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Micron Document

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<title>Rotationsenergie</title>

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

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Rotationsenergie ist die <a href="Kinetische_Energie" title="Kinetische Energie">kinetische Energie</a> eines <a href="Starrer_K%C3%B6rper" title="Starrer Körper">starren Körpers</a> (Beispiel: <a href="Schwungrad" title="Schwungrad">Schwungrad</a>), der um einen festen Punkt oder seinen (beweglichen) <a href="Massenmittelpunkt" title="Massenmittelpunkt">Massenmittelpunkt</a> <a href="Rotationsbewegung" class="mw-redirect" title="Rotationsbewegung">rotiert</a>. In diesen beiden Fällen lässt sich die kinetische Energie des Körpers in einen <a href="Translation_(Physik)" title="Translation (Physik)">translatorischen</a> und einen rotatorischen Anteil zerlegen. Diese <a href="Energie" title="Energie">Energie</a> ist abhängig vom <a href="Tr%C3%A4gheitsmoment" title="Trägheitsmoment">Trägheitsmoment</a> und der <a href="Winkelgeschwindigkeit" title="Winkelgeschwindigkeit">Winkelgeschwindigkeit</a> des Körpers: je mehr Masse von der Rotationsachse entfernt ist, desto mehr Energie gibt der Körper ab, wenn seine Rotation gestoppt wird.

Dies lässt sich durch folgendes Experiment verdeutlichen: Zwei gleich schwere Kugeln mit identischen Radien werden auf eine schiefe Ebene gelegt und rollen herunter, siehe <a href="Eulersche_Gleichungen_(Kreiseltheorie)#Ein_eine_schiefe_Ebene_hinabrollendes_Rad" title="Eulersche Gleichungen (Kreiseltheorie)">eine schiefe Ebene hinabrollendes Rad</a>. Eine Kugel besteht aus einem leichten Material wie Kunststoff und ist massiv gefertigt. Die andere Kugel jedoch ist hohl, besteht aber aus einem dichteren und somit schwereren Material als Kunststoff. Die hohle Kugel wird langsamer rollen, da bei ihr die gesamte Masse auf einer dünnen Schale mit gewissem Abstand zur Rotationsachse verteilt ist. Die massive Kugel mit derselben Masse rollt schneller, weil prozentual mehr Masse nahe der Rotationsachse liegt und sich daher langsamer auf der Kreisbahn bewegen muss. Daher wird weniger ihrer <a href="Lageenergie" class="mw-redirect" title="Lageenergie">Lageenergie</a> in Rotationsenergie und mehr in <a href="Translatorische_Energie" class="mw-redirect" title="Translatorische Energie">translatorische Energie</a> umgewandelt und sie rollt schneller.

Rotationsenergie ist unter anderem von Bedeutung bei: <a href="Turbine" title="Turbine">Turbinen</a>, <a href="Generator" class="mw-redirect" title="Generator">Generatoren</a>, <a href="Rad" title="Rad">Rädern</a> und <a href="Reifen" title="Reifen">Reifen</a>, <a href="Welle_(Mechanik)" class="mw-redirect" title="Welle (Mechanik)">Wellen</a>, <a href="Propeller" title="Propeller">Propellern</a>.

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

Ein Körper, der mit der Winkelgeschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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

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</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"> um die x-Achse rotiert, besitzt die Rotationsenergie

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<annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {rot} }={\frac {1}{2}}\cdot J_{x}\cdot \omega ^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6793366c5c888a7ea7a97dcd49a32dba01f0be77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.472ex; height:5.176ex;" alt="{\displaystyle E_{\mathrm {rot} }={\frac {1}{2}}\cdot J_{x}\cdot \omega ^{2}}" loading="lazy"></dd></dl>

mit

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/573c4eac7701ac45330a1677abe1c3a7fc054ae1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.463ex; height:2.509ex;" alt="{\displaystyle J_{x}}" loading="lazy">: <a href="Tr%C3%A4gheitsmoment" title="Trägheitsmoment">Trägheitsmoment</a> des Körpers um die x-Achse</li>

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

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</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">: <a href="Winkelgeschwindigkeit" title="Winkelgeschwindigkeit">Winkelgeschwindigkeit</a>.</li></ul>

Dies lässt sich allgemein ausdrücken als:

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<annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {rot} }={\frac {1}{2}}\;{\vec {\omega }}\cdot \mathbf {J} \cdot {\vec {\omega }}={\frac {1}{2}}\sum _{\alpha ,\beta =1}^{3}\,J_{\alpha \beta }\;\omega _{\alpha }\;\omega _{\beta }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8051ebadcb1337ec11c41b371fec47af19c85bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.393ex; height:7.509ex;" alt="{\displaystyle E_{\mathrm {rot} }={\frac {1}{2}}\;{\vec {\omega }}\cdot \mathbf {J} \cdot {\vec {\omega }}={\frac {1}{2}}\sum _{\alpha ,\beta =1}^{3}\,J_{\alpha \beta }\;\omega _{\alpha }\;\omega _{\beta }}" loading="lazy"></dd></dl>

mit

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7686846b1a6b756cb514954000004ab5e7b2a5ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.381ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} }" loading="lazy">:

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<td><a href="Tr%C3%A4gheitstensor" title="Trägheitstensor">Trägheitstensor</a>,

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1164b5e7c32d8e51cd44aaa0cfc3eba13a054903.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.516ex; height:2.843ex;" alt="{\displaystyle J_{\alpha \beta }}" loading="lazy">:

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<td><a href="Tr%C3%A4gheitsmoment" title="Trägheitsmoment">Trägheitsmomente</a> (α=β) und <a href="Deviationsmoment" title="Deviationsmoment">Deviationsmomente</a> (α≠β) bezüglich einer beliebigen <a href="Orthonormalbasis" title="Orthonormalbasis">Orthonormalbasis</a>, sowie

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15f097b57783e051c7f975bb4cf2c20830ae0c6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.73ex; height:2.009ex;" alt="{\displaystyle \omega _{\alpha }}" loading="lazy">:

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<td>die Komponenten der Winkelgeschwindigkeit bezüglich derselben Basis.

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In einem körperfesten System sind die Trägheits- und Deviationsmomente sowie der Trägheitstensor konstant, ansonsten meist zeitabhängig.

Um die Energie eines Körpers anzugeben, der um eine beliebige Achse rotiert (<a href="Einheitsvektor" title="Einheitsvektor">Einheitsvektor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49569db585c1b6306d5ffd91161775f67235fae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.343ex;" alt="{\displaystyle {\vec {n}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\vec {n}}\right|=1}">

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<annotation encoding="application/x-tex">{\displaystyle \left|{\vec {n}}\right|=1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c6e54e2f121b8665d5cfd6c92d82d86edde5aa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.949ex; height:2.843ex;" alt="{\displaystyle \left|{\vec {n}}\right|=1}" loading="lazy">), wird die Winkelgeschwindigkeit jeweils durch ihre Vektorkomponenten in x-, y- und z-Richtung ausgedrückt:

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<annotation encoding="application/x-tex">{\displaystyle {\vec {\omega }}=\omega \,{\vec {n}}=\omega \cdot {\begin{pmatrix}n_{x}\\n_{y}\\n_{z}\end{pmatrix}}=\omega \cdot {\begin{pmatrix}n_{1}\\n_{2}\\n_{3}\end{pmatrix}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3c8104f33e00d830c2dd8cf11f434b38827e6b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:34.871ex; height:9.509ex;" alt="{\displaystyle {\vec {\omega }}=\omega \,{\vec {n}}=\omega \cdot {\begin{pmatrix}n_{x}\\n_{y}\\n_{z}\end{pmatrix}}=\omega \cdot {\begin{pmatrix}n_{1}\\n_{2}\\n_{3}\end{pmatrix}}}" loading="lazy"></dd></dl>

wenn, wie üblich, die Koordinaten nach dem Schema x→1, y→2 und z→3 nummeriert werden. Für die Rotationsenergie gilt damit:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{\mathrm {rot} }&={\frac {1}{2}}\sum _{\alpha ,\beta =1}^{3}J_{\alpha \beta }\;n_{\alpha }\;n_{\beta }\;\omega ^{2}\\&={\frac {1}{2}}\cdot J_{n}\cdot \omega ^{2}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1af08ca520c488912abe812fa0f510577de82b57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.7ex; margin-bottom: -0.305ex; width:28.428ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}E_{\mathrm {rot} }&={\frac {1}{2}}\sum _{\alpha ,\beta =1}^{3}J_{\alpha \beta }\;n_{\alpha }\;n_{\beta }\;\omega ^{2}\\&={\frac {1}{2}}\cdot J_{n}\cdot \omega ^{2}\end{aligned}}}" loading="lazy"></dd></dl>

mit dem Trägheitsmoment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{n}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22b79b47e08e4e16510d309639ef56a24c28696c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.509ex;" alt="{\displaystyle J_{n}}" loading="lazy"> bezüglich einer beliebigen Achse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}}">

