\omega ^{2}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3470754db672dadd69220eac255cad81f10deda9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.258ex; height:2.343ex;" alt="{\displaystyle \scriptstyle \ E_{k}=I\omega ^{2}/2}"></span>, gdje je <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 \scriptstyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>I</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7784992569d3b9f73a37d4cff1f18443d0583397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.829ex; height:1.676ex;" alt="{\displaystyle \scriptstyle I}"></span> <a href="/wiki/Moment_inercije" title="Moment inercije">moment inercije</a> tijela oko čvrste osi, dok je <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 \scriptstyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>ω</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691e776df98eed20b0084e0cb2953e837b45dbc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.022ex; height:1.343ex;" alt="{\displaystyle \scriptstyle \omega }"></span> kutna brzina). </p> <div class="mw-heading mw-heading2"><h2 id="Izvori">Izvori</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rad_(fizika)&veaction=edit&section=12" title="Uredi odlomak: Izvori" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Rad_(fizika)&action=edit&section=12" title="Uredi kôd odjeljka Izvori"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r6541845">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><b>rad</b>, <a rel="nofollow" class="external autonumber" href="http://www.enciklopedija.hr/Natuknica.aspx?ID=51403">[1]</a> "Hrvatska enciklopedija", Leksikografski zavod Miroslav Krleža, www.enciklopedija.hr, 2015.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.scribd.com/doc/40272543/Levanat-Kinematika-i-dinamika">I. Levanat: Fizika za TVZ - Kinematika i dinamika</a> Tehničko veleučilište u Zagrebu (2010)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Young H. D., Freedman R. A., Sears and Zemansky University Physics, Addison-Wesley, San Francisco (2004)</span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Dobavljeno iz "<a dir="ltr" href="https://hr.wikipedia.org/w/index.php?title=Rad_(fizika)&oldid=6400248">https://hr.wikipedia.org/w/index.php?title=Rad_(fizika)&oldid=6400248</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Posebno:Kategorije" title="Posebno:Kategorije">Kategorije</a>: <ul><li><a href="/wiki/Kategorija:Fizikalne_veli%C4%8Dine" title="Kategorija:Fizikalne veličine">Fizikalne veličine</a></li><li><a href="/wiki/Kategorija:Klasi%C4%8Dna_mehanika" title="Kategorija:Klasična mehanika">Klasična mehanika</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Ova stranica posljednji je put uređivana 29. svibnja 2022. u 12:28.</li> <li id="footer-info-copyright">Tekst je dostupan pod licencijom <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.hr">Creative Commons: Imenuj autora/Dijeli pod istim uvjetima</a>; mogu se primjenjivati i dodatni uvjeti. 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