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<title>Tangentialkraft</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Tangentialkraft</h1>

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Die Tangentialkraft wirkt tangential zur Bahnkurve eines bewegten Körpers. Das heißt, sie wirkt entlang der Richtung, in die sich das Objekt gerade bewegt. Die Richtungsänderung ist für sie unbedeutend. Dies kann, wenn keine weiteren Kräfte wirken, zu einer Geschwindigkeitsänderung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta v}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta v}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e18b43e4225eeaafeeb25aefc4ee90bd86f004dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.063ex; height:2.176ex;" alt="{\displaystyle \Delta v}" loading="lazy"> in Richtung der Kraft führen<a href="#cite_note-1">[1]</a>.

Wirken in einer Ebene mehrere Kräfte auf die Beschleunigung eines Körpers, so lässt sich die Resultierende in die beiden senkrechten Komponenten der Tangentialkraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{||}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{||}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8f041ffc86c8e5958e67c1e797d329dfb49cf41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.918ex; height:3.676ex;" alt="{\displaystyle {\vec {F}}_{||}}" loading="lazy"> und der Normalkraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{\bot }}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{\bot }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f6d9b33ccfc25d69eca470e169b6abf81534ff02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.282ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{\bot }}" loading="lazy"> zerlegen. Die Tangentialkomponente verändert nur den Betrag der Geschwindigkeit und nicht die Richtung. Die Normalkraft ändert nur die Bewegungsrichtung in Abhängigkeit von der Geschwindigkeit des Körpers und der Bahnform.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Freier_Fall_im_homogenen_Schwerefeld">Freier Fall im homogenen Schwerefeld</h3></div>

Der <a href="Freier_Fall" title="Freier Fall">freie Fall</a> im homogenen Schwerefeld beschreibt die Bewegung eines Körpers, der nur durch die konstante Gravitationskraft beeinflusst wird, ohne dass Luftwiderstand oder andere Kräfte wirken. Im homogenen Schwerefeld ist die Gravitationskraft überall gleich groß und wirkt in die gleiche Richtung. Die Tangentialkraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{\|}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{\|}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11605d82d13fbacd2c86eb1c5d3b1a31ce277ed9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.825ex; height:3.676ex;" alt="{\displaystyle {\vec {F}}_{\|}}" loading="lazy"> verläuft parallel zur Gewichtskraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{\text{G}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{\text{G}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5e1c937e698431368fef7c2db5fd1674b3163b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.293ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{\text{G}}}" loading="lazy">.

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{\|}={\vec {F}}_{\text{G}}=m{\vec {g}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95c47797cd57dad91c0020698cbc462fd2941d62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.53ex; height:3.676ex;" alt="{\displaystyle {\vec {F}}_{\|}={\vec {F}}_{\text{G}}=m{\vec {g}}}" loading="lazy"></dd></dl>

Der Körper erfährt also eine konstante Beschleunigung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" 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}" loading="lazy">. Ohne Reibung und Luftwiderstand ist der zurückgelegte Weg <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

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<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> proportional dem Quadrat der Fallzeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\sim t^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle s\sim t^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2608e3bde1d3d6ae49f66551baa50273c44ba386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.083ex; height:2.676ex;" alt="{\displaystyle s\sim t^{2}}" loading="lazy"><a href="#cite_note-3">[3]</a>.

<div class="mw-heading mw-heading3"><h3 id="Hangabtriebskraft_an_der_schiefen_Ebene">Hangabtriebskraft an der schiefen Ebene</h3></div>

In einem homogenen Schwerefeld beschleunigt die Tangentialkraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{\|}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{\|}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11605d82d13fbacd2c86eb1c5d3b1a31ce277ed9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.825ex; height:3.676ex;" alt="{\displaystyle {\vec {F}}_{\|}}" loading="lazy"> einen Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" 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}" loading="lazy"> auf der <a href="Schiefe_Ebene" title="Schiefe Ebene">schiefen Ebene</a> ohne Wirkung der Reibung nach unten<a href="#cite_note-4">[4]</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{\|}={\vec {F}}_{G}+{\vec {F}}_{Z}\quad {\text{mit}}\quad |{\vec {F}}_{\|}|=|{\vec {F}}_{G}|\sin \alpha =mg\sin \alpha }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ec0896bc0573060af1c7a6988dd52b04fc41b60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:51.678ex; height:3.676ex;" alt="{\displaystyle {\vec {F}}_{\|}={\vec {F}}_{G}+{\vec {F}}_{Z}\quad {\text{mit}}\quad |{\vec {F}}_{\|}|=|{\vec {F}}_{G}|\sin \alpha =mg\sin \alpha }" loading="lazy"></dd></dl>

Die Beschleunigung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\sin \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle g\sin \alpha }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9823ad1c494283e9f896a3758bbd4b24dc657338.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.233ex; height:2.509ex;" alt="{\displaystyle g\sin \alpha }" loading="lazy"> durch die Hangabtriebskraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{\|}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11605d82d13fbacd2c86eb1c5d3b1a31ce277ed9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.825ex; height:3.676ex;" alt="{\displaystyle {\vec {F}}_{\|}}" loading="lazy"> ist um den Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \sin \alpha }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/071763dba322940a74766aaf79f4569c5954dcf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.73ex; height:2.176ex;" alt="{\displaystyle \sin \alpha }" loading="lazy"> kleiner als im freien Fall<a href="#cite_note-5">[5]</a>.

Die von der schiefen Ebene auf den Körper <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{Z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a85b45cd584c0e47cd1ead18d2e0257850fd0403.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.191ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{Z}}" loading="lazy"> ist betragsmäßig gleich der Normalkomponente der Gewichtskraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {F}}_{G\bot }|=mg\sin \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle |{\vec {F}}_{G\bot }|=mg\sin \alpha }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d14af064af774e18c7e18827ff61ae917226c93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.239ex; height:3.343ex;" alt="{\displaystyle |{\vec {F}}_{G\bot }|=mg\sin \alpha }" loading="lazy">, wirkt jedoch in entgegengesetzter Richtung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{Z}=-{\vec {F}}_{G\bot }}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{Z}=-{\vec {F}}_{G\bot }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c2401d1b6189bac9f3ecc6fbad3863798d08a37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.671ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{Z}=-{\vec {F}}_{G\bot }}" loading="lazy">. Es herrscht Kräftegleichgewicht senkrecht zur Ebene, so dass der Körper in dieser Richtung nicht beschleunigt wird<a href="#cite_note-6">[6]</a>.

<div class="mw-heading mw-heading3"><h3 id="Reibungskraft_zwischen_festen_Körpern">Reibungskraft zwischen festen Körpern</h3></div>

Ein Beispiel für eine Tangentialkraft ist die Kraft, die uns in den Sitz drückt, wenn wir im Auto Gas geben. Die Tangentialgeschwindigkeit ändert sich, wenn nicht andere Kräfte dies verhindern. Im realen Fall mit Reibung muss die Beschleunigungskraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\|}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{\|}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05426e137cfb963c851aa07e14643039f5314615.svg" 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_{\|}}" loading="lazy"> einen Schwellenwert überschreiten, damit sich ein Körper in Bewegung setzt. Diese Beschleunigungskraft muss immer größer sein als die <a href="Reibung" title="Reibung">Reibungskraft</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\text{R}}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{\text{R}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c34144611effde123228b6d09a6588a155145c39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.936ex; height:2.509ex;" alt="{\displaystyle F_{\text{R}}}" loading="lazy">, die als Tangentialkraft mit dem Einheitsvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{\|}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7251de8e23116c0b4cc34ea11e8495d794fbc476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.277ex; height:3.176ex;" alt="{\displaystyle {\vec {e}}_{\|}}" loading="lazy"> immer der Bewegungsrichtung entgegengerichtet ist. Mit der Normalkraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\bot }}">

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<annotation encoding="application/x-tex">{\displaystyle F_{\bot }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b001d6b1d388edc478fed1ca0379aed3adec284c.svg" 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_{\bot }}" loading="lazy"> des Körpers senkrecht zur Auflage beträgt die Reibungskraft erfahrungsgemäß<a href="#cite_note-7">[7]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{\text{R}}=-\mu \,F_{\bot }{\vec {e}}_{\|}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54f18c5776ae59317972a4df66ca05dbf666c670.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.191ex; height:3.676ex;" alt="{\displaystyle {\vec {F}}_{\text{R}}=-\mu \,F_{\bot }{\vec {e}}_{\|}}" loading="lazy"></dd></dl>

Die Reibungszahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" 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 }" loading="lazy"> steigt von der Rollreibung über die Gleitreibung zur Haftreibung an<a href="#cite_note-8">[8]</a>.

