# 力矩

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

$\tau = rF\sin \theta\,\!$

## 定义

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

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

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

## 力矩與角動量之間的關係

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

\begin{align}\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{align}\,\!

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

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

$\boldsymbol{\tau}_1 + \cdots + \boldsymbol{\tau}_n = \boldsymbol{\tau}_{\mathrm{net}} = \frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t}\,\!$

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

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

## 单位

$E= \tau\theta\,\!$

## 矩臂方程式

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

## 静力概念

$\sum F_x =0\,\!$
$\sum F_y =0\,\!$
$\sum \tau =0\,\!$

## 力矩、能量和功率之間的關係

$W = \int_{\theta_1}^{\theta_2} \tau\ \mathrm{d}\theta\,\!$

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

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

## 力矩原理

$(\mathbf{r}\times\mathbf{F}_1) + (\mathbf{r}\times\mathbf{F}_2) + \cdots = \mathbf{r}\times(\mathbf{F}_1+\mathbf{F}_2 + \cdots)\,\!$

## 参考文献

1. ^ Serway, R. A. and Jewett, Jr. J. W. (2003). Physics for Scientists and Engineers. 6th Ed. Brooks Cole. ISBN 0-53440-842-7.
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]
4. ^ SI brochure Ed. 8, Section 2.2.2, Bureau International des Poids et Mesures. 2006 [2007-04-01]
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 0-7167-0809-4.