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<annotation encoding="application/x-tex">{\displaystyle J_{n}=\sum _{\alpha ,\beta =1}^{3}J_{\alpha \beta }\;n_{\alpha }\;n_{\beta }={\vec {n}}\cdot \mathbf {J} \cdot {\vec {n}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1612cd22946c4935c8e999ce9a5a86e16dbdd812.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:31.227ex; height:7.509ex;" alt="{\displaystyle J_{n}=\sum _{\alpha ,\beta =1}^{3}J_{\alpha \beta }\;n_{\alpha }\;n_{\beta }={\vec {n}}\cdot \mathbf {J} \cdot {\vec {n}}}" loading="lazy"></dd></dl>

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

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

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

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</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"> und Masse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

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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"> hat das Trägheitsmoment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J={\tfrac {2}{5}}\,mr^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle J={\tfrac {2}{5}}\,mr^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8fccf99d508f831f742ebf7ff29bb05625d44cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.758ex; height:3.676ex;" alt="{\displaystyle J={\tfrac {2}{5}}\,mr^{2}}" loading="lazy">. Wenn sie mit der Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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</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"> auf der Ebene rollt, beträgt ihre Winkelgeschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\tfrac {v}{r}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3187c4a33b84e49fbdec92b51d17902c985777e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.178ex; height:3.009ex;" alt="{\displaystyle \omega ={\tfrac {v}{r}}}" loading="lazy"> und folglich ihre gesamte kinetische Energie:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{\mathrm {kin} }&=E_{\mathrm {trans} }+E_{\mathrm {rot} }\\&={\frac {1}{2}}mv^{2}+{\frac {1}{2}}\cdot {\frac {2}{5}}mr^{2}\cdot \omega ^{2}\\&={\frac {7}{10}}mv^{2}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/635aaa88a3fb6cf4717306a706146cb2e085b905.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:31.097ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}E_{\mathrm {kin} }&=E_{\mathrm {trans} }+E_{\mathrm {rot} }\\&={\frac {1}{2}}mv^{2}+{\frac {1}{2}}\cdot {\frac {2}{5}}mr^{2}\cdot \omega ^{2}\\&={\frac {7}{10}}mv^{2}\end{aligned}}}" loading="lazy"></dd></dl></dd></dl>

<ul><li>Ein Körper, der um die <a href="Diagonale_(Geometrie)" title="Diagonale (Geometrie)">Diagonale</a> durch seine xy-Fläche rotiert, hat die Winkelgeschwindigkeit:</li></ul>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d388fd298ab5235908c888bd38055c97574d6715.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.772ex; height:2.343ex;" alt="{\displaystyle {\vec {\omega }}=\omega \,{\vec {n}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {n}}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\1\\0\\\end{pmatrix}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {n}}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\1\\0\\\end{pmatrix}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1376cd0e0a4dc7a4462eb2cd496f5051cfa1475.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.408ex; height:9.176ex;" alt="{\displaystyle {\vec {n}}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\1\\0\\\end{pmatrix}}}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>Daraus folgt für das Trägheitsmoment bzgl. dieser Drehachse:</dd></dl>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J_{n}&={\vec {n}}\cdot \mathbf {J} \cdot {\vec {n}}\\&={\frac {1}{\sqrt {2}}}\left(1,1,0\right)\left({\begin{matrix}J_{11}&J_{12}&J_{13}\\J_{12}&J_{22}&J_{23}\\J_{13}&J_{23}&J_{33}\\\end{matrix}}\right){\frac {1}{\sqrt {2}}}\left({\begin{matrix}1\\1\\0\\\end{matrix}}\right)\\&={\frac {1}{2}}\cdot (J_{11}+J_{12}+J_{12}+J_{22})\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9adff7cc231b2d2b2c799b67cba07c3e56ab5c6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:48.084ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}J_{n}&={\vec {n}}\cdot \mathbf {J} \cdot {\vec {n}}\\&={\frac {1}{\sqrt {2}}}\left(1,1,0\right)\left({\begin{matrix}J_{11}&J_{12}&J_{13}\\J_{12}&J_{22}&J_{23}\\J_{13}&J_{23}&J_{33}\\\end{matrix}}\right){\frac {1}{\sqrt {2}}}\left({\begin{matrix}1\\1\\0\\\end{matrix}}\right)\\&={\frac {1}{2}}\cdot (J_{11}+J_{12}+J_{12}+J_{22})\end{aligned}}}" loading="lazy"></dd></dl></dd></dl>

<dl><dd>Die Rotationsenergie erhält man damit aus:</dd></dl>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{\mathrm {rot} }&={\frac {1}{2}}\cdot J_{n}\cdot \omega ^{2}\\&={\frac {1}{4}}\cdot (J_{11}+J_{12}+J_{12}+J_{22})\cdot \omega ^{2}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc61ed1a857377c1f6b2b418d9e98cf65f308d05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.497ex; margin-bottom: -0.174ex; width:38.757ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}E_{\mathrm {rot} }&={\frac {1}{2}}\cdot J_{n}\cdot \omega ^{2}\\&={\frac {1}{4}}\cdot (J_{11}+J_{12}+J_{12}+J_{22})\cdot \omega ^{2}\end{aligned}}}" loading="lazy"></dd></dl></dd></dl>