<div class="mw-heading mw-heading3"><h3 id="Luftwiderstand">Luftwiderstand</h3></div>

Der <a href="Str%C3%B6mungswiderstand" title="Strömungswiderstand">Luftwiderstand</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\text{Luft}}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{\text{Luft}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e44fc2c7a294f07b2ab4659a9342d0396cacb7a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.811ex; height:2.509ex;" alt="{\displaystyle F_{\text{Luft}}}" loading="lazy"> ist die Reibungskraft, die einem sich mit der Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle F_{\text{Luft}}={\textstyle {\frac {1}{2}}}C_{\text{W}}\cdot \rho \cdot A\cdot v^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5821423515aa870d0506dfe24159d91751658359.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.315ex; height:3.509ex;" alt="{\displaystyle F_{\text{Luft}}={\textstyle {\frac {1}{2}}}C_{\text{W}}\cdot \rho \cdot A\cdot v^{2}}" loading="lazy"></dd></dl>

Sie ist also eine Tangentialkraft, die nicht nur quadratisch mit der Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle v^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4faa98a21ac8133ab466999288849492be28b3d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.182ex; height:2.676ex;" alt="{\displaystyle v^{2}}" loading="lazy"> wächst, sondern auch proportional zum Widerstandsbeiwert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{\text{W}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{\text{W}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d1de700aa63b2b79f948a5e12af0f09462f85df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.583ex; height:2.509ex;" alt="{\displaystyle C_{\text{W}}}" loading="lazy">, zur Luftdichte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" 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 }" loading="lazy"> und zur Querschnittsfläche <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> ist<a href="#cite_note-9">[9]</a>.

<div class="mw-heading mw-heading3"><h3 id="Viskose_Reibung_in_Flüssigkeiten">Viskose Reibung in Flüssigkeiten</h3></div>

Eine Kugel mit dem Radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> sinkt mit konstant kleiner Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> durch eine Flüssigkeit (<a href="Reynolds-Zahl" title="Reynolds-Zahl">Reynolds-Zahl</a> Re<0,4<a href="#cite_note-10">[10]</a>). Die der Bewegung entgegenwirkende viskose Reibungskraft berechnet sich zu

<msub>

<mover>

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<mtext>viskos</mtext>

</mrow>

</msub>

<mover>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{\text{viskos}}=-6\pi \cdot \eta \cdot r\cdot {\vec {v}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23c7b237b809633cc9bc4ae2ede92b8ba32c6881.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.146ex; height:3.343ex;" alt="{\displaystyle {\vec {F}}_{\text{viskos}}=-6\pi \cdot \eta \cdot r\cdot {\vec {v}}}" loading="lazy"></dd></dl>

Die viskose Reibung oder <a href="Gesetz_von_Stokes" title="Gesetz von Stokes">Stokes-Reibung</a><a href="#cite_note-11">[11]</a> tritt in Flüssigkeiten auf und hängt von der dynamischen Viskosität <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> der Flüssigkeit ab. Sie ist eine Tangentialkraft. Bei konstanter Geschwindigkeit herrscht Kräftegleichgewicht zwischen Reibungskraft und Gewichtskraft:

<msub>

<mover>

</mover>

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</mrow>

</msub>

<msub>

<mover>

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<mtext>viskos</mtext>

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</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

<mtext>viskos</mtext>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

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<mrow>

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</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0={\vec {F}}_{\|}+{\vec {F}}_{\text{viskos}}={\vec {F}}_{\text{G}}+{\vec {F}}_{\text{viskos}}=m{\vec {g}}-6\pi \cdot \eta \cdot r\cdot {\vec {v}}\quad \Rightarrow \quad v={\frac {m\cdot g}{6\pi \cdot \eta \cdot r}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b8130d83d3e0cf16ced9153b7b71bb3de49a69e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:73.258ex; height:5.343ex;" alt="{\displaystyle 0={\vec {F}}_{\|}+{\vec {F}}_{\text{viskos}}={\vec {F}}_{\text{G}}+{\vec {F}}_{\text{viskos}}=m{\vec {g}}-6\pi \cdot \eta \cdot r\cdot {\vec {v}}\quad \Rightarrow \quad v={\frac {m\cdot g}{6\pi \cdot \eta \cdot r}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Tangentialkraft_am_Fadenpendel">Tangentialkraft am Fadenpendel</h3></div>

Beim <a href="Pendel" title="Pendel">Fadenpendel</a> schwingt ein Körper der Masse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" 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}" loading="lazy"> an einem masselosen Faden fester Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"> im homogenen Gravitationsfeld <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {g}}=-g\,{\vec {e}}_{z}}">

<mover>

</mover>

</mrow>

<msub>

<mover>

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</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {g}}=-g\,{\vec {e}}_{z}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dce14d0189a5dce4001bc6df5ba54730be03a646.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.809ex; height:2.676ex;" alt="{\displaystyle {\vec {g}}=-g\,{\vec {e}}_{z}}" loading="lazy"> auf einer Kreisbahn mit dem Auslenkungswinkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> hin und her<a href="#cite_note-12">[12]</a>. Die Kraft, die den Pendelkörper auf seiner Bahn beschleunigt, ist die Tangentialkraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{\|}=F_{\|}\,{\vec {e}}_{T}}">

<msub>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{\|}=F_{\|}\,{\vec {e}}_{T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec60c892b346c35a96e53ca7d96a3b2e46bee9b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.472ex; height:3.676ex;" alt="{\displaystyle {\vec {F}}_{\|}=F_{\|}\,{\vec {e}}_{T}}" loading="lazy">. Eine Kraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{Z}=F_{Z}\,{\vec {e}}_{N}}">

<msub>

<mover>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{Z}=F_{Z}\,{\vec {e}}_{N}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c8d8e5b0065cf1b2ae69ceca3b048f413cf6445.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.507ex; height:3.176ex;" alt="{\displaystyle {\vec {F}}_{Z}=F_{Z}\,{\vec {e}}_{N}}" loading="lazy"> senkrecht zur Bewegungsrichtung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{T}}">

<msub>

<mover>

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</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{T}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a2832ca12c9e012986bbba5e04f56cdf52eca82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.612ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{T}}" loading="lazy"> zwingt den Körper auf die Kreisbahn. Die Bewegung erfolgt in einer Ebene und wir brauchen die binormale Komponente der Zwangskraft entlang <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{B}={\vec {e}}_{T}\times {\vec {e}}_{N}}">

<msub>

<mover>

</mover>

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<msub>

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<msub>

<mover>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{B}={\vec {e}}_{T}\times {\vec {e}}_{N}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b15b069501390af84a1c7697744da2cf046a9620.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.168ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{B}={\vec {e}}_{T}\times {\vec {e}}_{N}}" loading="lazy"> nicht zu berücksichtigen. Die Newtonsche <a href="Bewegungsgleichung" title="Bewegungsgleichung">Bewegungsgleichung</a> mit einer Zwangskraft lautet<a href="#cite_note-13">[13]</a>:

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<annotation encoding="application/x-tex">{\displaystyle m\,{\vec {a}}=m\,{\vec {g}}+F_{Z}\,{\vec {e}}_{N}=-mg\,{\vec {e}}_{z}+F_{Z}\,{\vec {e}}_{N}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2f5423f87d46254a26e532b0517a2813c5456d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.146ex; height:2.676ex;" alt="{\displaystyle m\,{\vec {a}}=m\,{\vec {g}}+F_{Z}\,{\vec {e}}_{N}=-mg\,{\vec {e}}_{z}+F_{Z}\,{\vec {e}}_{N}}" loading="lazy"></dd></dl>

Mit <a href="Polarkoordinaten" title="Polarkoordinaten">Polarkoordinaten</a> und parametrisiert durch die Bogenlänge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=l\varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=l\varphi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/337111c8d3601d98b1f8b52ffd69a1e16bd87b01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.402ex; height:2.676ex;" alt="{\displaystyle s=l\varphi }" loading="lazy"> wird die Lage des Pendelkörpers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{P}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31e2edc7d32b8a10bb07ac6fc2c54eee95ee5cb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.69ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{P}}" loading="lazy"> beschrieben:

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</mtable>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{P}={\begin{pmatrix}x\\z\\\end{pmatrix}}={\begin{pmatrix}l\sin \varphi \\-l\cos \varphi \\\end{pmatrix}}={\begin{pmatrix}l\sin {\frac {s}{l}}\\-l\cos {\frac {s}{l}}\\\end{pmatrix}}=l\,{\vec {e}}_{r}\quad {\text{mit }}{\vec {e}}_{r}={\begin{pmatrix}\sin {\frac {s}{l}}\\-\cos {\frac {s}{l}}\\\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24eb72047156008ee13f7d63122d658693da8d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:71.788ex; height:7.509ex;" alt="{\displaystyle {\vec {r}}_{P}={\begin{pmatrix}x\\z\\\end{pmatrix}}={\begin{pmatrix}l\sin \varphi \\-l\cos \varphi \\\end{pmatrix}}={\begin{pmatrix}l\sin {\frac {s}{l}}\\-l\cos {\frac {s}{l}}\\\end{pmatrix}}=l\,{\vec {e}}_{r}\quad {\text{mit }}{\vec {e}}_{r}={\begin{pmatrix}\sin {\frac {s}{l}}\\-\cos {\frac {s}{l}}\\\end{pmatrix}}}" loading="lazy"></dd></dl>

Die Geschwindigkeit ist die erste Zeitableitung des Ortsvektors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\vec {r}}}_{P}={\vec {v}}_{P}}">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {r}}}_{P}={\vec {v}}_{P}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf367dcff6b20e8c5f10eaf8c65fb52463f60b04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.464ex; height:3.176ex;" alt="{\displaystyle {\dot {\vec {r}}}_{P}={\vec {v}}_{P}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v={\dot {s}}}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v={\dot {s}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/824f65aa4d340737cb71a897af704e135131cee1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.518ex; height:2.176ex;" alt="{\displaystyle v={\dot {s}}}" loading="lazy">, wobei die Zeitableitung durch einen Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{.}=\partial /\partial t}">

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<annotation encoding="application/x-tex">{\displaystyle ^{.}=\partial /\partial t}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e556b1861b4b17e15701454b7b97ff331a1b0c4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.426ex; height:2.843ex;" alt="{\displaystyle ^{.}=\partial /\partial t}" loading="lazy"> über der abzuleitenden Größe symbolisiert wird.

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<mtext>mit dem Tangenteneinheitsvektor </mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}={\dot {\vec {r}}}_{P}={\dot {s}}\,{\frac {{\text{d}}{\vec {r}}_{P}}{{\text{d}}s}}={\dot {s}}\,{\begin{pmatrix}\cos {\frac {s}{l}}\\\sin {\frac {s}{l}}\\\end{pmatrix}}=v\,{\vec {e}}_{T}\quad {\text{mit dem Tangenteneinheitsvektor }}{\vec {e}}_{T}={\frac {{\text{d}}{\vec {r}}_{P}}{{\text{d}}s}}={\begin{pmatrix}\cos {\frac {s}{l}}\\\sin {\frac {s}{l}}\\\end{pmatrix}}={\begin{pmatrix}\cos {\varphi }\\\sin {\varphi }\\\end{pmatrix}}={\vec {e}}_{\varphi }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf281c740f35d08827bdd537d917ab7421c52e7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:116.111ex; height:7.509ex;" alt="{\displaystyle {\vec {v}}={\dot {\vec {r}}}_{P}={\dot {s}}\,{\frac {{\text{d}}{\vec {r}}_{P}}{{\text{d}}s}}={\dot {s}}\,{\begin{pmatrix}\cos {\frac {s}{l}}\\\sin {\frac {s}{l}}\\\end{pmatrix}}=v\,{\vec {e}}_{T}\quad {\text{mit dem Tangenteneinheitsvektor }}{\vec {e}}_{T}={\frac {{\text{d}}{\vec {r}}_{P}}{{\text{d}}s}}={\begin{pmatrix}\cos {\frac {s}{l}}\\\sin {\frac {s}{l}}\\\end{pmatrix}}={\begin{pmatrix}\cos {\varphi }\\\sin {\varphi }\\\end{pmatrix}}={\vec {e}}_{\varphi }}" loading="lazy"></dd></dl>

Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{N}\cdot {\vec {e}}_{T}=0}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{\|}=F_{\|}\,{\vec {e}}_{T}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle F_{\|}=m\,{\vec {a}}\cdot {\vec {e}}_{T}=-mg\,{\vec {e}}_{z}\cdot {\vec {e}}_{T}+F_{Z}\,{\vec {e}}_{N}\cdot {\vec {e}}_{T}=-mg\,\sin {\varphi }\quad {\text{und}}\quad {\vec {F}}_{\|}=F_{\|}\,{\vec {e}}_{T}=-mg\,\sin {\varphi }\,{\vec {e}}_{\varphi }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ef3a2ed091193a5108bedb0f6067527ed902be5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:92.023ex; height:3.676ex;" alt="{\displaystyle F_{\|}=m\,{\vec {a}}\cdot {\vec {e}}_{T}=-mg\,{\vec {e}}_{z}\cdot {\vec {e}}_{T}+F_{Z}\,{\vec {e}}_{N}\cdot {\vec {e}}_{T}=-mg\,\sin {\varphi }\quad {\text{und}}\quad {\vec {F}}_{\|}=F_{\|}\,{\vec {e}}_{T}=-mg\,\sin {\varphi }\,{\vec {e}}_{\varphi }}" loading="lazy"></dd></dl>

Diese Kraft wirkt entlang der Bewegungsrichtung des Pendels und ist am größten, wenn das Pendel im Umkehrpunkt seiner Schwingung seine höchste Lage erreicht.

Für die Zwangskraft ist die Beschleunigung als zeitliche Ableitung der Geschwindigkeit zu berechnen:

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<mtext>mit dem Normaleneinheitsvektor </mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {a}}={\dot {\vec {v}}}={\ddot {s}}\,{\frac {{\text{d}}{\vec {r}}_{P}}{{\text{d}}s}}+{\dot {s}}^{2}\,{\frac {{\text{d}}^{2}{\vec {r}}_{P}}{{\text{d}}s^{2}}}={\ddot {s}}\,{\vec {e}}_{T}+{\frac {v^{2}}{l}}{\begin{pmatrix}-\sin {\frac {s}{l}}\\\cos {\frac {s}{l}}\\\end{pmatrix}}={\ddot {s}}\,{\vec {e}}_{T}+{\frac {v^{2}}{l}}\,{\vec {e}}_{N}\quad {\text{mit dem Normaleneinheitsvektor }}{\vec {e}}_{N}={\begin{pmatrix}-\sin {\frac {s}{l}}\\\cos {\frac {s}{l}}\\\end{pmatrix}}={\begin{pmatrix}-\sin {\varphi }\\\cos {\varphi }\\\end{pmatrix}}=-{\vec {e}}_{r}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55cf6744fa94d122a512ec040a3b8b6a300ab7ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:142.608ex; height:7.509ex;" alt="{\displaystyle {\vec {a}}={\dot {\vec {v}}}={\ddot {s}}\,{\frac {{\text{d}}{\vec {r}}_{P}}{{\text{d}}s}}+{\dot {s}}^{2}\,{\frac {{\text{d}}^{2}{\vec {r}}_{P}}{{\text{d}}s^{2}}}={\ddot {s}}\,{\vec {e}}_{T}+{\frac {v^{2}}{l}}{\begin{pmatrix}-\sin {\frac {s}{l}}\\\cos {\frac {s}{l}}\\\end{pmatrix}}={\ddot {s}}\,{\vec {e}}_{T}+{\frac {v^{2}}{l}}\,{\vec {e}}_{N}\quad {\text{mit dem Normaleneinheitsvektor }}{\vec {e}}_{N}={\begin{pmatrix}-\sin {\frac {s}{l}}\\\cos {\frac {s}{l}}\\\end{pmatrix}}={\begin{pmatrix}-\sin {\varphi }\\\cos {\varphi }\\\end{pmatrix}}=-{\vec {e}}_{r}}" loading="lazy"></dd></dl>

