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class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Sito"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Indice" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Indice</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">nascondi</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">Inizio</div> </a> </li> <li id="toc-Storia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Storia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Storia</span> </div> </a> <ul id="toc-Storia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tipologia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tipologia"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Tipologia</span> </div> </a> <button aria-controls="toc-Tipologia-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>Attiva/disattiva la sottosezione Tipologia</span> </button> <ul id="toc-Tipologia-sublist" class="vector-toc-list"> <li id="toc-Attrito_radente" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Attrito_radente"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Attrito radente</span> </div> </a> <ul id="toc-Attrito_radente-sublist" class="vector-toc-list"> <li id="toc-Calcolo_del_coefficiente_d'attrito_dinamico" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Calcolo_del_coefficiente_d'attrito_dinamico"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Calcolo del coefficiente d'attrito dinamico</span> </div> </a> <ul id="toc-Calcolo_del_coefficiente_d'attrito_dinamico-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Attrito_volvente" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Attrito_volvente"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Attrito volvente</span> </div> </a> <ul id="toc-Attrito_volvente-sublist" class="vector-toc-list"> <li id="toc-Calcolo_del_coefficiente_di_resistenza_al_rotolamento_e_della_forza_di_resistenza_al_rotolamento" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Calcolo_del_coefficiente_di_resistenza_al_rotolamento_e_della_forza_di_resistenza_al_rotolamento"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Calcolo del coefficiente di resistenza al rotolamento e della forza di resistenza al rotolamento</span> </div> </a> <ul id="toc-Calcolo_del_coefficiente_di_resistenza_al_rotolamento_e_della_forza_di_resistenza_al_rotolamento-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calcolo_del_coefficiente_d'attrito_volvente" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Calcolo_del_coefficiente_d'attrito_volvente"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Calcolo del coefficiente d'attrito volvente</span> </div> </a> <ul id="toc-Calcolo_del_coefficiente_d'attrito_volvente-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Attrito_viscoso" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Attrito_viscoso"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Attrito viscoso</span> </div> </a> <ul id="toc-Attrito_viscoso-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Anisotropia_dell'attrito" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Anisotropia_dell'attrito"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Anisotropia dell'attrito</span> </div> </a> <ul id="toc-Anisotropia_dell'attrito-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Effetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Effetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Effetti</span> </div> </a> <ul id="toc-Effetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" 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="Mostra/Nascondi l'indice" > <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">Mostra/Nascondi l'indice</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">Attrito</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="Vai a una voce in un'altra lingua. Disponibile in 86 lingue" > <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-86" 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">86 lingue</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/Wrywing" title="Wrywing - afrikaans" lang="af" hreflang="af" data-title="Wrywing" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D8%AD%D8%AA%D9%83%D8%A7%D9%83" title="احتكاك - arabo" lang="ar" hreflang="ar" data-title="احتكاك" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Resfreg%C3%B3n" title="Resfregón - asturiano" lang="ast" hreflang="ast" data-title="Resfregón" data-language-autonym="Asturianu" data-language-local-name="asturiano" 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/S%C3%BCrt%C3%BCnm%C9%99" title="Sürtünmə - azerbaigiano" lang="az" hreflang="az" data-title="Sürtünmə" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaigiano" 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%A1%D1%96%D0%BB%D0%B0_%D1%82%D1%80%D1%8D%D0%BD%D0%BD%D1%8F" title="Сіла трэння - bielorusso" lang="be" hreflang="be" data-title="Сіла трэння" data-language-autonym="Беларуская" data-language-local-name="bielorusso" 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%A6%D0%B5%D1%80%D1%86%D0%B5" title="Церце - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Церце" 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%A2%D1%80%D0%B8%D0%B5%D0%BD%D0%B5" title="Триене - bulgaro" lang="bg" hreflang="bg" data-title="Триене" data-language-autonym="Български" data-language-local-name="bulgaro" 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%98%E0%A6%B0%E0%A7%8D%E0%A6%B7%E0%A6%A3" title="ঘর্ষণ - bengalese" lang="bn" hreflang="bn" data-title="ঘর্ষণ" data-language-autonym="বাংলা" data-language-local-name="bengalese" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Trenje" title="Trenje - bosniaco" lang="bs" hreflang="bs" data-title="Trenje" data-language-autonym="Bosanski" data-language-local-name="bosniaco" 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/Fricci%C3%B3" title="Fricció - catalano" lang="ca" hreflang="ca" data-title="Fricció" data-language-autonym="Català" data-language-local-name="catalano" 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/%D9%84%DB%8E%DA%A9%D8%AE%D8%B4%D8%A7%D9%86%D8%AF%D9%86" title="لێکخشاندن - curdo centrale" lang="ckb" hreflang="ckb" data-title="لێکخشاندن" data-language-autonym="کوردی" data-language-local-name="curdo centrale" 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/T%C5%99en%C3%AD" title="Tření - ceco" lang="cs" hreflang="cs" data-title="Tření" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A1%C4%95%D1%80%D1%82%C4%95%D0%BD%D3%B3" title="Сĕртĕнӳ - ciuvascio" lang="cv" hreflang="cv" data-title="Сĕртĕнӳ" data-language-autonym="Чӑвашла" data-language-local-name="ciuvascio" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ffrithiant" title="Ffrithiant - gallese" lang="cy" hreflang="cy" data-title="Ffrithiant" data-language-autonym="Cymraeg" data-language-local-name="gallese" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Friktion" title="Friktion - danese" lang="da" hreflang="da" data-title="Friktion" data-language-autonym="Dansk" data-language-local-name="danese" 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/Reibung" title="Reibung - tedesco" lang="de" hreflang="de" data-title="Reibung" data-language-autonym="Deutsch" data-language-local-name="tedesco" 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%A4%CF%81%CE%B9%CE%B2%CE%AE" title="Τριβή - greco" lang="el" hreflang="el" data-title="Τριβή" data-language-autonym="Ελληνικά" data-language-local-name="greco" 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/Friction" title="Friction - inglese" lang="en" hreflang="en" data-title="Friction" data-language-autonym="English" data-language-local-name="inglese" 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/Froto" title="Froto - esperanto" lang="eo" hreflang="eo" data-title="Froto" 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/Fricci%C3%B3n" title="Fricción - spagnolo" lang="es" hreflang="es" data-title="Fricción" data-language-autonym="Español" data-language-local-name="spagnolo" 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/H%C3%B5%C3%B5re" title="Hõõre - estone" lang="et" hreflang="et" data-title="Hõõre" data-language-autonym="Eesti" data-language-local-name="estone" 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/Marruskadura-indar" title="Marruskadura-indar - basco" lang="eu" hreflang="eu" data-title="Marruskadura-indar" data-language-autonym="Euskara" data-language-local-name="basco" 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/%D8%A7%D8%B5%D8%B7%DA%A9%D8%A7%DA%A9" title="اصطکاک - persiano" lang="fa" hreflang="fa" data-title="اصطکاک" data-language-autonym="فارسی" data-language-local-name="persiano" 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/Kitka" title="Kitka - finlandese" lang="fi" hreflang="fi" data-title="Kitka" data-language-autonym="Suomi" data-language-local-name="finlandese" 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/Frottement" title="Frottement - francese" lang="fr" hreflang="fr" data-title="Frottement" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Wriuwing" title="Wriuwing - frisone occidentale" lang="fy" hreflang="fy" data-title="Wriuwing" data-language-autonym="Frysk" data-language-local-name="frisone occidentale" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Frithchuimilt" title="Frithchuimilt - irlandese" lang="ga" hreflang="ga" data-title="Frithchuimilt" data-language-autonym="Gaeilge" data-language-local-name="irlandese" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Fricci%C3%B3n" title="Fricción - galiziano" lang="gl" hreflang="gl" data-title="Fricción" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%99%D7%9B%D7%95%D7%9A" title="חיכוך - ebraico" lang="he" hreflang="he" data-title="חיכוך" data-language-autonym="עברית" data-language-local-name="ebraico" 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%98%E0%A4%B0%E0%A5%8D%E0%A4%B7%E0%A4%A3" title="घर्षण - hindi" lang="hi" hreflang="hi" data-title="घर्षण" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Trenje" title="Trenje - croato" lang="hr" hreflang="hr" data-title="Trenje" data-language-autonym="Hrvatski" data-language-local-name="croato" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Fwotman" title="Fwotman - creolo haitiano" lang="ht" hreflang="ht" data-title="Fwotman" data-language-autonym="Kreyòl ayisyen" data-language-local-name="creolo haitiano" 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/S%C3%BArl%C3%B3d%C3%A1s" title="Súrlódás - ungherese" lang="hu" hreflang="hu" data-title="Súrlódás" data-language-autonym="Magyar" data-language-local-name="ungherese" 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/%D5%87%D6%83%D5%B8%D6%82%D5%B4_(%D6%86%D5%AB%D5%A6%D5%AB%D5%AF%D5%A1)" title="Շփում (ֆիզիկա) - armeno" lang="hy" hreflang="hy" data-title="Շփում (ֆիզիկա)" data-language-autonym="Հայերեն" data-language-local-name="armeno" 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/Gaya_gesek" title="Gaya gesek - indonesiano" lang="id" hreflang="id" data-title="Gaya gesek" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" 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/N%C3%BAningskraftur" title="Núningskraftur - islandese" lang="is" hreflang="is" data-title="Núningskraftur" data-language-autonym="Íslenska" data-language-local-name="islandese" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-ja