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

Die Rotationsenergie kann auch durch den <a href="Drehimpuls" title="Drehimpuls">Drehimpuls</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}_{s}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98b270e3db2f27153bb24f8d065cdd1828c37023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.586ex; height:3.176ex;" alt="{\displaystyle {\vec {L}}_{s}}" loading="lazy"> um den Massenmittelpunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {s}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57ce74ecbbc0e198d0bb4b1b2ffc0cb9669b72e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {s}}}" loading="lazy"> ausgedrückt werden:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{\mathrm {rot} }&={\frac {1}{2}}\,{\vec {\omega }}\cdot \mathbf {J} _{s}\cdot {\vec {\omega }}\\&={\frac {1}{2}}\,{\vec {\omega }}\cdot {\vec {L}}_{s}\\&={\frac {1}{2}}\,{\vec {L}}_{s}\cdot \mathbf {J} _{s}^{-1}\cdot {\vec {L}}_{s}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c302ecec1122887f78495ea3f25e78030235862e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.192ex; margin-bottom: -0.313ex; width:22.533ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}E_{\mathrm {rot} }&={\frac {1}{2}}\,{\vec {\omega }}\cdot \mathbf {J} _{s}\cdot {\vec {\omega }}\\&={\frac {1}{2}}\,{\vec {\omega }}\cdot {\vec {L}}_{s}\\&={\frac {1}{2}}\,{\vec {L}}_{s}\cdot \mathbf {J} _{s}^{-1}\cdot {\vec {L}}_{s}\end{aligned}}}" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}_{s}=\mathbf {J} _{s}\cdot {\vec {\omega }}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {L}}_{s}=\mathbf {J} _{s}\cdot {\vec {\omega }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e75183f4f8fcace511cf7de61106713ed7527721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.194ex; height:3.176ex;" alt="{\displaystyle {\vec {L}}_{s}=\mathbf {J} _{s}\cdot {\vec {\omega }}}" loading="lazy"> und der <a href="Inverse_Matrix" title="Inverse Matrix"> Inversen</a> des Trägheitstensors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} _{s}^{-1}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{s}^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21967d082ba12642efa6fec3c9f6c99b0c4cb994.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.714ex; height:3.176ex;" alt="{\displaystyle \mathbf {J} _{s}^{-1}}" loading="lazy">, die immer existiert.

Es ist zu beachten, dass der Drehimpuls und die Winkelgeschwindigkeit im Allgemeinen nicht parallel zueinander stehen (außer bei Rotation um eine <a href="Haupttr%C3%A4gheitsachse" title="Hauptträgheitsachse">Hauptträgheitsachse</a>); siehe auch <a href="Tr%C3%A4gheitsellipsoid" title="Trägheitsellipsoid">Trägheitsellipsoid</a>.

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

Sei der <a href="Starrer_K%C3%B6rper" title="Starrer Körper">Starre Körper</a> durch einzelne <a href="Massenpunkt" title="Massenpunkt">Massenpunkte</a> mit Massen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95ec8e804f69706d3f5ad235f4f983220c8df7c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;" alt="{\displaystyle m_{i}}" loading="lazy"> an den Orten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{i}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de93eb4c8bca39012a94e9809c45d7fd677bf975.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.023ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{i}}" loading="lazy"> relativ zum Ursprung eines körperfesten <a href="Bezugssystem" title="Bezugssystem">Bezugssystems</a> gegeben, das sich am Ort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}(t)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2777e43b4adf37f8f087eb616a77edbb204150f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.743ex; height:3.343ex;" alt="{\displaystyle {\vec {b}}(t)}" loading="lazy"> im <a href="Inertialsystem" title="Inertialsystem">Inertialsystem</a> befindet. Bei der <a href="Starrer_K%C3%B6rper#Allgemeine_Bewegungen_starrer_Körper" title="Starrer Körper">allgemeinen Bewegung starrer Körper</a> gilt die eulersche Geschwindigkeitsgleichung:

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}({\vec {r}},t)={\dot {\vec {b}}}(t)+{\vec {\omega }}(t)\times {\vec {r}}\,.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b13f10c33fa6696d563976107be4d7b2376b8e1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.024ex; height:4.009ex;" alt="{\displaystyle {\vec {v}}({\vec {r}},t)={\dot {\vec {b}}}(t)+{\vec {\omega }}(t)\times {\vec {r}}\,.}" loading="lazy"></dd></dl>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4e066a68ceb355e3314fb2b97f1c0c421ca6074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.343ex;" alt="{\displaystyle {\vec {\omega }}}" loading="lazy"> die <a href="Winkelgeschwindigkeit" title="Winkelgeschwindigkeit">Winkelgeschwindigkeit</a> des starren Körpers (inklusive des körperfesten Bezugssystems), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\vec {b}}}}">