Die Zwangskraft ergibt sich aus der Normalkomponente der Newtonschen Bewegungsgleichung:

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<annotation encoding="application/x-tex">{\displaystyle m\,{\vec {a}}\cdot {\vec {e}}_{N}=m\,\left({\ddot {s}}\,{\vec {e}}_{T}+{\textstyle {\frac {v^{2}}{l}}}\,{\vec {e}}_{N}\right)\cdot {\vec {e}}_{N}=(m\,{\vec {g}}+F_{Z}\,{\vec {e}}_{N})\cdot {\vec {e}}_{N}=-mg\,{\vec {e}}_{z}\cdot {\vec {e}}_{N}+F_{Z}\,{\vec {e}}_{N}\cdot {\vec {e}}_{N}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d61a9d626d7700d0815862bc13ccf51e68396c3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:85.511ex; height:4.843ex;" alt="{\displaystyle m\,{\vec {a}}\cdot {\vec {e}}_{N}=m\,\left({\ddot {s}}\,{\vec {e}}_{T}+{\textstyle {\frac {v^{2}}{l}}}\,{\vec {e}}_{N}\right)\cdot {\vec {e}}_{N}=(m\,{\vec {g}}+F_{Z}\,{\vec {e}}_{N})\cdot {\vec {e}}_{N}=-mg\,{\vec {e}}_{z}\cdot {\vec {e}}_{N}+F_{Z}\,{\vec {e}}_{N}\cdot {\vec {e}}_{N}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{T}\cdot {\vec {e}}_{N}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57701041d36ffc55c4a83fc595f7ccc2a8478231.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.467ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{T}\cdot {\vec {e}}_{N}=0}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{N}\cdot {\vec {e}}_{N}=1}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{N}\cdot {\vec {e}}_{N}=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b2ca5774d573dbb46868cc68a41e668f78dfbc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.769ex; height:2.676ex;" alt="{\displaystyle {\vec {e}}_{N}\cdot {\vec {e}}_{N}=1}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{z}\cdot {\vec {e}}_{N}=\cos {\varphi }}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{z}\cdot {\vec {e}}_{N}=\cos {\varphi }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db16476a17b4732c346a48a8686570f350b1552c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.935ex; height:2.843ex;" alt="{\displaystyle {\vec {e}}_{z}\cdot {\vec {e}}_{N}=\cos {\varphi }}" loading="lazy"> gilt für die Zwangskraft

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<annotation encoding="application/x-tex">{\displaystyle F_{Z}=m{\textstyle {\frac {v^{2}}{l}}}+mg\cos {\varphi }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e35d56392a89d5f870fa25cfe177e4c0a6fdfea3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.921ex; height:4.176ex;" alt="{\displaystyle F_{Z}=m{\textstyle {\frac {v^{2}}{l}}}+mg\cos {\varphi }}" loading="lazy"></dd></dl>

Das Geschwindigkeitsquadrat <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle v^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4faa98a21ac8133ab466999288849492be28b3d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.182ex; height:2.676ex;" alt="{\displaystyle v^{2}}" loading="lazy"> folgt aus der <a href="Energieerhaltungssatz" title="Energieerhaltungssatz">Energieerhaltung</a><a href="#cite_note-14">[14]</a> beim Fadenpendel

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<annotation encoding="application/x-tex">{\displaystyle E_{\text{kin}}+E_{\text{pot}}={\textstyle {\frac {m}{2}}}v^{2}+mgl(1-\cos {\varphi })=mgl(1-\cos {\varphi }_{0})\quad \Rightarrow \quad v^{2}=2gl(\cos {\varphi }-\cos {\varphi }_{0})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7334c87a3a2a633c90c6209dda39519e1253e251.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:87.515ex; height:3.509ex;" alt="{\displaystyle E_{\text{kin}}+E_{\text{pot}}={\textstyle {\frac {m}{2}}}v^{2}+mgl(1-\cos {\varphi })=mgl(1-\cos {\varphi }_{0})\quad \Rightarrow \quad v^{2}=2gl(\cos {\varphi }-\cos {\varphi }_{0})}" loading="lazy"></dd></dl>

Für die Zwangskraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{Z}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{Z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/087cedf7a50b6bcf38c737465e266530c40d7131.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.915ex; height:2.509ex;" alt="{\displaystyle F_{Z}}" loading="lazy"> bedeutet dies<a href="#cite_note-15">[15]</a>

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<annotation encoding="application/x-tex">{\displaystyle F_{Z}=m{\textstyle {\frac {v^{2}}{l}}}+mg\cos {\varphi }=m{\textstyle {\frac {2gl(\cos {\varphi }-\cos {\varphi }_{0})}{l}}}+mg\cos {\varphi }=mg(3\cos {\varphi }-2\cos {\varphi }_{0})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7390fdc9227b566fa3183aeba12d8094c57fd4b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:77.208ex; height:4.343ex;" alt="{\displaystyle F_{Z}=m{\textstyle {\frac {v^{2}}{l}}}+mg\cos {\varphi }=m{\textstyle {\frac {2gl(\cos {\varphi }-\cos {\varphi }_{0})}{l}}}+mg\cos {\varphi }=mg(3\cos {\varphi }-2\cos {\varphi }_{0})}" loading="lazy"></dd></dl>

An den Umkehrpunkten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\varphi _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle F_{Z,{\text{min}}}=mg\cos {\varphi }_{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0340fce41bb9417856a6c773643aba6bb87c43a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.827ex; height:2.843ex;" alt="{\displaystyle F_{Z,{\text{min}}}=mg\cos {\varphi }_{0}}" loading="lazy"> und am tiefsten Punkt der Pendelschwingung mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{Z,{\text{max}}}=mg(3-2\cos {\varphi }_{0})}">

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<annotation encoding="application/x-tex">{\displaystyle F_{Z,{\text{max}}}=mg(3-2\cos {\varphi }_{0})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57caa5b739dc3af937f8b278b88de8ccca42983b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.12ex; height:3.009ex;" alt="{\displaystyle F_{Z,{\text{max}}}=mg(3-2\cos {\varphi }_{0})}" loading="lazy"> maximal. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {\varphi }_{0}=\cos {\text{48°}}\approx {\textstyle {\frac {2}{3}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \cos {\varphi }_{0}=\cos {\text{48°}}\approx {\textstyle {\frac {2}{3}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f0de0a0366012d15b31800f9820ea081ee53537.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.913ex; height:3.676ex;" alt="{\displaystyle \cos {\varphi }_{0}=\cos {\text{48°}}\approx {\textstyle {\frac {2}{3}}}}" loading="lazy"> erreicht <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{Z,{\text{max}}}\approx {\textstyle {\frac {5}{3}}}mg}">