badge-Q17437798 badge-goodarticle mw-list-item" title="voce di qualità"><a href="https://ja.wikipedia.org/wiki/%E6%91%A9%E6%93%A6" title="摩擦 - giapponese" lang="ja" hreflang="ja" data-title="摩擦" data-language-autonym="日本語" data-language-local-name="giapponese" 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%AE%E1%83%90%E1%83%AE%E1%83%A3%E1%83%9C%E1%83%98" title="ხახუნი - georgiano" lang="ka" hreflang="ka" data-title="ხახუნი" data-language-autonym="ქართული" data-language-local-name="georgiano" 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/%D2%AE%D0%B9%D0%BA%D0%B5%D0%BB%D1%96%D1%81_%D0%BA%D2%AF%D1%88%D1%96" title="Үйкеліс күші - kazako" lang="kk" hreflang="kk" data-title="Үйкеліс күші" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%98%E0%B2%B0%E0%B3%8D%E0%B2%B7%E0%B2%A3%E0%B3%86" title="ಘರ್ಷಣೆ - kannada" lang="kn" hreflang="kn" data-title="ಘರ್ಷಣೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" 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/%EB%A7%88%EC%B0%B0%EB%A0%A5" title="마찰력 - coreano" lang="ko" hreflang="ko" data-title="마찰력" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Attrito" title="Attrito - ligure" lang="lij" hreflang="lij" data-title="Attrito" data-language-autonym="Ligure" data-language-local-name="ligure" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/Friction_(%E0%BA%9F%E0%BA%B5%E0%BA%8A%E0%BA%B4%E0%BA%81)" title="Friction (ຟີຊິກ) - lao" lang="lo" hreflang="lo" data-title="Friction (ຟີຊິກ)" data-language-autonym="ລາວ" data-language-local-name="lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Trinties_j%C4%97ga" title="Trinties jėga - lituano" lang="lt" hreflang="lt" data-title="Trinties jėga" data-language-autonym="Lietuvių" data-language-local-name="lituano" 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/Berze" title="Berze - lettone" lang="lv" hreflang="lv" data-title="Berze" data-language-autonym="Latviešu" data-language-local-name="lettone" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mdf mw-list-item"><a href="https://mdf.wikipedia.org/wiki/%D0%A8%D0%BE%D0%B2%D0%B0%D0%BC%D0%B0" title="Шовама - moksha" lang="mdf" hreflang="mdf" data-title="Шовама" data-language-autonym="Мокшень" data-language-local-name="moksha" class="interlanguage-link-target"><span>Мокшень</span></a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Gesekan" title="Gesekan - menangkabau" lang="min" hreflang="min" data-title="Gesekan" data-language-autonym="Minangkabau" data-language-local-name="menangkabau" class="interlanguage-link-target"><span>Minangkabau</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%98%E0%B5%BC%E0%B4%B7%E0%B4%A3%E0%B4%82" title="ഘർഷണം - malayalam" lang="ml" hreflang="ml" data-title="ഘർഷണം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Geseran" title="Geseran - malese" lang="ms" hreflang="ms" data-title="Geseran" data-language-autonym="Bahasa Melayu" data-language-local-name="malese" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%81%D0%B7%D0%B0%D0%BC%D0%BE" title="Ёзамо - erzya" lang="myv" hreflang="myv" data-title="Ёзамо" data-language-autonym="Эрзянь" data-language-local-name="erzya" class="interlanguage-link-target"><span>Эрзянь</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%98%E0%A4%B0%E0%A5%8D%E0%A4%B7%E0%A4%A3" title="घर्षण - nepalese" lang="ne" hreflang="ne" data-title="घर्षण" data-language-autonym="नेपाली" data-language-local-name="nepalese" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wrijving" title="Wrijving - olandese" lang="nl" hreflang="nl" data-title="Wrijving" data-language-autonym="Nederlands" data-language-local-name="olandese" 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/Friksjon" title="Friksjon - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Friksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegese 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/Friksjon" title="Friksjon - norvegese bokmål" lang="nb" hreflang="nb" data-title="Friksjon" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%98%E0%AC%B0%E0%AD%8D%E0%AC%B7%E0%AC%A3" title="ଘର୍ଷଣ - odia" lang="or" hreflang="or" data-title="ଘର୍ଷଣ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="odia" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B0%E0%A8%97%E0%A9%9C" title="ਰਗੜ - punjabi" lang="pa" hreflang="pa" data-title="ਰਗੜ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabi" 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/Tarcie_(fizyka)" title="Tarcie (fizyka) - polacco" lang="pl" hreflang="pl" data-title="Tarcie (fizyka)" data-language-autonym="Polski" data-language-local-name="polacco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Atrito" title="Atrito - portoghese" lang="pt" hreflang="pt" data-title="Atrito" data-language-autonym="Português" data-language-local-name="portoghese" 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/Qhaquy" title="Qhaquy - quechua" lang="qu" hreflang="qu" data-title="Qhaquy" data-language-autonym="Runa Simi" data-language-local-name="quechua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Frecare" title="Frecare - rumeno" lang="ro" hreflang="ro" data-title="Frecare" data-language-autonym="Română" data-language-local-name="rumeno" 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%A2%D1%80%D0%B5%D0%BD%D0%B8%D0%B5" title="Трение - russo" lang="ru" hreflang="ru" data-title="Трение" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Munciuniata" title="Munciuniata - siciliano" lang="scn" hreflang="scn" data-title="Munciuniata" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Freection" title="Freection - scozzese" lang="sco" hreflang="sco" data-title="Freection" data-language-autonym="Scots" data-language-local-name="scozzese" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Trenje" title="Trenje - serbo-croato" lang="sh" hreflang="sh" data-title="Trenje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croato" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Friction" title="Friction - Simple English" lang="en-simple" hreflang="en-simple" data-title="Friction" 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/Trenie" title="Trenie - slovacco" lang="sk" hreflang="sk" data-title="Trenie" data-language-autonym="Slovenčina" data-language-local-name="slovacco" 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/Trenje" title="Trenje - sloveno" lang="sl" hreflang="sl" data-title="Trenje" data-language-autonym="Slovenščina" data-language-local-name="sloveno" 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/Rudzindi" title="Rudzindi - shona" lang="sn" hreflang="sn" data-title="Rudzindi" 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/F%C3%ABrkimi" title="Fërkimi - albanese" lang="sq" hreflang="sq" data-title="Fërkimi" data-language-autonym="Shqip" data-language-local-name="albanese" 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%A2%D1%80%D0%B5%D1%9A%D0%B5" title="Трење - serbo" lang="sr" hreflang="sr" data-title="Трење" data-language-autonym="Српски / srpski" data-language-local-name="serbo" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Friktion" title="Friktion - svedese" lang="sv" hreflang="sv" data-title="Friktion" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Msuguano" title="Msuguano - swahili" lang="sw" hreflang="sw" data-title="Msuguano" data-language-autonym="Kiswahili" data-language-local-name="swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%89%E0%AE%B0%E0%AE%BE%E0%AE%AF%E0%AF%8D%E0%AE%B5%E0%AF%81" title="உராய்வு - tamil" lang="ta" hreflang="ta" data-title="உராய்வு" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%98%E0%B0%B0%E0%B1%8D%E0%B0%B7%E0%B0%A3" title="ఘర్షణ - telugu" lang="te" hreflang="te" data-title="ఘర్షణ" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%A3%E0%B8%87%E0%B9%80%E0%B8%AA%E0%B8%B5%E0%B8%A2%E0%B8%94%E0%B8%97%E0%B8%B2%E0%B8%99" title="แรงเสียดทาน - thailandese" lang="th" hreflang="th" data-title="แรงเสียดทาน" data-language-autonym="ไทย" data-language-local-name="thailandese" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Kiskisan" title="Kiskisan - tagalog" lang="tl" hreflang="tl" data-title="Kiskisan" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/S%C3%BCrt%C3%BCnme_kuvveti" title="Sürtünme kuvveti - turco" lang="tr" hreflang="tr" data-title="Sürtünme kuvveti" data-language-autonym="Türkçe" data-language-local-name="turco" 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%A2%D0%B5%D1%80%D1%82%D1%8F" title="Тертя - ucraino" lang="uk" hreflang="uk" data-title="Тертя" data-language-autonym="Українська" data-language-local-name="ucraino" 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/%D8%B1%DA%AF%DA%91" title="رگڑ - urdu" lang="ur" hreflang="ur" data-title="رگڑ" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ishqalanish" title="Ishqalanish - uzbeco" lang="uz" hreflang="uz" data-title="Ishqalanish" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeco" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ma_s%C3%A1t" title="Ma sát - vietnamita" lang="vi" hreflang="vi" data-title="Ma sát" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%91%A9%E6%93%A6%E5%8A%9B" title="摩擦力 - wu" lang="wuu" hreflang="wuu" data-title="摩擦力" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%91%A9%E6%93%A6%E5%8A%9B" title="摩擦力 - cinese" lang="zh" hreflang="zh" data-title="摩擦力" data-language-autonym="中文" data-language-local-name="cinese" 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/%E6%91%A9%E6%93%A6%E5%8A%9B" title="摩擦力 - cinese classico" lang="lzh" hreflang="lzh" data-title="摩擦力" data-language-autonym="文言" data-language-local-name="cinese classico" 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/M%C3%B4%CD%98-chhat" title="Mô͘-chhat - min nan" lang="nan" hreflang="nan" data-title="Mô͘-chhat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan" 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/%E6%91%A9%E6%93%A6%E5%8A%9B" title="摩擦力 - cantonese" lang="yue" hreflang="yue" data-title="摩擦力" data-language-autonym="粵語" data-language-local-name="cantonese" 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/Q82580#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</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="Namespace"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs 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class="mw-content-ltr mw-parser-output" lang="it" dir="ltr"><p>In <a href="/wiki/Fisica" title="Fisica">fisica</a> l'<b>attrito</b> è una <a href="/wiki/Forza" title="Forza">forza</a> che si oppone al movimento o allo spostamento di un <a href="/wiki/Corpo_(fisica)" title="Corpo (fisica)">corpo</a> relativo alla <a href="/wiki/Superficie" title="Superficie">superficie</a> su cui si trova: se si manifesta tra superfici in quiete relativa si parla di <b>attrito statico</b>, se invece si manifesta tra superfici in moto relativo si parla di <b>attrito dinamico</b>. Si tratta di un fenomeno microscopico, sempre presente nel mondo reale, che presenta vantaggi e svantaggi a seconda del contesto di analisi e la cui origine fisica è fatta risalire alle forze di <a href="/wiki/Adesione" title="Adesione">adesione</a> o <a href="/wiki/Coesione" title="Coesione">coesione</a> tra <a href="/wiki/Materiale" title="Materiale">materiali</a> in interazione tra loro. Queste forze a loro volta derivano in ultima analisi dall'<a href="/wiki/Interazione_elettrostatica" class="mw-redirect" title="Interazione elettrostatica">interazione elettrostatica</a> tra i materiali in questione. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Storia">Storia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=1" title="Modifica la sezione Storia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=1" title="Edit section's source code: Storia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Aristotele" title="Aristotele">Aristotele</a> non isolò il fenomeno dell'attrito, legandolo inscindibilmente alla <a href="/wiki/Dinamica_(fisica)" class="mw-redirect" title="Dinamica (fisica)">dinamica</a> di un corpo: nel suo modello per principio un corpo tenderebbe naturalmente a fermarsi se non mosso da qualche <a href="/wiki/Forza" title="Forza">forza</a>, in accordo con le proprie osservazioni del mondo quotidiano. Fu invece <a href="/wiki/Galilei" class="mw-redirect" title="Galilei">Galilei</a> a rendersi conto grazie agli esperimenti sul <a href="/wiki/Piano_inclinato" title="Piano inclinato">piano inclinato</a> che era un fenomeno variabile in base al tipo di contatto tra i corpi, e che non era quindi "proprio" dei corpi stessi. </p><p><a href="/wiki/Charles_Augustin_de_Coulomb" title="Charles Augustin de Coulomb">Coulomb</a> proseguì lo studio fino ad arrivare all'enunciazione di tre leggi classiche riguardanti in particolare l'attrito radente: questo dipende linearmente dal carico di <a href="/wiki/Compressione_(meccanica)" title="Compressione (meccanica)">compressione</a> delle superfici, non dipende dall'estensione della superficie di contatto tra i due corpi, ed infine non dipende dalla <a href="/w/index.php?title=Velocit%C3%A0_relativa&action=edit&redlink=1" class="new" title="Velocità relativa (la pagina non esiste)">velocità relativa</a> di strisciamento di un corpo sull'altro. </p><p>Queste tre "leggi" sono in realtà approssimazioni<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>, in quanto valide solo sotto particolari ipotesi riduttive. In particolare l'ultima legge è valida solo per velocità di strisciamento piuttosto ridotte, poiché all'aumentare della velocità il coefficiente di attrito diminuisce con legge non lineare; la seconda legge è valida per superfici mediamente piane e non eccessivamente ridotte, mentre al loro tendere al minimo (forza concentrata) il coefficiente di attrito può diminuire; infine la prima legge (di linearità) è valida fintanto che i materiali a contatto siano sufficientemente isotropi, esibiscano comportamenti elastici e poco viscosi e che l'intervallo di tempo in cui la superficie è a contatto sia sufficientemente lungo, mentre in caso di materiale viscoso, anisotropo e/o plastico (come alcuni terreni) o con tempi di contatto ridotti (come per ruote pneumatiche che ruotino velocemente, anche se senza scorrimenti) il legame carico di compressione/attrito diviene non-lineare. </p><p>Un esempio quotidiano della non validità dell'ultima legge si manifesta oggi nei freni automobilistici: la forza frenante non dipende solo dal coefficiente μ<sub>rd</sub> e dalla forza che preme il tamburo sul cerchione o la pastiglia sul disco (mentre è sostanzialmente indipendente dall'estensione dell'area di quest'ultimo), ma dipende anche dalla velocità di strisciamento tra pastiglia e disco, e di conseguenza una frenata di intensità costante ha una efficacia che aumenta al diminuire della velocità, causando il classico effetto di "contraccolpo" al momento dell'arresto. </p><p>La causa dell'attrito radente fu però in passato individuata nelle asperità tra le superfici a contatto finché <a href="/wiki/Heinrich_Rudolf_Hertz" title="Heinrich Rudolf Hertz">Hertz</a> invece dimostrò come l'attrito radente sia dovuto soprattutto a fenomeni di adesione (legami chimici) tra le superfici a contatto, e modificò quindi il modello matematico del fenomeno. Si osserva in particolare che lastre metalliche lucidate a specchio in condizioni di vuoto spinto possiedono un coefficiente di attrito enorme. </p><p>Infine la spiegazione quantistica dell'attrito ne lega le cause all'<a href="/wiki/Interazione_elettrostatica" class="mw-redirect" title="Interazione elettrostatica">interazione elettrostatica</a> attrattiva tra le molecole delle superfici di contatto, come evidente nel <a href="/wiki/Modello_di_Tomlinson" title="Modello di Tomlinson">modello di Tomlinson</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Tipologia">Tipologia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=2" title="Modifica la sezione Tipologia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=2" title="Edit section's source code: Tipologia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Secondo l'interpretazione classica, esistono tre diversi tipi di attrito: </p> <div class="mw-heading mw-heading3"><h3 id="Attrito_radente">Attrito radente</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=3" title="Modifica la sezione Attrito radente" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=3" title="Edit section's source code: Attrito radente"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Froto.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Froto.svg/220px-Froto.svg.png" decoding="async" width="220" height="63" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Froto.svg/330px-Froto.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Froto.svg/440px-Froto.svg.png 2x" data-file-width="1339" data-file-height="381" /></a><figcaption>Rappresentazione del fenomeno dell'attrito radente tra due superfici a livello microscopico.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Attrito_radente.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Attrito_radente.png/350px-Attrito_radente.png" decoding="async" width="350" height="292" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Attrito_radente.png/525px-Attrito_radente.png 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a1/Attrito_radente.png 2x" data-file-width="600" data-file-height="500" /></a><figcaption>Grafico del valore della forza di attrito radente in funzione della forza applicata. Si noti il passaggio da attrito statico ad attrito dinamico, coincidente con l'inizio del moto del corpo</figcaption></figure> <p>L'attrito radente è dovuto allo strisciamento (ad esempio, l'interazione tra due superfici piane che rimangono a contatto mentre scorrono l'una rispetto all'altra).<sup id="cite_ref-Ard580_2-0" class="reference"><a href="#cite_note-Ard580-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Si esercita tra le superfici di corpi solidi a contatto ed è espresso dalla formula:<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <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 {F}_{r}={\mu _{r}}\cdot {F}_{\perp }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F}_{r}={\mu _{r}}\cdot {F}_{\perp }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0693450ca7fe1ed0dbb12541081cc9f799e9a1d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.627ex; height:2.676ex;" alt="{\displaystyle {F}_{r}={\mu _{r}}\cdot {F}_{\perp }}"></span></dd></dl> <p>dove <i>F<sub>r</sub></i> è la forza di attrito radente, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e2dc5a760017c379bd66f0043d213db98bf77b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.375ex; height:2.176ex;" alt="{\displaystyle \mu _{r}}"></span> il <b>coefficiente di attrito radente</b> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {F}_{\perp }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F}_{\perp }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d057d58f1749a39ded020326df64a78446a0442f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.005ex; height:2.509ex;" alt="{\displaystyle {F}_{\perp }}"></span> la componente perpendicolare al piano di appoggio della forza esercitata dal piano di appoggio sul corpo. Per un corpo appoggiato su un piano orizzontale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {F}_{\perp }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F}_{\perp }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d057d58f1749a39ded020326df64a78446a0442f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.005ex; height:2.509ex;" alt="{\displaystyle {F}_{\perp }}"></span> è semplicemente uguale 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 F_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2662434791194da8930c0a1c6d194e189a4470b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.554ex; height:2.843ex;" alt="{\displaystyle F_{p}}"></span>, <a href="/wiki/Forza_peso" title="Forza peso">forza peso</a> del corpo; per un corpo appoggiato su un piano inclinato di un angolo <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> rispetto all'orizzontale risulta invece </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 {F}_{\perp }={F}_{p}\cos \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F}_{\perp }={F}_{p}\cos \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/176236e88b4405a6f60a959873d77f5bfc2f17ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.031ex; height:2.843ex;" alt="{\displaystyle {F}_{\perp }={F}_{p}\cos \alpha }"></span></dd></dl> <p>Il coefficiente d'attrito è una grandezza adimensionale e dipende dai materiali delle due superfici a contatto e dal modo in cui sono state lavorate. Esso corrisponde al rapporto tra la forza di attrito tra due corpi (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {F}_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F}_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/227a623a9d70139cb9a325160e90ff3c8af59bef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.468ex; height:2.509ex;" alt="{\displaystyle {F}_{r}}"></span>) e la forza che li tiene in contatto (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {F}_{\perp }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F}_{\perp }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d057d58f1749a39ded020326df64a78446a0442f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.005ex; height:2.509ex;" alt="{\displaystyle {F}_{\perp }}"></span>). Il coefficiente di attrito statico <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{rs}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{rs}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/241463f01aa368decc125ea3a94ec4748ea1b246" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.147ex; height:2.176ex;" alt="{\displaystyle \mu _{rs}}"></span> è sempre maggiore o uguale al coefficiente d'attrito dinamico <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{rd}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{rd}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad22647f9c2d030f09234993bd8253c7c10a3b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.235ex; height:2.176ex;" alt="{\displaystyle \mu _{rd}}"></span> per le medesime superfici. Dal punto di vista microscopico, esso è dovuto alle forze di interazione tra gli atomi dei materiali a contatto. Questo implica che la forza necessaria al primo distacco (cioè per far sì che i corpi inizino a strisciare) è superiore a quella necessaria a tenerli in strisciamento. Il coefficiente di attrito statico è uguale alla