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6d4de2423072620ebc7c41415ae6d01c0a70ef8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:3.509ex;" alt="{\displaystyle {\dot {\vec {b}}}}" loading="lazy"> die Geschwindigkeit von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c9ef58be7103eb0b2bfcb460df23430f6a36216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}" loading="lazy">, und beide dürfen von der Zeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

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

</semantics>

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

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}({\vec {r}},t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dba26b2ca160d0499477d5abb6a9550c83ac987f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.081ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}({\vec {r}},t)}" loading="lazy"> ist zur Zeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

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

</semantics>

</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"> die Geschwindigkeit des Massenpunkts am Ort <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aec3c9ce13b53e9e24c98e7cce4212627884c91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.223ex; height:2.343ex;" alt="{\displaystyle {\vec {r}}}" loading="lazy">; die Position des Massenpunkts im Raum ist also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}={\vec {b}}+{\vec {r}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f13547ca2aa531dd9ffda334c69eb4d2cfa414e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.586ex; height:3.009ex;" alt="{\displaystyle {\vec {x}}={\vec {b}}+{\vec {r}}}" loading="lazy">.

Die kinetische Energie des Körpers ist dann gegeben durch<a href="#cite_note-thph2-1">[1]</a>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msub>

<mover>

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</mrow>

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</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msup>

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

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<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

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</msub>

<msup>

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</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</msub>

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</munder>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<mover>

</mover>

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<msub>

<mover>

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</mrow>

</msub>

<msup>

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</msup>

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<msub>

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</msub>

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<mover>

</mover>

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<mover>

</mover>

</mrow>

<mover>

<mover>

</mover>

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</mover>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{\rm {kin}}=&{\frac {1}{2}}\sum _{i}m_{i}{\vec {v}}({\vec {r}}_{i},t)^{2}={\frac {1}{2}}\sum _{i}m_{i}({\dot {\vec {b}}}+{\vec {\omega }}\times {\vec {r}}_{i})^{2}\\=&{\frac {1}{2}}\sum _{i}m_{i}{\dot {\vec {b}}}^{2}+{\frac {1}{2}}\sum _{i}m_{i}2{\dot {\vec {b}}}\cdot ({\vec {\omega }}\times {\vec {r}}_{i})+{\frac {1}{2}}\sum _{i}m_{i}({\vec {\omega }}\times {\vec {r}}_{i})^{2}\\=&\underbrace {{\frac {1}{2}}m{\dot {\vec {b}}}^{2}} _{E_{\rm {trans}}}+\underbrace {{\frac {1}{2}}\sum _{i}m_{i}({\vec {\omega }}\times {\vec {r}}_{i})^{2}} _{E_{\rm {rot}}}+m{\vec {\omega }}\cdot ({\vec {s}}\times {\dot {\vec {b}}})\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1daa746a6a2ef4aa2a60bbe42731c025acb66e73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.338ex; width:63.487ex; height:23.843ex;" alt="{\displaystyle {\begin{aligned}E_{\rm {kin}}=&{\frac {1}{2}}\sum _{i}m_{i}{\vec {v}}({\vec {r}}_{i},t)^{2}={\frac {1}{2}}\sum _{i}m_{i}({\dot {\vec {b}}}+{\vec {\omega }}\times {\vec {r}}_{i})^{2}\\=&{\frac {1}{2}}\sum _{i}m_{i}{\dot {\vec {b}}}^{2}+{\frac {1}{2}}\sum _{i}m_{i}2{\dot {\vec {b}}}\cdot ({\vec {\omega }}\times {\vec {r}}_{i})+{\frac {1}{2}}\sum _{i}m_{i}({\vec {\omega }}\times {\vec {r}}_{i})^{2}\\=&\underbrace {{\frac {1}{2}}m{\dot {\vec {b}}}^{2}} _{E_{\rm {trans}}}+\underbrace {{\frac {1}{2}}\sum _{i}m_{i}({\vec {\omega }}\times {\vec {r}}_{i})^{2}} _{E_{\rm {rot}}}+m{\vec {\omega }}\cdot ({\vec {s}}\times {\dot {\vec {b}}})\end{aligned}}}" loading="lazy"></dd></dl>

