# 力矩

${\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} \,\!}$

${\displaystyle \tau =rF\sin \theta \,\!}$

## 定義

${\displaystyle {\boldsymbol {\tau }}\ {\stackrel {def}{=}}\ \mathbf {r} \times \mathbf {F} \,\!}$

${\displaystyle \tau =|\mathbf {r} ||\mathbf {F} |\sin \theta \,\!}$

${\displaystyle \tau =rF_{\perp }\,\!}$

## 力矩與角動量之間的關係

${\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} \,\!}$

{\displaystyle {\begin{aligned}{\frac {d\mathbf {L} }{dt}}&=\mathbf {r} \times {\frac {d\mathbf {p} }{dt}}+{\frac {d\mathbf {r} }{dt}}\times \mathbf {p} \\&=\mathbf {r} \times m{\frac {d\mathbf {v} }{dt}}+\mathbf {v} \times m\mathbf {v} \\&=\mathbf {r} \times m\mathbf {a} \\\end{aligned}}\,\!}

${\displaystyle {\frac {d\mathbf {L} }{dt}}=\mathbf {r} \times \mathbf {F} \,\!}$

${\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}\,\!}$

${\displaystyle {\boldsymbol {\tau }}_{1}+\cdots +{\boldsymbol {\tau }}_{n}={\boldsymbol {\tau }}_{\mathrm {net} }={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}\,\!}$

${\displaystyle \mathbf {L} =I{\boldsymbol {\omega }}\,\!}$

${\displaystyle {\boldsymbol {\tau }}_{\mathrm {net} }={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}={\frac {\mathrm {d} (I{\boldsymbol {\omega }})}{\mathrm {d} t}}=I{\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}=I{\boldsymbol {\alpha }}\,\!}$

## 單位

${\displaystyle E=\tau \theta \,\!}$

## 矩臂方程式

${\displaystyle {\boldsymbol {\tau }}=({\text{moment arm}})\cdot {\textrm {force}}\,\!}$

## 靜力概念

${\displaystyle \sum F_{x}=0\,\!}$
${\displaystyle \sum F_{y}=0\,\!}$
${\displaystyle \sum \tau =0\,\!}$

## 力矩、能量和功率之間的關係

${\displaystyle W=\int _{\theta _{1}}^{\theta _{2}}\tau \ \mathrm {d} \theta \,\!}$

${\displaystyle K_{\mathrm {rot} }={\tfrac {1}{2}}I\omega ^{2}\,\!}$

${\displaystyle P={\boldsymbol {\tau }}\cdot {\boldsymbol {\omega }}\,\!}$

## 力矩原理

${\displaystyle (\mathbf {r} \times \mathbf {F} _{1})+(\mathbf {r} \times \mathbf {F} _{2})+\cdots =\mathbf {r} \times (\mathbf {F} _{1}+\mathbf {F} _{2}+\cdots )\,\!}$

## 參考文獻

1. ^ Serway, R. A. and Jewett, Jr. J. W. (2003). Physics for Scientists and Engineers. 6th Ed. Brooks Cole. ISBN 978-0-534-40842-8.
2. ^ *喬治亞州州立大學Georgia State University）線上物理網頁：力矩的右手定則, [2007-09-08]
3. ^ SI brochure Ed. 8, Section 5.1, Bureau International des Poids et Mesures, 2006 [2007-04-01], （原始內容存檔於2007-05-19）
4. ^ SI brochure Ed. 8, Section 2.2.2, Bureau International des Poids et Mesures, 2006 [2007-04-01], （原始內容存檔於2005-03-16）
5. ^ Engineering Mechanics: Equilibrium, by C. Hartsuijker, J. W. Welleman, page 64
• Tipler, Paul. Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. 2004. ISBN 978-0-7167-0809-4.