tangente dell'angolo massimo raggiungibile tra le due forze prima che uno dei due corpi cominci a scivolare lungo l'altro (angolo di attrito). </p><p>La forza di attrito, definita dalla prima delle due formule scritte sopra, rappresenta la <i>forza di attrito massima</i> che si manifesta nel contatto tra due superfici. Se la forza motrice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc15d41d3176d0fb9b4474762c53d49add76fbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.17ex; height:2.509ex;" alt="{\displaystyle F_{m}}"></span> è minore di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{rs}F_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{rs}F_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80c6fb32168779704ec7c839d104dc34a961d147" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.7ex; height:2.843ex;" alt="{\displaystyle \mu _{rs}F_{p}}"></span>, allora l'attrito è pari 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 F_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc15d41d3176d0fb9b4474762c53d49add76fbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.17ex; height:2.509ex;" alt="{\displaystyle F_{m}}"></span> e il corpo non si muove; 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 F_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc15d41d3176d0fb9b4474762c53d49add76fbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.17ex; height:2.509ex;" alt="{\displaystyle F_{m}}"></span> supera <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{rs}F_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>s</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{rs}F_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80c6fb32168779704ec7c839d104dc34a961d147" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.7ex; height:2.843ex;" alt="{\displaystyle \mu _{rs}F_{p}}"></span>, il corpo inizia a muoversi; per valori di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc15d41d3176d0fb9b4474762c53d49add76fbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.17ex; height:2.509ex;" alt="{\displaystyle F_{m}}"></span> ancora maggiori, l'attrito (dinamico) è sempre costante e pari 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 \mu _{rd}F_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>d</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{rd}F_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1ba22ccb9583227de1f65ab55e6516415c75ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.789ex; height:2.843ex;" alt="{\displaystyle \mu _{rd}F_{p}}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Calcolo_del_coefficiente_d'attrito_dinamico"><span id="Calcolo_del_coefficiente_d.27attrito_dinamico"></span>Calcolo del coefficiente d'attrito dinamico</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=4" title="Modifica la sezione Calcolo del coefficiente d'attrito dinamico" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=4" title="Edit section's source code: Calcolo del coefficiente d'attrito dinamico"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si può calcolare il coefficiente d'attrito dinamico di un materiale su un altro attraverso un piano inclinato, facendolo strisciare su di esso. Lasciando andare il corpo, esso si muoverà di moto rettilineo uniformemente accelerato, con accelerazione pari 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 a={\frac {2s}{t^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>s</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {2s}{t^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042dd282a427d28b416876942203a026ee551818" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.417ex; height:5.509ex;" alt="{\displaystyle a={\frac {2s}{t^{2}}}}"></span></dd></dl> <p>dove a è l'accelerazione, <i>s</i> è lo spazio percorso e <i>t</i> è il tempo trascorso. La forza che muove il corpo è pari 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 F_{m}=F_{\|}-{F}_{d}=mg\sin {\alpha }-mg{\mu _{rd}}\cos \alpha =mg(\sin {\alpha }-{\mu _{rd}}\cos \alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mi>g</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>−</mo> <mi>m</mi> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>d</mi> </mrow> </msub> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo>=</mo> <mi>m</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>d</mi> </mrow> </msub> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{m}=F_{\|}-{F}_{d}=mg\sin {\alpha }-mg{\mu _{rd}}\cos \alpha =mg(\sin {\alpha }-{\mu _{rd}}\cos \alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b53c6928eb9e57fc206f0b0c9efa191be07c037b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:64.464ex; height:3.176ex;" alt="{\displaystyle F_{m}=F_{\|}-{F}_{d}=mg\sin {\alpha }-mg{\mu _{rd}}\cos \alpha =mg(\sin {\alpha }-{\mu _{rd}}\cos \alpha )}"></span></dd></dl> <p>Dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afc15d41d3176d0fb9b4474762c53d49add76fbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.17ex; height:2.509ex;" alt="{\displaystyle F_{m}}"></span> è la forza che muove il corpo, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\|}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05426e137cfb963c851aa07e14643039f5314615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.549ex; height:3.009ex;" alt="{\displaystyle F_{\|}}"></span> è la forza parallela al piano inclinato, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5984c180b42afaefecd3a8568953401af8c8889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.587ex; height:2.509ex;" alt="{\displaystyle F_{d}}"></span> è l'attrito dinamico, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> è la massa, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> è l'accelerazione di gravità, <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> è l'angolo d'inclinazione del piano e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{rd}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{rd}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad22647f9c2d030f09234993bd8253c7c10a3b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.235ex; height:2.176ex;" alt="{\displaystyle \mu _{rd}}"></span> è il coefficiente d'attrito dinamico. Dividendo la forza risultante per la massa del corpo si ottiene l'accelerazione: </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 a={\frac {F_{m}}{m}}={\frac {mg(\sin {\alpha }-{\mu _{rd}}\cos \alpha )}{m}}=g(\sin {\alpha }-{\mu _{rd}}\cos \alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>m</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>g</mi> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>d</mi> </mrow> </msub> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mi>m</mi> </mfrac> </mrow> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>d</mi> </mrow> </msub> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {F_{m}}{m}}={\frac {mg(\sin {\alpha }-{\mu _{rd}}\cos \alpha )}{m}}=g(\sin {\alpha }-{\mu _{rd}}\cos \alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adf7bab6457ae5ee45be8c587061408795b53238" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.616ex; height:5.676ex;" alt="{\displaystyle a={\frac {F_{m}}{m}}={\frac {mg(\sin {\alpha }-{\mu _{rd}}\cos \alpha )}{m}}=g(\sin {\alpha }-{\mu _{rd}}\cos \alpha )}"></span></dd></dl> <p>Ora si mettono a confronto le due formule dell'accelerazione: </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 {\frac {2s}{t^{2}}}=g(\sin {\alpha }-{\mu _{rd}}\cos \alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>s</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>d</mi> </mrow> </msub> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2s}{t^{2}}}=g(\sin {\alpha }-{\mu _{rd}}\cos \alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da916dcbf0667a9e69d0fe05ab756dc76cd8eb17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.292ex; height:5.509ex;" alt="{\displaystyle {\frac {2s}{t^{2}}}=g(\sin {\alpha }-{\mu _{rd}}\cos \alpha )}"></span></dd></dl> <p>e risolvendo l'equazione, si trova: </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 {\mu _{rd}}=\tan {\alpha }-{\frac {2s\sec {\alpha }}{gt^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>d</mi> </mrow> </msub> </mrow> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>s</mi> <mi>sec</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </mrow> <mrow> <mi>g</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mu _{rd}}=\tan {\alpha }-{\frac {2s\sec {\alpha }}{gt^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10e5a827d8f10f63e3c715666b90aee657d8b8c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.74ex; height:5.843ex;" alt="{\displaystyle {\mu _{rd}}=\tan {\alpha }-{\frac {2s\sec {\alpha }}{gt^{2}}}}"></span></dd></dl> <table class="wikitable" style="text-align: right;"> <caption><i>Alcuni valori del coefficiente di <b>attrito radente</b></i>.<sup id="cite_ref-aa_4-0" class="reference"><a href="#cite_note-aa-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr> <th>Superfici</th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{rs}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{rs}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/241463f01aa368decc125ea3a94ec4748ea1b246" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.147ex; height:2.176ex;" alt="{\displaystyle \mu _{rs}}"></span> (statico)</th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{rd}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{rd}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad22647f9c2d030f09234993bd8253c7c10a3b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.235ex; height:2.176ex;" alt="{\displaystyle \mu _{rd}}"></span> (dinamico) </th></tr> <tr> <td>Legno - legno</td> <td>0,25 - 0,50</td> <td>0,20 </td></tr> <tr> <td>Acciaio - acciaio</td> <td>0,74</td> <td>0,57 </td></tr> <tr> <td>Acciaio - acciaio lubrificato</td> <td>0,11</td> <td>0,05 </td></tr> <tr> <td>Acciaio - alluminio</td> <td>0,61</td> <td>0,47 </td></tr> <tr> <td>Acciaio - ottone</td> <td>0,51</td> <td>0,44 </td></tr> <tr> <td>Acciaio - teflon</td> <td>0,04</td> <td>0,04 </td></tr> <tr> <td>Acciaio - ghiaccio</td> <td>0,027</td> <td>0,014 </td></tr> <tr> <td>Acciaio - aria</td> <td>0,001</td> <td>0,001 </td></tr> <tr> <td>Acciaio - piombo</td> <td>0,90</td> <td>n.d. </td></tr> <tr> <td>Acciaio - ghisa</td> <td>0,40</td> <td>n.d. </td></tr> <tr> <td>Acciaio - grafite</td> <td>0,10</td> <td>n.d. </td></tr> <tr> <td>Acciaio - plexiglas</td> <td>0,80</td> <td>n.d. </td></tr> <tr> <td>Acciaio - polistirene</td> <td>0,50</td> <td>n.d. </td></tr> <tr> <td>Rame - acciaio</td> <td>0,53</td> <td>0,36 </td></tr> <tr> <td>Rame - vetro</td> <td>0,68</td> <td>0,53 </td></tr> <tr> <td>Gomma - asfalto (asciutto)</td> <td>1,0</td> <td>0,8 </td></tr> <tr> <td>Gomma - asfalto (bagnato)</td> <td>0,7</td> <td>0,6 </td></tr> <tr> <td>Vetro - vetro</td> <td>0,9 - 1,0</td> <td>0,4 </td></tr> <tr> <td>Legno <a href="/wiki/Sciolina" title="Sciolina">sciolinato</a> - neve</td> <td>0,10</td> <td>0,05 </td></tr> <tr> <td>legno - cartone</td> <td>0,32</td> <td>0,23 </td></tr> <tr> <td>Teflon - Teflon </td> <td>0,04 </td> <td>0,04 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Attrito_volvente">Attrito volvente</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=5" title="Modifica la sezione Attrito volvente" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=5" title="Edit section's source code: Attrito volvente"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="thumb tright"><div class="thumbinner" style="width:308px;flex-direction:row"><div class="tsingle" style="float:left;margin:1px;width:202px;padding-left:0px;padding-right:0px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Attrito_volvente.