Darin ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle m:=\sum _{i}m_{i}}">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle m:=\sum _{i}m_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3eea51271548cd9e323280ee6afc29869ee8e692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.266ex; height:3.009ex;" alt="{\displaystyle \textstyle m:=\sum _{i}m_{i}}" loading="lazy"> die Gesamtmasse des Körpers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\rm {trans}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\rm {trans}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52709ecc57a0f2185bfdc6e076ac10a11c9f322b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.616ex; height:2.509ex;" alt="{\displaystyle E_{\rm {trans}}}" loading="lazy"> seine translatorische Energie, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\rm {rot}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\rm {rot}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6abb11e32bb9ae58dfbe22194b1c8651e00da368.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.054ex; height:2.509ex;" alt="{\displaystyle E_{\rm {rot}}}" loading="lazy"> seine Rotationsenergie, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\vec {s}}:={\frac {1}{m}}\sum _{i}m_{i}{\vec {r}}_{i}}">

<mover>

</mover>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\vec {s}}:={\frac {1}{m}}\sum _{i}m_{i}{\vec {r}}_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e15351e21f60fee90980d0002cd82e22cb632a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.138ex; height:3.343ex;" alt="{\displaystyle \textstyle {\vec {s}}:={\frac {1}{m}}\sum _{i}m_{i}{\vec {r}}_{i}}" loading="lazy"> sein Massenmittelpunkt im körperfesten Bezugssystem und es wurde ausgenutzt, dass im <a href="Spatprodukt" title="Spatprodukt">Spatprodukt</a> dreier Vektoren deren Reihenfolge zyklisch vertauscht werden darf. Der dritte Summand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\vec {\omega }}\cdot ({\vec {s}}\times {\dot {\vec {b}}})}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m{\vec {\omega }}\cdot ({\vec {s}}\times {\dot {\vec {b}}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16e7cfcbba2f6c57a41f5e96a22b30487ab8d7ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.201ex; height:4.009ex;" alt="{\displaystyle m{\vec {\omega }}\cdot ({\vec {s}}\times {\dot {\vec {b}}})}" loading="lazy"> verschwindet unter vier Bedingungen:

<ol><li>Wenn im körperfesten Bezugssystem der <a href="Massenmittelpunkt" title="Massenmittelpunkt">Massenmittelpunkt</a> im Ursprung (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {s}}={\vec {0}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\vec {s}}={\vec {0}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18d105a8acbac4f49549bc9881fc2baa59948b77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.484ex; height:2.843ex;" alt="{\displaystyle {\vec {s}}={\vec {0}}}" loading="lazy">) oder auf der Drehachse liegt (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}\parallel {\vec {s}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/459fb6a21a026982c28bc4c39a195a70f162c922.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.122ex; height:2.843ex;" alt="{\displaystyle {\vec {\omega }}\parallel {\vec {s}}}" loading="lazy">), die Rotation also um den Massenmittelpunkt stattfindet.</li>

<li>Wenn das körperfeste System ruht (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\vec {b}}}={\vec {0}}}">

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

<mover>

</mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dot {\vec {b}}}={\vec {0}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d710c16581fff9e3b9c70a8d967123c6b2c91e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.423ex; height:3.509ex;" alt="{\displaystyle {\dot {\vec {b}}}={\vec {0}}}" loading="lazy">) oder sich entlang der Drehachse bewegt (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\omega }}\parallel {\dot {\vec {b}}}}">

<mover>

</mover>

</mrow>

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b019ce108f8c81fb89e786322d8f914bb68b6c35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.061ex; height:4.009ex;" alt="{\displaystyle {\vec {\omega }}\parallel {\dot {\vec {b}}}}" loading="lazy">), was sich durch geeignete Wahl des Bezugspunkts immer einrichten lässt,<a href="#cite_note-thph2-1">[1]</a> siehe <a href="Schraubung#Schraubung_von_Starrkörpern" title="Schraubung">Schraubung</a>.</li>

<li>Wenn sich der Bezugspunkt in Richtung des Massenmittelpunkts bewegt (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {\vec {b}}}\parallel {\vec {s}}}">

<mover>

<mover>

</mover>

</mrow>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3b7e730487d413d362cd848383d12fd99a4c290.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.838ex; height:4.009ex;" alt="{\displaystyle {\dot {\vec {b}}}\parallel {\vec {s}}}" loading="lazy">), was einem <a href="Balancieren" title="Balancieren">Balanceakt</a> gleichkommt.</li>

<li>Der triviale Fall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\vec {\omega }}={\vec {0}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m{\vec {\omega }}={\vec {0}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/478beae229dfa333b686a6e39a11b8341217c546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.747ex; height:2.843ex;" alt="{\displaystyle m{\vec {\omega }}={\vec {0}}}" loading="lazy"> wird hier nicht weiter betrachtet.</li></ol>