<msub>

</mrow>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{Z,{\text{max}}}\approx {\textstyle {\frac {5}{3}}}mg}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21d8ddf721edc17e969b02f667bda0bfc9ec739f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.344ex; height:3.676ex;" alt="{\displaystyle F_{Z,{\text{max}}}\approx {\textstyle {\frac {5}{3}}}mg}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Tangentialkraft_auf_ein_Objekt,_das_sich_entlang_einer_Zykloide_bewegt">Tangentialkraft auf ein Objekt, das sich entlang einer Zykloide bewegt</h3></div>

Ein Kreis mit dem Radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> rollt ohne Schlupf auf einer Geraden ab. Die Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> des Kreismittelpunktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> sei konstant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{M}=r\omega }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{M}=r\omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac7d35ed06da0f5fb78249a41c1865f11241d775.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.68ex; height:2.009ex;" alt="{\displaystyle v_{M}=r\omega }" loading="lazy"> mit der Kreisfrequenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">. Ein Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" 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}" loading="lazy"> auf dem Kreisumfang bewegt sich auf einer gewöhnlichen <a href="Zykloide" title="Zykloide">Zykloide</a><a href="#cite_note-16">[16]</a>. Die Gleichungen der Zykloide in einem kartesischen Koordinatensystem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5902cdb03ce350ff2f806116a56d5a660833894.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.261ex; height:2.843ex;" alt="{\displaystyle (x,z)}" loading="lazy"> lauten mit dem Drehwinkel<a href="#cite_note-17">[17]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\omega t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi =\omega t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4224e277c0ad8bc028bdc00fce8e2dd2227e075c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.904ex; height:2.509ex;" alt="{\displaystyle \varphi =\omega t}" loading="lazy">:

<msub>

<mover>

</mover>

</mrow>

</mrow>

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<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

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</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

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</mtr>

<mtr>

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</mtr>

</mtable>

</mrow>

<mrow>

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<mtr>

<mtd>

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</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{P}={\begin{pmatrix}x\\z\\\end{pmatrix}}={\begin{pmatrix}r\varphi -r\sin \varphi \\r-r\cos \varphi \\\end{pmatrix}}=r{\begin{pmatrix}\omega t-\sin \omega t\\1-\cos \omega t\\\end{pmatrix}}=r{\begin{pmatrix}\omega t\\1\\\end{pmatrix}}-r{\begin{pmatrix}\sin \omega t\\\cos \omega t\\\end{pmatrix}}={\overrightarrow {OM}}-{\overrightarrow {PM}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/feaba36e6f20e74b86cfba51fc496f821282c60d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:88.44ex; height:6.176ex;" alt="{\displaystyle {\vec {r}}_{P}={\begin{pmatrix}x\\z\\\end{pmatrix}}={\begin{pmatrix}r\varphi -r\sin \varphi \\r-r\cos \varphi \\\end{pmatrix}}=r{\begin{pmatrix}\omega t-\sin \omega t\\1-\cos \omega t\\\end{pmatrix}}=r{\begin{pmatrix}\omega t\\1\\\end{pmatrix}}-r{\begin{pmatrix}\sin \omega t\\\cos \omega t\\\end{pmatrix}}={\overrightarrow {OM}}-{\overrightarrow {PM}}}" loading="lazy"></dd></dl>

Die Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85820588abd7333ef4d0c56539cb31c20e730753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}" loading="lazy"> des Punktes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

<msub>

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {r}}}_{P}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72c0ca753b2731843f2f3f8bd32e5cdf9e1e4577.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:3.176ex;" alt="{\displaystyle {\dot {\vec {r}}}_{P}}" loading="lazy">. (Symbol: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{.}=\partial /\partial t}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ^{.}=\partial /\partial t}</annotation>

</semantics>

<mover>

</mover>

</mrow>

<msub>

<mover>

<mover>

</mover>

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</mover>

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</msub>

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<mrow>

<mtr>

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</mtr>

<mtr>

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</mrow>

<msub>

<mover>

</mover>

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</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}={\dot {\vec {r}}}_{P}={\begin{pmatrix}{\dot {x}}\\{\dot {z}}\\\end{pmatrix}}=r\omega {\begin{pmatrix}1-\cos \omega t\\\sin \omega t\\\end{pmatrix}}=r\omega {\begin{pmatrix}2\sin ^{2}(\omega t/2)\\2\sin(\omega t/2)\cos(\omega t/2)=\\\end{pmatrix}}=2r\omega \sin(\omega t/2){\begin{pmatrix}\sin(\omega t/2)\\\cos(\omega t/2)\\\end{pmatrix}}=v{\vec {e}}_{\|}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5f3cbd9e108331396859faca010e9610f240228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:104.736ex; height:6.509ex;" alt="{\displaystyle {\vec {v}}={\dot {\vec {r}}}_{P}={\begin{pmatrix}{\dot {x}}\\{\dot {z}}\\\end{pmatrix}}=r\omega {\begin{pmatrix}1-\cos \omega t\\\sin \omega t\\\end{pmatrix}}=r\omega {\begin{pmatrix}2\sin ^{2}(\omega t/2)\\2\sin(\omega t/2)\cos(\omega t/2)=\\\end{pmatrix}}=2r\omega \sin(\omega t/2){\begin{pmatrix}\sin(\omega t/2)\\\cos(\omega t/2)\\\end{pmatrix}}=v{\vec {e}}_{\|}}" loading="lazy"></dd></dl>

mit dem Tangenteneinheitsvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{\|}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{\|}}</annotation>

</semantics>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{\|}={\begin{pmatrix}\sin(\omega t/2)\\\cos(\omega t/2)\\\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a035299a3de081a2316e08dcc73716d808b2da73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.079ex; height:6.176ex;" alt="{\displaystyle {\vec {e}}_{\|}={\begin{pmatrix}\sin(\omega t/2)\\\cos(\omega t/2)\\\end{pmatrix}}}" loading="lazy"></dd></dl>

und dem Geschwindigkeitsbetrag

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v=2r\omega \sin(\omega t/2).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd43018e63e95bd5683403db18d1b040297a3aa8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.192ex; height:2.843ex;" alt="{\displaystyle v=2r\omega \sin(\omega t/2).}" loading="lazy"></dd></dl>

Deutliche Vereinfachungen<a href="#cite_note-18">[18]</a> ermöglichten die Beziehungen der halben Argumente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}(\omega t/2)={\textstyle {\frac {1}{2}}}(1-\cos \omega t)}">

<msup>

</mrow>

</msup>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin ^{2}(\omega t/2)={\textstyle {\frac {1}{2}}}(1-\cos \omega t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df9ed694e6e8f6671e4ca603381d1c2dbab2fc19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.682ex; height:3.509ex;" alt="{\displaystyle \sin ^{2}(\omega t/2)={\textstyle {\frac {1}{2}}}(1-\cos \omega t)}" loading="lazy"> und der doppelten Argumente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \omega t=2\sin(\omega t/2)\cos(\omega t/2)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin \omega t=2\sin(\omega t/2)\cos(\omega t/2)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b1bfab0b384ce5cd4c320bd967f1b4d6b5ab94b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.369ex; height:2.843ex;" alt="{\displaystyle \sin \omega t=2\sin(\omega t/2)\cos(\omega t/2)}" loading="lazy">.