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Attrito_volvente.svg/200px-Attrito_volvente.svg.png" decoding="async" width="200" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Attrito_volvente.svg/300px-Attrito_volvente.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Attrito_volvente.svg/400px-Attrito_volvente.svg.png 2x" data-file-width="2000" data-file-height="1414" /></a></span></div></div><div class="tsingle" style="float:left;margin:1px;width:102px;padding-left:0px;padding-right:0px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Rolling_Resistance.PNG" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Rolling_Resistance.PNG/100px-Rolling_Resistance.PNG" decoding="async" width="100" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Rolling_Resistance.PNG/150px-Rolling_Resistance.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Rolling_Resistance.PNG/200px-Rolling_Resistance.PNG 2x" data-file-width="246" data-file-height="329" /></a></span></div></div><div style="clear:left"></div><div class="thumbcaption" style="clear:left;text-align:left">La resistenza prodotta dall'attrito volvente è in generale molto minore rispetto a quella dovuta all'attrito radente. Da ciò derivano le applicazioni di ruote o rulli per il trasporto di oggetti pesanti che, se trascinati, richiederebbero una forza molto maggiore per essere spostati, e l'interposizione di cuscinetti a sfere tra perni e supporti.<sup id="cite_ref-Ard580_2-1" class="reference"><a href="#cite_note-Ard580-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></div></div></div> <p>Il <a href="/wiki/Moto_di_puro_rotolamento" title="Moto di puro rotolamento">rotolamento</a> di norma è reso possibile dalla presenza di attrito radente statico tra la ruota e il terreno; se questo attrito non ci fosse, o fosse molto piccolo (come nel caso di un terreno ghiacciato), la ruota striscerebbe senza riuscire a compiere un rotolamento puro, nel qual caso entrerebbe subito in gioco l'attrito radente dinamico che si oppone allo slittamento, riducendo progressivamente la velocità relativa fra i corpi striscianti, tende a ripristinare le condizioni di puro rotolamento. Un caso in cui il puro rotolamento può avvenire senza l'aiuto dell'attrito statico si ha quando una ruota che sta già rotolando su un piano orizzontale con velocità angolare <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =V/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =V/r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be865ad0f24f1d2b149ba400d5882e6dcbb101f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.543ex; height:2.843ex;" alt="{\displaystyle \omega =V/r}"></span>, dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> è la velocità del centro di massa della ruota, viene lasciata a sé stessa: in tal caso l'attrito statico assume il valore zero e solo l'attrito volvente può frenare il rotolamento, riducendo simultaneamente e armonicamente sia la velocità di traslazione sia quella di rotazione della ruota in modo che il puro rotolamento si conservi fino a fine corsa. </p><p>Se si applica un <a href="/wiki/Momento_meccanico" title="Momento meccanico">momento</a> alla ruota, essa inizia a rotolare senza strisciare fintanto che il momento applicato è minore di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\cdot \mu _{rs}\cdot F_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>⋅</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>s</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\cdot \mu _{rs}\cdot F_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/672f78810457de3569d33d12e92d1052e2fd2481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.822ex; height:2.843ex;" alt="{\displaystyle R\cdot \mu _{rs}\cdot F_{p}}"></span>, dove <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> è il raggio della ruota. Se il momento supera questo valore, la forza motrice applicata alla superficie della ruota supera l'attrito statico massimo e la ruota slitta mentre rotola; è la classica "sgommata" ottenuta accelerando da fermi in modo repentino. </p><p>L'effetto dell'attrito volvente si può descrivere spostando leggermente in indietro, nel senso opposto al moto, la reazione vincolare (in genere non perfettamente normale) esercitata dal piano di rotolamento sul corpo rotolante, di modo che tale reazione vincolare abbia non solo una componente contraria al moto traslatorio, ma anche un momento di forza rispetto all'asse di rotazione della ruota che si oppone al moto rotatorio. Una siffatta reazione vincolare è la sintesi schematica del campo di sforzi che sorgono e si distribuiscono sull'intera area di contatto (che non è mai veramente puntiforme o riducibile ad un segmento) tra la ruota e il terreno: la rotazione causa di fatto una deformazione dell'area di contatto e quindi una distribuzione delle forze di <a href="/wiki/Pressione" title="Pressione">pressione</a>, dovute alla forza peso, non uniforme su tutta la superficie di contatto; il risultato di queste interazioni si può riassumere dicendo che il piano di rotolamento esercita sulla ruota una forza vincolare quasi-normale, rivolta verso l'alto e all'indietro rispetto al moto, la cui linea di applicazione di norma non passa per l'asse della ruota, di modo che tale forza produce sia una debole resistenza al moto traslatorio sia un debole momento torcente opposto al senso del rotolamento in atto. Quantitativamente, questo tipo di attrito è espresso da un'equazione simile alla precedente, </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 {F}_{v}={\frac {{\mu _{v}}\cdot {F}_{\perp }}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {F}_{v}={\frac {{\mu _{v}}\cdot {F}_{\perp }}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/115a28791e2dcb0962ecfdc2d98c25ce89115616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.575ex; height:5.343ex;" alt="{\displaystyle {F}_{v}={\frac {{\mu _{v}}\cdot {F}_{\perp }}{r}}}"></span></dd></dl> <p>A parità delle altre condizioni, infatti, la resistenza opposta dall'attrito volvente è tanto minore quanto maggiore è il <a href="/wiki/Raggio_di_curvatura" class="mw-redirect" title="Raggio di curvatura">raggio di curvatura</a> del corpo che rotola. </p> <div class="mw-heading mw-heading4"><h4 id="Calcolo_del_coefficiente_di_resistenza_al_rotolamento_e_della_forza_di_resistenza_al_rotolamento">Calcolo del coefficiente di resistenza al rotolamento e della forza di resistenza al rotolamento</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=6" title="Modifica la sezione Calcolo del coefficiente di resistenza al rotolamento e della forza di resistenza al rotolamento" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=6" title="Edit section's source code: Calcolo del coefficiente di resistenza al rotolamento e della forza di resistenza al rotolamento"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Resistance_roulement_force_ou_couple.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Resistance_roulement_force_ou_couple.svg/220px-Resistance_roulement_force_ou_couple.svg.png" decoding="async" width="220" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Resistance_roulement_force_ou_couple.svg/330px-Resistance_roulement_force_ou_couple.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Resistance_roulement_force_ou_couple.svg/440px-Resistance_roulement_force_ou_couple.svg.png 2x" data-file-width="410" data-file-height="196" /></a><figcaption>Variazione della forza resistenze in base quando il movimento è dato da una spinta (al centro) o una coppia (a destra)</figcaption></figure> <p>La "resistenza al rotolamento" è definita secondo la seguente equazione:<sup id="cite_ref-NAS286_5-0" class="reference"><a href="#cite_note-NAS286-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ R=C_{rr}P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>R</mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ R=C_{rr}P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a5415ef2997d9b5a69ffe04888828b06660686" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.566ex; height:2.509ex;" alt="{\displaystyle \ R=C_{rr}P}"></span> dove: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> è la forza resistente al rotolamento,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span> parametro adimensionale <b>coefficiente resistenza rotolamento</b> o <b>coefficiente di frizione al rotolamento</b></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> è la <a href="/wiki/Forza_normale" class="mw-redirect" title="Forza normale">forza normale</a>, forza perpendicolare alla superficie su cui la ruota si muove.</li></ul> <p>Il "coefficiente di resistenza al rotolamento" per una ruota rigida lenta su una superficie perfettamente elastica, non corretta per la velocità, può essere calcolato da <sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}={\sqrt {z/d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}={\sqrt {z/d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57b3c71bdeb68ae2f1f02894c090b21ee0a10513" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.266ex; height:4.843ex;" alt="{\displaystyle C_{rr}={\sqrt {z/d}}}"></span> dove: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> è l'affondamento nella superficie</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> è il diametro della ruota rigida</li></ul> <div class="mw-heading mw-heading4"><h4 id="Calcolo_del_coefficiente_d'attrito_volvente"><span id="Calcolo_del_coefficiente_d.27attrito_volvente"></span>Calcolo del coefficiente d'attrito volvente</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=7" title="Modifica la sezione Calcolo del coefficiente d'attrito volvente" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=7" title="Edit section's source code: Calcolo del coefficiente d'attrito volvente"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Il coefficiente d'attrito volvente ha le dimensioni di una lunghezza. Similmente all'attrito radente, è possibile calcolare il coefficiente d'attrito volvente di un materiale su un altro attraverso un piano inclinato, facendolo rotolare su di esso. Lasciando andare il corpo, esso si muoverà di moto rettilineo uniformemente accelerato, con accelerazione pari 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 a={\frac {2s}{t^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>s</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {2s}{t^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042dd282a427d28b416876942203a026ee551818" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.417ex; height:5.509ex;" alt="{\displaystyle a={\frac {2s}{t^{2}}}}"></span></dd></dl> <p>La forza che muove il corpo è pari 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 F_{m}=F_{\|}-{F}_{v}=mg\sin {\alpha }-{\frac {mg{\mu _{v}}\cos \alpha }{r}}=mg\left(\sin {\alpha }-{\frac {{\mu _{v}}\cos \alpha }{r}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mi>m</mi> <mi>g</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{m}=F_{\|}-{F}_{v}=mg\sin {\alpha }-{\frac {mg{\mu _{v}}\cos \alpha }{r}}=mg\left(\sin {\alpha }-{\frac {{\mu _{v}}\cos \alpha }{r}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/250a8193b5b94a9e05b2f1086fb6695396aa804f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:65.819ex; height:4.843ex;" alt="{\displaystyle F_{m}=F_{\|}-{F}_{v}=mg\sin {\alpha }-{\frac {mg{\mu _{v}}\cos \alpha }{r}}=mg\left(\sin {\alpha }-{\frac {{\mu _{v}}\cos \alpha }{r}}\right)}"></span></dd></dl> <p>Dividendo la forza risultante per la massa del corpo si ottiene l'accelerazione: </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 a={\frac {F_{m}}{m}}={\frac {mg\left(\sin {\alpha }-{\frac {{\mu _{v}}\cos \alpha }{r}}\right)}{m}}=g\left(\sin {\alpha }-{\frac {{\mu _{v}}\cos \alpha }{r}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>m</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mi>m</mi> </mfrac> </mrow> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {F_{m}}{m}}={\frac {mg\left(\sin {\alpha }-{\frac {{\mu _{v}}\cos \alpha }{r}}\right)}{m}}=g\left(\sin {\alpha }-{\frac {{\mu _{v}}\cos \alpha }{r}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d837d9e5f9df5da4762a0f62a4b25a7d6503f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:56.411ex; height:7.509ex;" alt="{\displaystyle a={\frac {F_{m}}{m}}={\frac {mg\left(\sin {\alpha }-{\frac {{\mu _{v}}\cos \alpha }{r}}\right)}{m}}=g\left(\sin {\alpha }-{\frac {{\mu _{v}}\cos \alpha }{r}}\right)}"></span></dd></dl> <p>Ora si mettono a confronto le due formule dell'accelerazione: </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 {\frac {2s}{t^{2}}}=g\left(\sin {\alpha }-{\frac {{\mu _{v}}\cos \alpha }{r}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>s</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2s}{t^{2}}}=g\left(\sin {\alpha }-{\frac {{\mu _{v}}\cos \alpha }{r}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9699705fe97b01637af4608f74d7d7d0117a79f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.677ex; height:5.509ex;" alt="{\displaystyle {\frac {2s}{t^{2}}}=g\left(\sin {\alpha }-{\frac {{\mu _{v}}\cos \alpha }{r}}\right)}"></span></dd></dl> <p>e risolvendo l'equazione, si trova: </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 {\mu _{v}}=r\left(\tan {\alpha }-{\frac {2s\sec {\alpha }}{gt^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mrow> <mo>=</mo> <mi>r</mi> <mrow> <mo>(</mo> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>s</mi> <mi>sec</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </mrow> <mrow> <mi>g</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mu _{v}}=r\left(\tan {\alpha }-{\frac {2s\sec {\alpha }}{gt^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad62f54182832716258330a4a0c57c9ceb9944c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.793ex; height:6.176ex;" alt="{\displaystyle {\mu _{v}}=r\left(\tan {\alpha }-{\frac {2s\sec {\alpha }}{gt^{2}}}\right)}"></span></dd></dl> <style data-mw-deduplicate="TemplateStyles:r119427926">.mw-parser-output .responsive-columns{display:flex;flex-wrap:wrap;justify-content:space-between;margin-bottom:-20px}.mw-parser-output .responsive-columns>div{flex:1 1;margin:0 20px 20px 0}</style><div class="responsive-columns" style=""> <div> <table class="wikitable"> <caption>Alcuni valori del coefficiente di <b>attrito volvente</b><sup id="cite_ref-aa_4-1" class="reference"><a href="#cite_note-aa-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr> <th>Superfici</th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b289c5f6da072c6522509f7c649abdf4c6f61a82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.431ex; height:2.176ex;" alt="{\displaystyle \mu _{v}}"></span>(m) </th></tr> <tr> <td>Legno su legno</td> <td>0,0015  </td></tr> <tr> <td>Acciaio su acciaio</td> <td>0,0005 </td></tr> <tr> <td>Legno su acciaio</td> <td>0,0012  </td></tr> <tr> <td>Gomma su calcestruzzo</td> <td>0,025 </td></tr> <tr> <td>Sfere rotolanti (cuscinetti)</td> <td>0,001÷0,005 R<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </td></tr> </tbody></table> </div> <div> <table class="wikitable"> <caption>Alcuni valori del coefficiente di <b>attrito volvente</b> dedicato ai mezzi di trasporto<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr> <th>Ruote su superfici</th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b289c5f6da072c6522509f7c649abdf4c6f61a82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.431ex; height:2.176ex;" alt="{\displaystyle \mu _{v}}"></span>(m) </th></tr> <tr> <td>Ruote in ferro su binari</td> <td>0,0002÷0,0010 </td></tr> <tr> <td>Pneumatici auto su strada</td> <td>0,01÷0,035 </td></tr> <tr> <td>Pneumatici auto a risparmio energetico</td> <td>0,006÷0,009 </td></tr> <tr> <td>Tubolare 22 mm 8 bar</td> <td>0,003 </td></tr> </tbody></table> </div> </div> <p>Si noti che i valori evidenziati nelle tabelle per l'attrito volvente devono essere utilizzati solo per calcoli approssimativi. </p><p>Più in generale il coefficiente di attrito volvente è all'incirca direttamente proporzionale al coefficiente di attrito statico e inversamente proporzionale al raggio della ruota. </p><p>L'attrito statico è sempre maggiore dell'attrito dinamico e l'attrito radente è sempre maggiore dell'attrito volvente, da cui il successo dell'<a href="/wiki/Invenzione" title="Invenzione">invenzione</a> della <a href="/wiki/Ruota" title="Ruota">ruota</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Attrito_viscoso">Attrito viscoso</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=8" title="Modifica la sezione Attrito viscoso" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=8" title="Edit section's source code: Attrito viscoso"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><span class="noviewer hatnote-icon" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /></span></span> <span class="hatnote-text">Lo stesso argomento in dettaglio: <b><a href="/wiki/Attrito_viscoso" class="mw-redirect" title="Attrito viscoso">Attrito viscoso</a></b>.</span></div> </div> <p>L'attrito viscoso si riferisce alla resistenza al movimento di un fluido (liquido o gas) quando le molecole del fluido scivolano l'una sull'altra. Questo tipo di attrito è causato dalla viscosità del fluido, che è la sua resistenza interna al flusso. </p><p>Quando uno strato di fluido scorre su un altro strato adiacente, le molecole del fluido interagiscono tra loro rallentando il movimento. La forza di attrito viscoso è proporzionale alla velocità del fluido e all'area attraverso cui scorre. Un esempio comune di attrito viscoso è il movimento di un oggetto attraverso un fluido viscoso, come l'aria o l'acqua, dove la resistenza dipende dalla viscosità del fluido. </p><p>L'attrito viscoso è spesso descritto dalla legge di Newton per l'attrito viscoso, che afferma che la forza di attrito viscoso è proporzionale alla velocità del moto relativo tra gli strati del fluido. </p><p>Quando un corpo si muove all'interno di un fluido (liquido o gas) è soggetto ad una forza di attrito dovuta all'interazione del corpo con le molecole del fluido. </p><p>Tale forza di attrito è legata ad un numero adimensionale detto <b><a href="/wiki/Numero_di_Reynolds" title="Numero di Reynolds">numero di Reynolds</a></b>: </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 \mathrm {Re} ={\frac {\rho \,v\,R_{s}}{\mu }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi>v</mi> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> <mi>μ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} ={\frac {\rho \,v\,R_{s}}{\mu }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/387b29e7eed9ce192475fc0e2d5e21f5a510213d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.196ex; height:5.843ex;" alt="{\displaystyle \mathrm {Re} ={\frac {\rho \,v\,R_{s}}{\mu }},}"></span></dd></dl> <p>in cui <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19598a1629ee9016913aae8f0d3215dc4a030e9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.767ex; height:2.509ex;" alt="{\displaystyle R_{s}}"></span> è la dimensione caratteristica dell'oggetto, nel caso di un sistema isotropo il raggio della sfera, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=\vert {\vec {v}}\vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mo fence="false" stretchy="false">|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo fence="false" stretchy="false">|</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=\vert {\vec {v}}\vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89e7b796d85c057eae00326d20e9cd8ea400fbb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.695ex; height:2.843ex;" alt="{\displaystyle v=\vert {\vec {v}}\vert }"></span> la sua velocità scalare, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> la <a href="/wiki/Densit%C3%A0" title="Densità">densità</a> del fluido e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> la <a href="/wiki/Viscosit%C3%A0_dinamica" class="mw-redirect" title="Viscosità dinamica">viscosità dinamica</a> del fluido. </p><p>Se il corpo si muove a bassa velocità, così che nel flusso prevalgano le forze di viscosità rispetto a quelle d'inerzia (<i>regime di Stokes</i>) ovvero per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hbox{Re}}<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Re</mtext> </mstyle> </mrow> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hbox{Re}}<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46fb2170cec1fc1bae9e3e8d61bd963197365898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.004ex; height:2.176ex;" alt="{\displaystyle {\hbox{Re}}<1}"></span>, allora la forza di attrito è proporzionale alla velocità del corpo nel fluido; nel caso di una sfera, la forza di attrito è data in questo caso dalla legge di Stokes, </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 {\vec {F}}_{a}=-6\pi \,\mu \,R_{s}\,{\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>6</mn> <mi>π</mi> <mspace width="thinmathspace" /> <mi>μ</mi> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{a}=-6\pi \,\mu \,R_{s}\,{\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef0c684d4af5bb453dea3e13a51e1b72b27706d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.779ex; height:3.343ex;" alt="{\displaystyle {\vec {F}}_{a}=-6\pi \,\mu \,R_{s}\,{\vec {v}}}"></span></dd></dl> <p>Se la velocità del corpo è superiore (<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 {\hbox{Re}}>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Re</mtext> </mstyle> </mrow> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hbox{Re}}>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dfa47b961deedf671418c07f6280d9071b9ace4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.004ex; height:2.176ex;" alt="{\displaystyle {\hbox{Re}}>1}"></span>), le forze d'inerzia prevalgono rispetto alla viscosità ed il moto relativo del fluido è detto <b>laminare</b> (fino 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 {\hbox{Re}}=10^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Re</mtext> </mstyle> </mrow> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hbox{Re}}=10^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51429ef73784806455d3a5daa4019c2cea023e63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.221ex; height:2.676ex;" alt="{\displaystyle {\hbox{Re}}=10^{4}}"></span>) oppure <b>turbolento</b> (per <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hbox{Re}}>10^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>Re</mtext> </mstyle> </mrow> <mo>></mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hbox{Re}}>10^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3c88c772e83718f248a553309e049b02ca0100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.221ex; height:2.676ex;" alt="{\displaystyle {\hbox{Re}}>10^{4}}"></span>). </p><p>In tale caso è possibile approssimare la forza di attrito con la formula </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 {\vec {F}}_{a}=-{\frac {1}{2}}\,\rho \,S\,c_{r}\,v\,{\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi>S</mi> <mspace width="thinmathspace" /> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>v</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{a}=-{\frac {1}{2}}\,\rho \,S\,c_{r}\,v\,{\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d675cf76182b169ca0528c89948ad9b2af8e6887" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.698ex; height:5.176ex;" alt="{\displaystyle {\vec {F}}_{a}=-{\frac {1}{2}}\,\rho \,S\,c_{r}\,v\,{\vec {v}}}"></span></dd></dl> <p>dove <i>S</i> è l'area della sezione frontale del corpo e <i>c<sub>r</sub></i> un coefficiente aerodinamico di resistenza (adimensionale) che tiene conto della forma e del profilo del corpo in moto nel fluido. I valori di <i>c<sub>r</sub></i> riportati per una sfera variano tra 0,4 e 0,5, mentre si hanno valori maggiori di 1 per oggetti di forma irregolare. Per un profilo alare <i>c<sub>r</sub></i> può anche essere significativamente minore di 0,1. La velocità in questo caso è lungo l'asse della direzione di avanzamento principale e <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 c_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f0d4b75418b6deb56b7a9d66396b3f610e9007" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.981ex; height:2.009ex;" alt="{\displaystyle c_{r}}"></span> è proporzionale all'angolo di attacco del mezzo. </p> <div class="mw-heading mw-heading2"><h2 id="Anisotropia_dell'attrito"><span id="Anisotropia_dell.27attrito"></span>Anisotropia dell'attrito</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=9" title="Modifica la sezione Anisotropia dell'attrito" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=9" title="Edit section's source code: Anisotropia dell'attrito"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gli effetti di <a href="/wiki/Anisotropia" title="Anisotropia">anisotropia</a> sull'attrito dinamico includono la variazione dell'intensità dell'attrito con la direzione di scivolamento e la comparsa di componenti della forza di attrito trasversali alla direzione di scivolamento. La trattazione matematica dell'anisotropia dell'attrito richiede di trasformare il coefficiente di attrito da quantità <a href="/wiki/Scalare_(matematica)" title="Scalare (matematica)">scalare</a> a quantità tensoriale in uno spazio bidimensionale. Per costruire il <a href="/wiki/Tensore" title="Tensore">tensore</a> dei coefficienti di attrito occorre individuare le direzioni lungo le quali l'attrito si comporta coerentemente con l'interpretazione classica, cioè lungo le quali esso ha interamente componente longitudinale alla direzione di scivolamento. Queste direzioni prendono il nome di <b>direzioni principali di attrito</b>. Nel caso di attrito isotropo ogni direzione del piano di scivolamento è direzione principale di attrito e il tensore di attrito si rappresenta algebricamente con una matrice 2×2 diagonale con due coefficienti identici (coincidenti con il coefficiente di attrito radente). L'espressione classica si sostituisce con la seguente: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{r}=-{F}_{\perp }{\begin{pmatrix}\mu _{r}&0\\0&\mu _{r}\end{pmatrix}}\cdot {\widehat {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{r}=-{F}_{\perp }{\begin{pmatrix}\mu _{r}&0\\0&\mu _{r}\end{pmatrix}}\cdot {\widehat {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a69df21245c50d7e82e798c19c670c186c7a2814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.809ex; height:6.176ex;" alt="{\displaystyle {\vec {F}}_{r}=-{F}_{\perp }{\begin{pmatrix}\mu _{r}&0\\0&\mu _{r}\end{pmatrix}}\cdot {\widehat {v}}}"></span> </p><p>dove <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 {\widehat {v}}={\binom {{\hat {i}}\cos \theta }{{\hat {j}}\sin \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {v}}={\binom {{\hat {i}}\cos \theta }{{\hat {j}}\sin \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51d396bd0d07ff86e442c77c98ab400eb2c156b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.885ex; height:6.509ex;" alt="{\displaystyle {\widehat {v}}={\binom {{\hat {i}}\cos \theta }{{\hat {j}}\sin \theta }}}"></span> è il versore di scivolamento in coordinate polari (<i>θ</i> è l'angolo compreso tra la direzione di scivolamento e l'asse delle ascisse) con <i>î</i> e <i>ĵ</i> versori rispettivamente degli assi <i>x</i> e <i>y</i>. </p><p>Nel caso di attrito anisotropo il tensore di attrito rimane rappresentabile con una matrice 2×2 se il numero di direzioni principali è inferiore o uguale a due. Inoltre i coefficienti della matrice a seguito del <a href="/wiki/Secondo_principio_della_termodinamica" title="Secondo principio della termodinamica">secondo principio della termodinamica</a> devono rispettare le seguente relazioni: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{11}\geq 0;\mu _{11}\mu _{22}-\mu _{12}\mu _{21}\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>≥</mo> <mn>0</mn> <mo>;</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{11}\geq 0;\mu _{11}\mu _{22}-\mu _{12}\mu _{21}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5f61d4a2d9470507fa005e531c33e2f9eb4dcbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.786ex; height:2.676ex;" alt="{\displaystyle \mu _{11}\geq 0;\mu _{11}\mu _{22}-\mu _{12}\mu _{21}\geq 0}"></span>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Le superfici che danno origine a effetti di anisotropia dell'attrito sono superfici strutturate a diverse scale dal punto di vista morfologico. Possono essere superfici ingegnerizzate o superfici cristalline.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Effetti">Effetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=10" title="Modifica la sezione Effetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=10" title="Edit section's source code: Effetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gli effetti dell'attrito sono la dispersione dell'energia meccanica (<a href="/wiki/Energia_cinetica" title="Energia cinetica">energia cinetica</a>) in calore, il che riduce il rendimento del movimento, ma in alcuni casi questo attrito può essere utile, qualora non si cerchi un movimento ma un'adesione/controllo, soprattutto in ambito stradale, o nelle attività fisiche, permettendo gli spostamenti e azioni che altrimenti non sarebbero possibili, difatti la tenuta stradale e la camminata/passeggiata, sono possibili anche grazie all'attrito con il suolo. </p><p>L'anisotropia dell'attrito aggiunge a questi effetti anche possibili deviazioni delle traiettorie di scivolamento dovute all'azione delle componenti trasversali dell'attrito. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=11" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=11" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Giusto Bellavitis, <a rel="nofollow" class="external text" href="https://gutenberg.beic.it/webclient/DeliveryManager?pid=9277486&search_terms=DTL32"><span style="font-style:italic;">Alcune considerazioni sugli effetti dell'attrito e sul modo di calcolarli</span></a>, in <span style="font-style:italic;">Atti delle adunanze dell'Imperial Regio Istituto veneto di scienze, lettere ed arti</span>, 1846-1847.</cite></span> </li> <li id="cite_note-Ard580-2"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-Ard580_2-0">a</a></i></sup> <sup><i><a href="#cite_ref-Ard580_2-1">b</a></i></sup></span> <span class="reference-text"><cite class="citation cita" style="font-style:normal"><a href="#CITEREFArduino">Arduino</a>, p. 580</cite>.</span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text"><cite class="citation testo" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://goldbook.iupac.org/F02530.html"><span style="font-style:italic;">IUPAC Gold Book, "friction factor"</span></a>.</cite></span> </li> <li id="cite_note-aa-4"><span class="mw-cite-backlink"><b>^</b> <sup><i><a href="#cite_ref-aa_4-0">a</a></i></sup> <sup><i><a href="#cite_ref-aa_4-1">b</a></i></sup></span> <span class="reference-text">Per una lista più completa si veda <cite class="citation testo" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190201171526/http://www.roymech.co.uk/Useful_Tables/Tribology/co_of_frict.htm#coef"><span style="font-style:italic;">www.roymech.co.uk</span></a> <small>(archiviato dall'<abbr title="http://www.roymech.co.uk/Useful_Tables/Tribology/co_of_frict.htm#coef">url originale</abbr> il 1º febbraio 2019)</small>.</cite></span> </li> <li id="cite_note-NAS286-5"><a href="#cite_ref-NAS286_5-0"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://onlinepubs.trb.org/onlinepubs/sr/sr286.pdf"><span style="font-style:italic;">Tires and Passenger Vehicle Fuel Economy: Informing Consumers, Improving Performance -- Special Report 286. National Academy of Sciences, Transportation Research Board, 2006</span></a> (<span style="font-weight: bolder; font-size:80%"><abbr title="documento in formato PDF">PDF</abbr></span>), su <span style="font-style:italic;">onlinepubs.trb.org</span>. <small>URL consultato l'11 agosto 2007</small>.</cite></span> </li> <li id="cite_note-6"><a href="#cite_ref-6"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Massimo Guiggiani, <span style="font-style:italic;">The Science of Vehicle Dynamics</span>, Springer Cham, 5 maggio 2018, p. 22, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-3-319-73220-6" title="Speciale:RicercaISBN/978-3-319-73220-6">978-3-319-73220-6</a>.</cite></span> </li> <li id="cite_note-7"><a href="#cite_ref-7"><b>^</b></a> <span class="reference-text">R è il raggio del corpo volvente. Cfr. <cite class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181111154454/http://www.roymech.co.uk/Useful_Tables/Tribology/Bearing%20Friction.html"><span style="font-style:italic;">Rolling Bearing Friction</span></a>, su <span style="font-style:italic;">Roymech.co.uk</span>. <small>URL consultato il 14 novembre 2018</small> <small>(archiviato dall'<abbr title="http://www.roymech.co.uk/Useful_Tables/Tribology/Bearing%20Friction.html">url originale</abbr> l'11 novembre 2018)</small>.</cite></span> </li> <li id="cite_note-8"><a href="#cite_ref-8"><b>^</b></a> <span class="reference-text"><cite class="citation testo" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://web.iitd.ac.in/~pmvs/courses/mel713/mel713-4.ppt"><span style="font-style:italic;">Energy Consumption & Power Requirements of A Vehicle</span></a> (<span style="font-weight: bolder; font-size:80%"><abbr title="presentazione PowerPoint 97-2003">PPT</abbr></span>).