24959:
21300:
24954:{\displaystyle {\begin{aligned}-\sum _{i=1}^{n}m_{i}&=-\sum _{i=1}^{n}m_{i}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}\right)\right)\;\ldots {\text{ cross-product as matrix multiplication}}\\&=-\sum _{i=1}^{n}m_{i}\left({\begin{bmatrix}0&-\Delta r_{3,i}&\Delta r_{2,i}\\\Delta r_{3,i}&0&-\Delta r_{1,i}\\-\Delta r_{2,i}&\Delta r_{1,i}&0\end{bmatrix}}{\begin{bmatrix}-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3}\\+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3}\\-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2}\end{bmatrix}}\right)\\&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-\Delta r_{3,i}(+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3})+\Delta r_{2,i}(-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2})\\+\Delta r_{3,i}(-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3})-\Delta r_{1,i}(-\Delta r_{2,i}\,u_{1}+\Delta r_{1,i}\,u_{2})\\-\Delta r_{2,i}(-\Delta r_{3,i}\,u_{2}+\Delta r_{2,i}\,u_{3})+\Delta r_{1,i}(+\Delta r_{3,i}\,u_{1}-\Delta r_{1,i}\,u_{3})\end{bmatrix}}\\&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-\Delta r_{3,i}^{2}\,u_{1}+\Delta r_{1,i}\Delta r_{3,i}\,u_{3}-\Delta r_{2,i}^{2}\,u_{1}+\Delta r_{1,i}\Delta r_{2,i}\,u_{2}\\-\Delta r_{3,i}^{2}\,u_{2}+\Delta r_{2,i}\Delta r_{3,i}\,u_{3}+\Delta r_{2,i}\Delta r_{1,i}\,u_{1}-\Delta r_{1,i}^{2}\,u_{2}\\+\Delta r_{3,i}\Delta r_{2,i}\,u_{2}-\Delta r_{2,i}^{2}\,u_{3}+\Delta r_{3,i}\Delta r_{1,i}\,u_{1}-\Delta r_{1,i}^{2}\,u_{3}\end{bmatrix}}\\&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-(\Delta r_{2,i}^{2}+\Delta r_{3,i}^{2})\,u_{1}+\Delta r_{1,i}\Delta r_{2,i}\,u_{2}+\Delta r_{1,i}\Delta r_{3,i}\,u_{3}\\+\Delta r_{2,i}\Delta r_{1,i}\,u_{1}-(\Delta r_{1,i}^{2}+\Delta r_{3,i}^{2})\,u_{2}+\Delta r_{2,i}\Delta r_{3,i}\,u_{3}\\+\Delta r_{3,i}\Delta r_{1,i}\,u_{1}+\Delta r_{3,i}\Delta r_{2,i}\,u_{2}-(\Delta r_{1,i}^{2}+\Delta r_{2,i}^{2})\,u_{3}\end{bmatrix}}\\&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-(\Delta r_{2,i}^{2}+\Delta r_{3,i}^{2})&\Delta r_{1,i}\Delta r_{2,i}&\Delta r_{1,i}\Delta r_{3,i}\\\Delta r_{2,i}\Delta r_{1,i}&-(\Delta r_{1,i}^{2}+\Delta r_{3,i}^{2})&\Delta r_{2,i}\Delta r_{3,i}\\\Delta r_{3,i}\Delta r_{1,i}&\Delta r_{3,i}\Delta r_{2,i}&-(\Delta r_{1,i}^{2}+\Delta r_{2,i}^{2})\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}\\&=-\sum _{i=1}^{n}m_{i}^{2}\mathbf {u} \\-\sum _{i=1}^{n}m_{i}&=\left(-\sum _{i=1}^{n}m_{i}^{2}\right)\mathbf {u} \;\ldots \;\mathbf {u} {\text{ is not characteristic of }}P_{i}\end{aligned}}}
17185:
16240:
17180:{\displaystyle {\begin{aligned}0&=\Delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i})\times \Delta \mathbf {r} _{i})+({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i})\times (\Delta \mathbf {r} _{i}\times {\boldsymbol {\omega }})\\&=\Delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i})\times \Delta \mathbf {r} _{i})+({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i})\times -({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i})\;\ldots {\text{ cross-product anticommutativity}}\\&=\Delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i})\times \Delta \mathbf {r} _{i})+-\;\ldots {\text{ cross-product scalar multiplication}}\\&=\Delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i})\times \Delta \mathbf {r} _{i})+-\;\ldots {\text{ self cross-product}}\\0&=\Delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i})\times \Delta \mathbf {r} _{i})\end{aligned}}}