Die Beschleunigung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/546e6615827e17295718741fd0b86f639a947f16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:2.343ex;" alt="{\displaystyle {\vec {a}}}" loading="lazy"> ist eine weitere Zeitableitung der Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation>

</semantics>

<mover>

</mover>

</mrow>

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

<mrow>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

<mover>

<mrow>

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</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {a}}={\dot {\vec {v}}}={\begin{pmatrix}{\ddot {x}}\\{\ddot {z}}\\\end{pmatrix}}=r\omega ^{2}{\begin{pmatrix}\sin \omega t\\\cos \omega t\\\end{pmatrix}}||{\overrightarrow {PM}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db5c35b0a33805397f51c8505596a0a699ec7e35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.476ex; height:6.176ex;" alt="{\displaystyle {\vec {a}}={\dot {\vec {v}}}={\begin{pmatrix}{\ddot {x}}\\{\ddot {z}}\\\end{pmatrix}}=r\omega ^{2}{\begin{pmatrix}\sin \omega t\\\cos \omega t\\\end{pmatrix}}||{\overrightarrow {PM}}}" loading="lazy"></dd></dl>

Der Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" 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}" loading="lazy"> auf dem Kreis wird mit der konstanten Beschleunigung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=|{\vec {a}}|=r\omega ^{2}}">

</mrow>

<mover>

</mover>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=|{\vec {a}}|=r\omega ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b2eea9b9ef1b45748164eaee47eb133f47aaa52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.499ex; height:3.176ex;" alt="{\displaystyle a=|{\vec {a}}|=r\omega ^{2}}" loading="lazy"> hin zum Kreismittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> gezogen<a href="#cite_note-19">[19]</a>. Der Vektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {DP}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {DP}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d607997fce83a476eed354bf1171375be0f547c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-top: -0.449ex; width:3.964ex; height:3.843ex;" alt="{\displaystyle {\overrightarrow {DP}}}" loading="lazy"> von der momentanen Drehachse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> hin zum Punkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<msub>

<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {DP}}=-{\overrightarrow {OD}}+{\vec {r}}_{P}={\begin{pmatrix}-r\omega t\\0\\\end{pmatrix}}+{\begin{pmatrix}r\omega t-r\sin \omega t\\r-r\cos \omega t\\\end{pmatrix}}=r{\begin{pmatrix}-\sin \omega t\\1-\cos \omega t\\\end{pmatrix}}=r{\begin{pmatrix}-2\sin(\omega t/2)\cos(\omega t/2)\\2\sin ^{2}(\omega t/2)\\\end{pmatrix}}=2r\sin(\omega t/2){\begin{pmatrix}-\cos(\omega t/2)\\\sin(\omega t/2)\\\end{pmatrix}}\,\bot \ {\vec {v}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/523c942ca66037edf22e6c1ec3ce8626f7dff934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:134.592ex; height:6.509ex;" alt="{\displaystyle {\overrightarrow {DP}}=-{\overrightarrow {OD}}+{\vec {r}}_{P}={\begin{pmatrix}-r\omega t\\0\\\end{pmatrix}}+{\begin{pmatrix}r\omega t-r\sin \omega t\\r-r\cos \omega t\\\end{pmatrix}}=r{\begin{pmatrix}-\sin \omega t\\1-\cos \omega t\\\end{pmatrix}}=r{\begin{pmatrix}-2\sin(\omega t/2)\cos(\omega t/2)\\2\sin ^{2}(\omega t/2)\\\end{pmatrix}}=2r\sin(\omega t/2){\begin{pmatrix}-\cos(\omega t/2)\\\sin(\omega t/2)\\\end{pmatrix}}\,\bot \ {\vec {v}}}" loading="lazy"></dd></dl>

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Die Kraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}=m{\vec {a}}=m\omega ^{2}r{\begin{pmatrix}\sin \omega t\\\cos \omega t\\\end{pmatrix}}=F_{\|}{\vec {e}}_{\|}+F_{\bot }{\vec {e}}_{\bot }\quad {\text{mit }}{\vec {e}}_{\bot }={\begin{pmatrix}-\cos(\omega t/2)\\\sin(\omega t/2)\\\end{pmatrix}}\,\bot \ {\vec {e}}_{\|}={\begin{pmatrix}\sin(\omega t/2)\\\cos(\omega t/2)\\\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da4c27aa3f8214ce17eb80c38f73fae391761fd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:93.264ex; height:6.176ex;" alt="{\displaystyle {\vec {F}}=m{\vec {a}}=m\omega ^{2}r{\begin{pmatrix}\sin \omega t\\\cos \omega t\\\end{pmatrix}}=F_{\|}{\vec {e}}_{\|}+F_{\bot }{\vec {e}}_{\bot }\quad {\text{mit }}{\vec {e}}_{\bot }={\begin{pmatrix}-\cos(\omega t/2)\\\sin(\omega t/2)\\\end{pmatrix}}\,\bot \ {\vec {e}}_{\|}={\begin{pmatrix}\sin(\omega t/2)\\\cos(\omega t/2)\\\end{pmatrix}}}" loading="lazy"></dd></dl>

Die Tangentialkraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{\|}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11605d82d13fbacd2c86eb1c5d3b1a31ce277ed9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.825ex; height:3.676ex;" alt="{\displaystyle {\vec {F}}_{\|}}" loading="lazy"> weist ebenfalls nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle F_{\|}={\vec {F}}_{\|}\cdot {\vec {e}}_{\|}=m\omega ^{2}r{\begin{pmatrix}\sin \omega t\\\cos \omega t\\\end{pmatrix}}\cdot {\begin{pmatrix}\sin(\omega t/2)\\\cos(\omega t/2)\\\end{pmatrix}}=m\omega ^{2}r[\sin \omega t\sin(\omega t/2)+\cos \omega t\cos(\omega t/2)]=m\omega ^{2}r\cos[\omega t-(\omega t/2)]=m\omega ^{2}r\cos(\omega t/2)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f14331587f050969cb83fc06191ef58e04f69ba2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:133.799ex; height:6.176ex;" alt="{\displaystyle F_{\|}={\vec {F}}_{\|}\cdot {\vec {e}}_{\|}=m\omega ^{2}r{\begin{pmatrix}\sin \omega t\\\cos \omega t\\\end{pmatrix}}\cdot {\begin{pmatrix}\sin(\omega t/2)\\\cos(\omega t/2)\\\end{pmatrix}}=m\omega ^{2}r[\sin \omega t\sin(\omega t/2)+\cos \omega t\cos(\omega t/2)]=m\omega ^{2}r\cos[\omega t-(\omega t/2)]=m\omega ^{2}r\cos(\omega t/2)}" loading="lazy"></dd></dl>

Die Normalkraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{\bot }}">

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<annotation encoding="application/x-tex">{\displaystyle F_{\bot }={\vec {F}}_{\bot }\cdot {\vec {e}}_{\bot }=m\omega ^{2}r{\begin{pmatrix}\sin \omega t\\\cos \omega t\\\end{pmatrix}}\cdot {\begin{pmatrix}-\cos(\omega t/2)\\\sin(\omega t/2)\\\end{pmatrix}}=m\omega ^{2}r[\cos \omega t\sin(\omega t/2)-\sin \omega t\cos(\omega t/2)]=-m\omega ^{2}r\sin[\omega t-(\omega t/2)]=-m\omega ^{2}r\sin(\omega t/2)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/107a4364c12c279e37bb1342e711c4f484390e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:140.469ex; height:6.176ex;" alt="{\displaystyle F_{\bot }={\vec {F}}_{\bot }\cdot {\vec {e}}_{\bot }=m\omega ^{2}r{\begin{pmatrix}\sin \omega t\\\cos \omega t\\\end{pmatrix}}\cdot {\begin{pmatrix}-\cos(\omega t/2)\\\sin(\omega t/2)\\\end{pmatrix}}=m\omega ^{2}r[\cos \omega t\sin(\omega t/2)-\sin \omega t\cos(\omega t/2)]=-m\omega ^{2}r\sin[\omega t-(\omega t/2)]=-m\omega ^{2}r\sin(\omega t/2)}" loading="lazy"></dd></dl>