</cite></span> </li> <li id="cite_note-9"><a href="#cite_ref-9"><b>^</b></a> <span class="reference-text"><cite class="citation libro" style="font-style:normal"> Zmitrowicz, Alfred., <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/827732454"><span style="font-style:italic;">Constutive modelling of anisotropic phenomena of friction, wear and frictional heat</span></a>, Instytut Maszyn Przepływowych PAN, 1993, <a href="/wiki/Online_Computer_Library_Center" title="Online Computer Library Center">OCLC</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/oclc/827732454">827732454</a>. <small>URL consultato l'11 aprile 2020</small>.</cite></span> </li> <li id="cite_note-10"><a href="#cite_ref-10"><b>^</b></a> <span class="reference-text"><cite class="citation pubblicazione" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) M. Campione e E. Fumagalli, <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevLett.105.166103"><span style="font-style:italic;">Friction Anisotropy of the Surface of Organic Crystals and Its Impact on Scanning Force Microscopy</span></a>, in <span style="font-style:italic;">Physical Review Letters</span>, vol. 105, n. 16, 12 ottobre 2010, p. 166103, <a href="/wiki/Digital_object_identifier" title="Digital object identifier">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1103%2FPhysRevLett.105.166103">10.1103/PhysRevLett.105.166103</a>. <small>URL consultato l'11 aprile 2020</small>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=12" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=12" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFFunaioli" class="citation libro" style="font-style:normal"> Ettore Funaioli, Alberto Maggiore, Umberto Meneghetti, <span style="font-style:italic;">Lezioni di <a href="/wiki/Meccanica_applicata" title="Meccanica applicata">meccanica applicata</a> alle macchine</span>, 1ª ed., Pàtron, 2008, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/88-555-2829-7" title="Speciale:RicercaISBN/88-555-2829-7">88-555-2829-7</a>.</cite></li> <li><cite id="CITEREFArduino" class="citation libro" style="font-style:normal"> Gianni Arduino, Renata Moggi, <span style="font-style:italic;">Educazione tecnica</span>, 1ª ed., Lattes, 1990.</cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=13" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=13" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Resistenza_passiva" title="Resistenza passiva">Resistenza passiva</a></li> <li><a href="/wiki/Resistenza_fluidodinamica" title="Resistenza fluidodinamica">Resistenza fluidodinamica</a></li> <li><a href="/wiki/Ferodo" title="Ferodo">Ferodo</a></li> <li><a href="/wiki/Teoria_di_Hertz" title="Teoria di Hertz">Teoria di Hertz</a></li> <li><a href="/wiki/Modello_di_Tomlinson" title="Modello di Tomlinson">Modello di Tomlinson</a></li> <li><a href="/wiki/Attrito_negativo" class="mw-redirect" title="Attrito negativo">Attrito negativo</a></li> <li><a href="/wiki/Lubrificante" title="Lubrificante">Lubrificante</a></li> <li><a href="/wiki/Tribologia" title="Tribologia">Tribologia</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=14" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=14" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><a href="https://it.wiktionary.org/wiki/attrito" class="extiw" title="wikt:attrito">Wikizionario</a></li> <li class="" title=""><span class="plainlinks" title="commons:Category:Friction"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Friction?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://it.wiktionary.org/wiki/" title="Collabora a Wikizionario"><img alt="Collabora a Wikizionario" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/18px-Wiktionary_small.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/27px-Wiktionary_small.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/36px-Wiktionary_small.svg.png 2x" data-file-width="350" data-file-height="350" /></a></span> <a href="https://it.wiktionary.org/wiki/" class="extiw" title="wikt:">Wikizionario</a> contiene il lemma di dizionario «<b><a href="https://it.wiktionary.org/wiki/attrito" class="extiw" title="wikt:attrito">attrito</a></b>»</li> <li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file sull'<b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Friction?uselang=it">attrito</a></span></b></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Attrito&veaction=edit&section=15" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Attrito&action=edit&section=15" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFTreccani.it" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/attrito"><span style="font-style:italic;">Attrito</span></a>, su <span style="font-style:italic;">Treccani.it – Enciclopedie on line</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82580#P3365" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFEnciclopedia_Italiana_1930" class="citation libro" style="font-style:normal"> Enrico Fermi, <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/attrito_(Enciclopedia-Italiana)/"><span style="font-style:italic;">ATTRITO</span></a>, in <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Treccani" title="Enciclopedia Treccani">Enciclopedia Italiana</a></span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 1930.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82580#P4223" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFEnciclopedia_Italiana_1938" class="citation libro" style="font-style:normal"> Antonio Capetti, <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/attrito_res-347a0046-8b74-11dc-8e9d-0016357eee51_(Enciclopedia-Italiana)/"><span style="font-style:italic;">ATTRITO</span></a>, in <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Treccani" title="Enciclopedia Treccani">Enciclopedia Italiana</a></span>, I Appendice, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 1938.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82580#P4223" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFDizionario_delle_scienze_fisiche" class="citation libro" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/attrito_(Dizionario-delle-Scienze-Fisiche)/"><span style="font-style:italic;">Attrito</span></a>, in <span style="font-style:italic;">Dizionario delle scienze fisiche</span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 1996.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82580#P10017" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFVocabolario_Treccani" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/vocabolario/attrito2"><span style="font-style:italic;">Attrito²</span></a>, su <span style="font-style:italic;"><a href="/wiki/Vocabolario_Treccani" title="Vocabolario Treccani">Vocabolario Treccani</a></span>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82580#P5844" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFSapere.it" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.sapere.it/enciclopedia/attrito+.html"><span style="font-style:italic;">attrito</span></a>, su <span style="font-style:italic;">sapere.it</span>, <a href="/wiki/De_Agostini" title="De Agostini">De Agostini</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82580#P6706" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFBritannica.com" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://www.britannica.com/science/friction"><span style="font-style:italic;">friction</span></a> / <a rel="nofollow" class="external text" href="https://www.britannica.com/science/static-friction"><span style="font-style:italic;">static friction</span></a>, su <span style="font-style:italic;"><a href="/wiki/Enciclopedia_Britannica" title="Enciclopedia Britannica">Enciclopedia Britannica</a></span>, Encyclopædia Britannica, Inc.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82580#P1417" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite id="CITEREFOpen_Library" class="citation web" style="font-style:normal">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="https://openlibrary.org/subjects/friction"><span style="font-style:italic;">Opere riguardanti Friction</span></a>, su <span style="font-style:italic;"><a href="/wiki/Open_Library" class="mw-redirect" title="Open Library">Open Library</a></span>, <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>.</cite> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q82580#P3847" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></li> <li><cite class="citation web" style="font-style:normal"> Puerari, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160313234909/http://www-3.unipv.it/electric/azionamenti/Puerari/Cap_6.pdf"><span style="font-style:italic;">L'attrito</span></a> (<span style="font-weight: bolder; font-size:80%"><abbr title="documento in formato PDF">PDF</abbr></span>), su <span style="font-style:italic;">Università di Pavia</span>. <small>URL consultato il 19 dicembre 2020</small> <small>(archiviato dall'<abbr title="http://www-1.unipv.it/electric/azionamenti/Puerari/Cap_6.pdf">url originale</abbr> il 13 marzo 2016)</small>.</cite></li> <li><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://www.itchiavari.org/chimica/tabelle/attrito.html"><span style="font-style:italic;">Coefficienti di attrito statico e dinamico</span></a>, su <span style="font-style:italic;">itchiavari.org</span>.</cite></li></ul> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA th{background-color:#f2f2f2;font-weight:bold;width:20%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .CdA th{background-color:#202122}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .CdA th{background-color:#202122}}</style><table class="CdA"><tbody><tr><th><a href="/wiki/Aiuto:Controllo_di_autorit%C3%A0" title="Aiuto:Controllo di autorità">Controllo di autorità</a></th><td><a href="/wiki/Nuovo_soggettario" title="Nuovo soggettario">Thesaurus BNCF</a> <span class="uid"><a rel="nofollow" class="external text" href="https://thes.bncf.firenze.sbn.it/termine.php?id=31340">31340</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr></span>) <a rel="nofollow" class="external text" href="http://id.loc.gov/authorities/subjects/sh85051971">sh85051971</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="tedesco">DE</abbr></span>) <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4049098-1">4049098-1</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Francia" title="Biblioteca nazionale di Francia">BNF</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="francese">FR</abbr></span>) <a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11941523z">cb11941523z</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11941523z">(data)</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_nazionale_di_Israele" title="Biblioteca nazionale di Israele">J9U</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr title="inglese">EN</abbr>, <abbr title="ebraico">HE</abbr></span>) <a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007553002905171">987007553002905171</a></span><span style="font-weight:bold;"> ·</span> <a href="/wiki/Biblioteca_della_Dieta_nazionale_del_Giappone" title="Biblioteca della Dieta nazionale del Giappone">NDL</a> <span class="uid">(<span style="font-weight:bolder; font-size:80%"><abbr 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