In den ersten drei Fällen spaltet sich die kinetische Energie in die translatorische und rotatorische auf, aber nur die ersten beiden Fälle sind für die <a href="Kreiseltheorie" title="Kreiseltheorie">Kreiseltheorie</a> interessant. Mit der <a href="Kreuzprodukt#Lagrange-Identität" title="Kreuzprodukt">Lagrange-Identität</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {\omega }}\times {\vec {r}}_{i})^{2}=({\vec {\omega }}\cdot {\vec {\omega }})({\vec {r}}_{i}\cdot {\vec {r}}_{i})-({\vec {\omega }}\cdot {\vec {r}}_{i})^{2}}">

<mover>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

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</msub>

<msup>

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</msup>

<mover>

</mover>

</mrow>

<mover>

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<msub>

<mover>

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<msub>

<mover>

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<mover>

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<msub>

<mover>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle ({\vec {\omega }}\times {\vec {r}}_{i})^{2}=({\vec {\omega }}\cdot {\vec {\omega }})({\vec {r}}_{i}\cdot {\vec {r}}_{i})-({\vec {\omega }}\cdot {\vec {r}}_{i})^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e425deabdfba3c46182cdfc24f98b1288edd333f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.037ex; height:3.176ex;" alt="{\displaystyle ({\vec {\omega }}\times {\vec {r}}_{i})^{2}=({\vec {\omega }}\cdot {\vec {\omega }})({\vec {r}}_{i}\cdot {\vec {r}}_{i})-({\vec {\omega }}\cdot {\vec {r}}_{i})^{2}}" loading="lazy"> berechnet sich unter Ausnutzung der Eigenschaften des <a href="Dyadisches_Produkt" title="Dyadisches Produkt">dyadischen Produkts</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

</mstyle>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de29098f5a34ee296a505681a0d5e875070f2aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }" loading="lazy"><a href="#cite_note-dyade-2">[2]</a> die Rotationsenergie zu:

<mtr>

<mtd>

<msub>

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<mtd>

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<msub>

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<mover>

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<msub>

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</msup>

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<mtr>

<mtd>

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<mtd>

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<msub>

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<mtr>

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<msub>

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<mfrac>

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<mover>

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<msub>

<mover>

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</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}2E_{\rm {rot}}=&\sum _{i}m_{i}({\vec {\omega }}\times {\vec {r}}_{i})^{2}=\sum _{i}m_{i}[({\vec {\omega }}\cdot {\vec {\omega }})({\vec {r}}_{i}\cdot {\vec {r}}_{i})-({\vec {\omega }}\cdot {\vec {r}}_{i})^{2}]\\=&{\vec {\omega }}\cdot \sum _{i}m_{i}[({\vec {r}}_{i}\cdot {\vec {r}}_{i}){\vec {\omega }}-({\vec {\omega }}\cdot {\vec {r}}_{i}){\vec {r}}_{i}]={\vec {\omega }}\cdot \sum _{i}m_{i}[({\vec {r}}_{i}\cdot {\vec {r}}_{i})\mathbf {1} -{\vec {r}}_{i}\otimes {\vec {r}}_{i}]\cdot {\vec {\omega }}\\=&{\vec {\omega }}\cdot \mathbf {\Theta } _{b}\cdot {\vec {\omega }}\\\Rightarrow E_{\rm {rot}}=&{\frac {1}{2}}{\vec {\omega }}\cdot \mathbf {\Theta } _{b}\cdot {\vec {\omega }}={\frac {1}{2}}{\vec {\omega }}\cdot {\vec {L}}_{b}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/532a05ada5b7a01ebc9bcc60b6e643de47ecedcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:77.849ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}2E_{\rm {rot}}=&\sum _{i}m_{i}({\vec {\omega }}\times {\vec {r}}_{i})^{2}=\sum _{i}m_{i}[({\vec {\omega }}\cdot {\vec {\omega }})({\vec {r}}_{i}\cdot {\vec {r}}_{i})-({\vec {\omega }}\cdot {\vec {r}}_{i})^{2}]\\=&{\vec {\omega }}\cdot \sum _{i}m_{i}[({\vec {r}}_{i}\cdot {\vec {r}}_{i}){\vec {\omega }}-({\vec {\omega }}\cdot {\vec {r}}_{i}){\vec {r}}_{i}]={\vec {\omega }}\cdot \sum _{i}m_{i}[({\vec {r}}_{i}\cdot {\vec {r}}_{i})\mathbf {1} -{\vec {r}}_{i}\otimes {\vec {r}}_{i}]\cdot {\vec {\omega }}\\=&{\vec {\omega }}\cdot \mathbf {\Theta } _{b}\cdot {\vec {\omega }}\\\Rightarrow E_{\rm {rot}}=&{\frac {1}{2}}{\vec {\omega }}\cdot \mathbf {\Theta } _{b}\cdot {\vec {\omega }}={\frac {1}{2}}{\vec {\omega }}\cdot {\vec {L}}_{b}\end{aligned}}}" loading="lazy"></dd></dl>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathbf {\Theta } _{b}:=\sum _{i}m_{i}[({\vec {r}}_{i}\cdot {\vec {r}}_{i})\mathbf {1} -{\vec {r}}_{i}\otimes {\vec {r}}_{i}]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b2cb8968b8fa97bd1066dc47894227d9db2c62e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.132ex; height:3.009ex;" alt="{\displaystyle \textstyle \mathbf {\Theta } _{b}:=\sum _{i}m_{i}[({\vec {r}}_{i}\cdot {\vec {r}}_{i})\mathbf {1} -{\vec {r}}_{i}\otimes {\vec {r}}_{i}]}" loading="lazy"> der <a href="Tr%C3%A4gheitstensor" title="Trägheitstensor">Trägheitstensor</a> des starren Körpers bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}">