Die Kraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}=m{\vec {a}}={\vec {F}}_{\|}+{\vec {F}}_{\bot }=m\omega ^{2}r\cos(\omega t/2){\begin{pmatrix}\sin(\omega t/2)\\\cos(\omega t/2)\\\end{pmatrix}}-m\omega ^{2}r\sin(\omega t/2){\begin{pmatrix}-\cos(\omega t/2)\\\sin(\omega t/2)\\\end{pmatrix}}=m\omega ^{2}r{\begin{pmatrix}2\sin(\omega t/2)\cos(\omega t/2)\\\cos ^{2}(\omega t/2)-\sin ^{2}(\omega t/2)\\\end{pmatrix}}=m\omega ^{2}r{\begin{pmatrix}\sin \omega t\\\cos \omega t\\\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3bfa2f7833f255104b43d85f51498b521d8511fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:141.743ex; height:6.509ex;" alt="{\displaystyle {\vec {F}}=m{\vec {a}}={\vec {F}}_{\|}+{\vec {F}}_{\bot }=m\omega ^{2}r\cos(\omega t/2){\begin{pmatrix}\sin(\omega t/2)\\\cos(\omega t/2)\\\end{pmatrix}}-m\omega ^{2}r\sin(\omega t/2){\begin{pmatrix}-\cos(\omega t/2)\\\sin(\omega t/2)\\\end{pmatrix}}=m\omega ^{2}r{\begin{pmatrix}2\sin(\omega t/2)\cos(\omega t/2)\\\cos ^{2}(\omega t/2)-\sin ^{2}(\omega t/2)\\\end{pmatrix}}=m\omega ^{2}r{\begin{pmatrix}\sin \omega t\\\cos \omega t\\\end{pmatrix}}}" loading="lazy"></dd></dl>

mit dem Betrag <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\vec {F}}|=|{\vec {F}}_{\|}+{\vec {F}}_{\bot }|=m\omega ^{2}r}">

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<annotation encoding="application/x-tex">{\displaystyle |{\vec {F}}|=|{\vec {F}}_{\|}+{\vec {F}}_{\bot }|=m\omega ^{2}r}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eeb093dd4e384ce4f9d0d3192ae0280d283b523d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.092ex; height:3.676ex;" alt="{\displaystyle |{\vec {F}}|=|{\vec {F}}_{\|}+{\vec {F}}_{\bot }|=m\omega ^{2}r}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Arbeit_der_Tangentialkraft">Arbeit der Tangentialkraft</h2></div>

Die mechanische Arbeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{d}}A}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{d}}A}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a3878c5a86c63f752762b7fa12601a760fa798b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.036ex; height:2.176ex;" alt="{\displaystyle {\text{d}}A}" loading="lazy"> ist definiert als Kraftkomponente mal Weg oder Kraft mal Wegkomponente und ist definiert als<a href="#cite_note-20">[20]</a>:

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<annotation encoding="application/x-tex">{\displaystyle {\text{d}}A={\vec {F}}\cdot {\text{d}}{\vec {s}}=F\,{\text{d}}s\cos \measuredangle ({\vec {F}},{\vec {s}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ba05d6361cd06df33daf6e7dd355f8f4c5634b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.11ex; height:3.343ex;" alt="{\displaystyle {\text{d}}A={\vec {F}}\cdot {\text{d}}{\vec {s}}=F\,{\text{d}}s\cos \measuredangle ({\vec {F}},{\vec {s}})}" loading="lazy"></dd></dl>

Wirkt eine Tangentialkraft auf einen Körper, so verrichtet sie Arbeit und ändert dessen Energie. Dieser Beitrag zur mechanischen Arbeit wird für eine reine Tangentialkraft <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}_{\|}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11605d82d13fbacd2c86eb1c5d3b1a31ce277ed9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.825ex; height:3.676ex;" alt="{\displaystyle {\vec {F}}_{\|}}" loading="lazy"> maximal zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{d}}A_{\text{max}}=F_{\|}\,{\text{d}}s}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{d}}A_{\text{max}}=F_{\|}\,{\text{d}}s}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d6ca0e60dea72b7c021c3e34b2af5cfd8e73c08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.744ex; height:3.009ex;" alt="{\displaystyle {\text{d}}A_{\text{max}}=F_{\|}\,{\text{d}}s}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Rainer Müller: <cite style="font-style:italic">Klassische Mechanik - Vom Weitsprung zum Marsflug</cite>. 2. Auflage. de Gruyter, Berlin / New York 2010, ISBN 978-3-11-025002-2, S. 61.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Rainer+M%C3%BCller&rft.btitle=Klassische+Mechanik+-+Vom+Weitsprung+zum+Marsflug&rft.date=2010&rft.edition=2.&rft.genre=book&rft.isbn=9783110250022&rft.pages=61&rft.place=Berlin+%2F+New+York&rft.pub=de+Gruyter" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Arnold Sommerfeld: <cite style="font-style:italic">Mechanik</cite>. 8. Auflage. Harri Deutsch, Thun und Frankfurt/Main 1977, ISBN 3-87144-374-3, S. 30.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Arnold+Sommerfeld&rft.btitle=Mechanik&rft.date=1977&rft.edition=8.&rft.genre=book&rft.isbn=3871443743&rft.pages=30&rft.place=Thun+und+Frankfurt%2FMain&rft.pub=Harri+Deutsch" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Hund, Friedrich: <cite style="font-style:italic">Einführung in der Theoretische Physik - Band I: Mechanik</cite>. 1. Auflage. Bibliographisches Institut, Leipzig 1945, S. 53.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Hund%2C+Friedrich&rft.btitle=Einf%C3%BChrung+in+der+Theoretische+Physik+-+Band+I%3A+Mechanik&rft.date=1945&rft.edition=1.&rft.genre=book&rft.pages=53&rft.place=Leipzig&rft.pub=Bibliographisches+Institut" style="display:none">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> Hund, Friedrich: <cite style="font-style:italic">Einführung in der Theoretische Physik - Band I: Mechanik</cite>. 1. Auflage. Bibliographisches Institut, Leipzig 1945, S. 53.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Hund%2C+Friedrich&rft.btitle=Einf%C3%BChrung+in+der+Theoretische+Physik+-+Band+I%3A+Mechanik&rft.date=1945&rft.edition=1.&rft.genre=book&rft.pages=53&rft.place=Leipzig&rft.pub=Bibliographisches+Institut" style="display:none">

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Christian Gerthsen, H. O. Kneser, Helmut Vogel: <cite style="font-style:italic">Physik</cite>. 12. Auflage. Springer-Verlag, Berlin Heidelberg New York 1974, ISBN 3-540-06336-6, S. 14.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Christian+Gerthsen%2C+H.+O.+Kneser%2C+Helmut+Vogel&rft.btitle=Physik&rft.date=1974&rft.edition=12.&rft.genre=book&rft.isbn=3540063366&rft.pages=14&rft.place=Berlin+Heidelberg+New+York&rft.pub=Springer-Verlag" style="display:none">

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> Rainer Müller: <cite style="font-style:italic">Klassische Mechanik - Vom Weitsprung zum Marsflug</cite>. 2. Auflage. de Gruyter, Berlin / New York 2010, ISBN 978-3-11-025002-2, S. 423.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Rainer+M%C3%BCller&rft.btitle=Klassische+Mechanik+-+Vom+Weitsprung+zum+Marsflug&rft.date=2010&rft.edition=2.&rft.genre=book&rft.isbn=9783110250022&rft.pages=423&rft.place=Berlin+%2F+New+York&rft.pub=de+Gruyter" style="display:none">

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> Christian Gerthsen, H. O. Kneser, Helmut Vogel: <cite style="font-style:italic">Physik</cite>. 12. Auflage. Springer-Verlag, Berlin Heidelberg New York 1974, ISBN 3-540-06336-6, S. 90.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Christian+Gerthsen%2C+H.+O.+Kneser%2C+Helmut+Vogel&rft.btitle=Physik&rft.date=1974&rft.edition=12.&rft.genre=book&rft.isbn=3540063366&rft.pages=90&rft.place=Berlin+Heidelberg+New+York&rft.pub=Springer-Verlag" style="display:none">

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> Anton Hammer, Hildegard Hammer, Karl Hammer: <cite style="font-style:italic">Taschenbuch der Physik</cite>. 9. Auflage. Lindauer, München 2004, ISBN 3-87488-094-X, S. 22.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Anton+Hammer%2C+Hildegard+Hammer%2C+Karl+Hammer&rft.btitle=Taschenbuch+der+Physik&rft.date=2004&rft.edition=9.&rft.genre=book&rft.isbn=387488094X&rft.pages=22&rft.place=M%C3%BCnchen&rft.pub=Lindauer" style="display:none">

