# 轉動慣量列表

## 常见物理模型的转动惯量

${\displaystyle I_{x}=I_{y}={\frac {1}{12}}m\left[3\left({r_{1}}^{2}+{r_{2}}^{2}\right)+h^{2}\right]}$

${\displaystyle I_{x}=I_{y}={\frac {1}{12}}m\left(3r^{2}+h^{2}\right)\,\!}$

${\displaystyle I_{x}=I_{y}={\frac {mr^{2}}{4}}\,\!}$

${\displaystyle I_{x}=I_{y}={\frac {mr^{2}}{2}}\,\!}$

${\displaystyle I_{b}={\frac {1}{5}}m\left(a^{2}+c^{2}\right)\,\!}$
${\displaystyle I_{c}={\frac {1}{5}}m\left(a^{2}+b^{2}\right)\,\!}$

${\displaystyle I_{x}=I_{y}={\frac {3}{20}}m\left({r^{2}}+{4}h^{2}\right)\,\!}$ [2]

${\displaystyle I_{w}={\frac {1}{12}}m\left(h^{2}+d^{2}\right)\,\!}$
${\displaystyle I_{d}={\frac {1}{12}}m\left(h^{2}+w^{2}\right)\,\!}$

${\displaystyle I_{\mathrm {hollow} }={\frac {1}{12}}ms^{2}\,\!}$ [3]
“solid”意为实心，“hollow”意为空心，下同。

${\displaystyle I_{x,\mathrm {solid} }=I_{y,\mathrm {solid} }=I_{z,\mathrm {solid} }={\frac {1}{10}}ms^{2}\,\!}$ [3]

## 常見物理模型的三維慣量張量

${\displaystyle \mathbf {n} \cdot \mathbf {I} \cdot \mathbf {n} \equiv n_{i}I_{ij}n_{j}\,,}$

${\displaystyle I={\begin{bmatrix}{\frac {2}{3}}mr^{2}&0&0\\0&{\frac {2}{3}}mr^{2}&0\\0&0&{\frac {2}{3}}mr^{2}\end{bmatrix}}}$

${\displaystyle I={\begin{bmatrix}{\frac {1}{3}}ml^{2}&0&0\\0&0&0\\0&0&{\frac {1}{3}}ml^{2}\end{bmatrix}}}$

${\displaystyle I={\begin{bmatrix}{\frac {1}{12}}ml^{2}&0&0\\0&0&0\\0&0&{\frac {1}{12}}ml^{2}\end{bmatrix}}}$

${\displaystyle I={\begin{bmatrix}{\frac {1}{12}}m(3r^{2}+h^{2})&0&0\\0&{\frac {1}{12}}m(3r^{2}+h^{2})&0\\0&0&{\frac {1}{2}}mr^{2}\end{bmatrix}}}$

${\displaystyle I={\begin{bmatrix}{\frac {1}{12}}m(3(r_{1}^{2}+r_{2}^{2})+h^{2})&0&0\\0&{\frac {1}{12}}m(3(r_{1}^{2}+r_{2}^{2})+h^{2})&0\\0&0&{\frac {1}{2}}m(r_{1}^{2}+r_{2}^{2})\end{bmatrix}}}$

## 參考資料

1. Raymond A. Serway. Physics for Scientists and Engineers, second ed.. Saunders College Publishing. 1986: 202. ISBN 0-03-004534-7.
2. Ferdinand P. Beer and E. Russell Johnston, Jr. Vector Mechanics for Engineers, fourth ed.. McGraw-Hill. 1984: 911. ISBN 0-07-004389-2.
3. Satterly, John. The Moments of Inertia of Some Polyhedra. The Mathematical Gazette (Mathematical Association). 1958, 42 (339): 11–13. JSTOR 3608345. doi:10.2307/3608345.
4. ^ Eric W. Weisstein. Moment of Inertia — Ring. Wolfram Research. [2010-03-25]. （原始内容存档于2013-07-13）.
5. ^ David Morin. Introduction to Classical Mechanics: With Problems and Solutions; first edition (8 january 2010). Cambridge University Press. 2010: 320. ISBN 0521876222.