<mover>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c9ef58be7103eb0b2bfcb460df23430f6a36216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {L}}_{b}:=\mathbf {\Theta } _{b}\cdot {\vec {\omega }}}">

<msub>

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {L}}_{b}:=\mathbf {\Theta } _{b}\cdot {\vec {\omega }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58709ca7f28a384c0de7fcb3af3d3b231e1d3fbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.406ex; height:3.176ex;" alt="{\displaystyle {\vec {L}}_{b}:=\mathbf {\Theta } _{b}\cdot {\vec {\omega }}}" loading="lazy"> sein Eigen<a href="Drehimpuls" title="Drehimpuls">drehimpuls</a> bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {b}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c9ef58be7103eb0b2bfcb460df23430f6a36216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.094ex; height:2.843ex;" alt="{\displaystyle {\vec {b}}}" loading="lazy"> und 1 der <a href="Einheitstensor" title="Einheitstensor">Einheitstensor</a>. Im körperfesten System ist der Trägheitstensor konstant, im Inertialsystem jedoch meist nicht, wenn sich der Körper dreht.

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Potentielle_Energie" title="Potentielle Energie">Potentielle Energie</a></li>

<li><a href="Rollersatzmasse" title="Rollersatzmasse">Rollersatzmasse</a></li>

<li><a href="Schwungradspeicherung" title="Schwungradspeicherung">Schwungradspeicherung</a></li>

<li><a href="Schwungrad-Speicherkraftwerk" title="Schwungrad-Speicherkraftwerk">Schwungrad-Speicherkraftwerk</a></li></ul>

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

<li id="cite_note-thph2-1">↑ <a href="#cite_ref-thph2_1-0">a</a> <a href="#cite_ref-thph2_1-1">b</a> Institut für Physik an der Universität Rostock (Hrsg.): <cite style="font-style:italic">Theoretische Physik II – Theoretische Mechanik</cite>. Kapitel 5 – Starrer Körper und Kreiseltheorie. (<a rel="nofollow" class="external text" href="http://www.qms.uni-rostock.de/fileadmin/Physik_Festkoerpertheorie/Lehre_Scheel/Theoretische_Physik_II_-_Theoretische_Mechanik/Theor_Phy_II_Kapitel_5_-_Starrer_Koerper_und_Kreiseltheorie.pdf">uni-rostock.de</a> [PDF; abgerufen am 6. Juni 2017]).<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:Rotationsenergie&rft.btitle=Theoretische+Physik+II+-+Theoretische+Mechanik&rft.genre=book" style="display:none">

</li>

<li id="cite_note-dyade-2"><a href="#cite_ref-dyade_2-0">↑</a> Das dyadische Produkt ist mit drei beliebigen Vektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}},{\vec {b}},{\vec {c}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/befa941df6751fd0d377ccd7d31130ef372a3b8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.615ex; height:3.176ex;" alt="{\displaystyle {\vec {a}},{\vec {b}},{\vec {c}}}" loading="lazy"> definiert durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {a}}\otimes {\vec {b}})\cdot {\vec {c}}:=({\vec {b}}\cdot {\vec {c}}){\vec {a}}}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\vec {a}}\otimes {\vec {b}})\cdot {\vec {c}}:=({\vec {b}}\cdot {\vec {c}}){\vec {a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39713e45ef853d9e883cbf46847908dcf28b03eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.656ex; height:3.343ex;" alt="{\displaystyle ({\vec {a}}\otimes {\vec {b}})\cdot {\vec {c}}:=({\vec {b}}\cdot {\vec {c}}){\vec {a}}}" loading="lazy">

</li>

</ol></div><div><div class="zim-footer">

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