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> Ludwig Prandtl: <cite style="font-style:italic">Führer durch die Strömungslehre</cite>. 2. Auflage. Friedrich Vieweg & Sohn, Braunschweig 1944, S. 159.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Ludwig+Prandtl&rft.btitle=F%C3%BChrer+durch+die+Str%C3%B6mungslehre&rft.date=1944&rft.edition=2.&rft.genre=book&rft.pages=159&rft.place=Braunschweig&rft.pub=Friedrich+Vieweg+%26+Sohn" style="display:none">

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> Ludwig Prandtl: <cite style="font-style:italic">Führer durch die Strömungslehre</cite>. 2. Auflage. Friedrich Vieweg & Sohn, Braunschweig 1944, S. 172.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Ludwig+Prandtl&rft.btitle=F%C3%BChrer+durch+die+Str%C3%B6mungslehre&rft.date=1944&rft.edition=2.&rft.genre=book&rft.pages=172&rft.place=Braunschweig&rft.pub=Friedrich+Vieweg+%26+Sohn" style="display:none">

</li>

<li id="cite_note-11"><a href="#cite_ref-11">↑</a> Friedhelm Kuypers: <cite style="font-style:italic">Physik in den Ingenieur- und Naturwissenschaften - Band 1: Mechanik und Thermodynamik</cite>. 4. Auflage. WILEY-VCH, Weinheim 2023, ISBN 978-3-527-41398-0, S. 192.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Friedhelm+Kuypers&rft.btitle=Physik+in+den+Ingenieur-+und+Naturwissenschaften+-+Band+1%3A+Mechanik+und+Thermodynamik&rft.date=2023&rft.edition=4.&rft.genre=book&rft.isbn=9783527413980&rft.pages=192&rft.place=Weinheim&rft.pub=WILEY-VCH" style="display:none">

</li>

<li id="cite_note-12"><a href="#cite_ref-12">↑</a> Arnold Sommerfeld: <cite style="font-style:italic">Mechanik</cite>. 8. Auflage. Harri Deutsch, Thun und Frankfurt/Main 1977, ISBN 3-87144-374-3, S. 76.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Arnold+Sommerfeld&rft.btitle=Mechanik&rft.date=1977&rft.edition=8.&rft.genre=book&rft.isbn=3871443743&rft.pages=76&rft.place=Thun+und+Frankfurt%2FMain&rft.pub=Harri+Deutsch" style="display:none">

</li>

<li id="cite_note-13"><a href="#cite_ref-13">↑</a> Rainer Müller: <cite style="font-style:italic">Klassische Mechanik - Vom Weitsprung zum Marsflug</cite>. 2. Auflage. de Gruyter, Berlin / New York 2010, ISBN 978-3-11-025002-2, S. 420.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Rainer+M%C3%BCller&rft.btitle=Klassische+Mechanik+-+Vom+Weitsprung+zum+Marsflug&rft.date=2010&rft.edition=2.&rft.genre=book&rft.isbn=9783110250022&rft.pages=420&rft.place=Berlin+%2F+New+York&rft.pub=de+Gruyter" style="display:none">

</li>

<li id="cite_note-14"><a href="#cite_ref-14">↑</a> Arnold Sommerfeld: <cite style="font-style:italic">Mechanik</cite>. 8. Auflage. Harri Deutsch, Thun und Frankfurt/Main 1977, ISBN 3-87144-374-3, S. 76.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Arnold+Sommerfeld&rft.btitle=Mechanik&rft.date=1977&rft.edition=8.&rft.genre=book&rft.isbn=3871443743&rft.pages=76&rft.place=Thun+und+Frankfurt%2FMain&rft.pub=Harri+Deutsch" style="display:none">

</li>

<li id="cite_note-15"><a href="#cite_ref-15">↑</a> Keck, Wilhelm: <cite style="font-style:italic">Vorträge über Mechanik als Grundlage für das Bau- und Maschinenwesen - Teil 1: Mechanik starrer Körper</cite>. 2. Auflage. Helwingsche Verlagsbuchhandlung, Hannover 1900, S. 76.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Keck%2C+Wilhelm&rft.btitle=Vortr%C3%A4ge+%C3%BCber+Mechanik+als+Grundlage+f%C3%BCr+das+Bau-+und+Maschinenwesen+-+Teil+1%3A+Mechanik+starrer+K%C3%B6rper&rft.date=1900&rft.edition=2.&rft.genre=book&rft.pages=76&rft.place=Hannover&rft.pub=Helwingsche+Verlagsbuchhandlung" style="display:none">

</li>

<li id="cite_note-16"><a href="#cite_ref-16">↑</a> Arnold Sommerfeld: <cite style="font-style:italic">Mechanik</cite>. 8. Auflage. Harri Deutsch, Thun und Frankfurt/Main 1977, ISBN 3-87144-374-3, S. 83.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Arnold+Sommerfeld&rft.btitle=Mechanik&rft.date=1977&rft.edition=8.&rft.genre=book&rft.isbn=3871443743&rft.pages=83&rft.place=Thun+und+Frankfurt%2FMain&rft.pub=Harri+Deutsch" style="display:none">

</li>

<li id="cite_note-17"><a href="#cite_ref-17">↑</a> I.N. Bronstein, K. A. Semendjajew: <cite style="font-style:italic">Taschenbuch der Mathematik</cite>. 19. Auflage. Harri Deutsch, Thun und Frankfurt/Main 1980, ISBN 3-87144-492-8, S. 143.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=I.N.+Bronstein%2C+K.+A.+Semendjajew&rft.btitle=Taschenbuch+der+Mathematik&rft.date=1980&rft.edition=19.&rft.genre=book&rft.isbn=3871444928&rft.pages=143&rft.place=Thun+und+Frankfurt%2FMain&rft.pub=Harri+Deutsch" style="display:none">

</li>

<li id="cite_note-18"><a href="#cite_ref-18">↑</a> I.N. Bronstein, K. A. Semendjajew: <cite style="font-style:italic">Taschenbuch der Mathematik</cite>. 19. Auflage. Harri Deutsch, Thun und Frankfurt/Main 1980, ISBN 3-87144-492-8, S. 233.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=I.N.+Bronstein%2C+K.+A.+Semendjajew&rft.btitle=Taschenbuch+der+Mathematik&rft.date=1980&rft.edition=19.&rft.genre=book&rft.isbn=3871444928&rft.pages=233&rft.place=Thun+und+Frankfurt%2FMain&rft.pub=Harri+Deutsch" style="display:none">

</li>

<li id="cite_note-19"><a href="#cite_ref-19">↑</a> Günther Holzmann, Heinz Meyer, Georg Schumpich: <cite style="font-style:italic">Technische Mechanik - Teil 2: Kinematik und Kinetik</cite>. 3. Auflage. B. G. Teubner, Stuttgart 1976, ISBN 3-519-26506-0, S. 48.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=G%C3%BCnther+Holzmann%2C+Heinz+Meyer%2C+Georg+Schumpich&rft.btitle=Technische+Mechanik+-+Teil+2%3A+Kinematik+und+Kinetik&rft.date=1976&rft.edition=3.&rft.genre=book&rft.isbn=3519265060&rft.pages=48&rft.place=Stuttgart&rft.pub=B.+G.+Teubner" style="display:none">

</li>

<li id="cite_note-20"><a href="#cite_ref-20">↑</a> Arnold Sommerfeld: <cite style="font-style:italic">Mechanik</cite>. 8. Auflage. Harri Deutsch, Thun und Frankfurt/Main 1977, ISBN 3-87144-374-3, S. 7.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Tangentialkraft&rft.au=Arnold+Sommerfeld&rft.btitle=Mechanik&rft.date=1977&rft.edition=8.&rft.genre=book&rft.isbn=3871443743&rft.pages=7&rft.place=Thun+und+Frankfurt%2FMain&rft.pub=Harri+Deutsch" style="display:none">

</li>

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