34574:
33291:
18170:
21271:
34569:{\displaystyle {\begin{aligned}I_{xy}=I_{yx}\ &{\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{N}m_{k}x_{k}y_{k},\\I_{xz}=I_{zx}\ &{\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{N}m_{k}x_{k}z_{k},\\I_{yz}=I_{zy}\ &{\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{N}m_{k}y_{k}z_{k},\\\mathbf {I} ={\begin{bmatrix}I_{11}&I_{12}&I_{13}\\I_{21}&I_{22}&I_{23}\\I_{31}&I_{32}&I_{33}\end{bmatrix}}&={\begin{bmatrix}I_{xx}&-I_{xy}&-I_{xz}\\-I_{yx}&I_{yy}&-I_{yz}\\-I_{zx}&-I_{zy}&I_{zz}\end{bmatrix}}={\begin{bmatrix}\sum _{k=1}^{N}m_{k}\left(y_{k}^{2}+z_{k}^{2}\right)&-\sum _{k=1}^{N}m_{k}x_{k}y_{k}&-\sum _{k=1}^{N}m_{k}x_{k}z_{k}\\-\sum _{k=1}^{N}m_{k}x_{k}y_{k}&\sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+z_{k}^{2}\right)&-\sum _{k=1}^{N}m_{k}y_{k}z_{k}\\-\sum _{k=1}^{N}m_{k}x_{k}z_{k}&-\sum _{k=1}^{N}m_{k}y_{k}z_{k}&\sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+y_{k}^{2}\right)\end{bmatrix}}.\end{aligned}}}
7648:: When a body moves parallel to a ground plane, the trajectories of all the points in the body lie in planes parallel to this ground plane. This means that any rotation that the body undergoes must be around an axis perpendicular to this plane. Planar movement is often presented as projected onto this ground plane so that the axis of rotation appears as a point. In this case, the angular velocity and angular acceleration of the body are scalars and the fact that they are vectors along the rotation axis is ignored. This is usually preferred for introductions to the topic. But in the case of moment of inertia, the combination of mass and geometry benefits from the geometric properties of the cross product. For this reason, in this section on planar movement the angular velocity and accelerations of the body are vectors perpendicular to the ground plane, and the cross product operations are the same as used for the study of spatial rigid body movement.
29440:
14020:
17196:
19301:
28878:
13333:
18165:{\displaystyle {\begin{aligned}\Delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}))&=-\\&=-\;\ldots {\text{ vector triple product}}\\&=-\;\ldots {\text{ scalar triple product}}\\&=-\;\ldots {\text{ self cross-product}}\\&=-\\&=-\;\ldots {\text{ cross-product scalar multiplication}}\\&={\boldsymbol {\omega }}\times -(\Delta \mathbf {r} _{i}({\boldsymbol {\omega }}\cdot \Delta \mathbf {r} _{i}))\;\ldots {\text{ cross-product scalar multiplication}}\\\Delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}))&={\boldsymbol {\omega }}\times -(\Delta \mathbf {r} _{i}(\Delta \mathbf {r} _{i}\cdot {\boldsymbol {\omega }}))\;\ldots {\text{ dot-product commutativity}}\\\end{aligned}}}
21266:{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}\;\ldots {\text{ cross-product distributivity over addition}}\\&=\sum _{i=1}^{n}m_{i}\;\ldots {\text{ cross-product scalar multiplication}}\\&=\sum _{i=1}^{n}m_{i}\;\ldots {\text{ self cross-product}}\\&=\sum _{i=1}^{n}m_{i}\\&=\sum _{i=1}^{n}m_{i}\;\ldots {\text{ vector triple product}}\\&=\sum _{i=1}^{n}m_{i}\;\ldots {\text{ cross-product anticommutativity}}\\&=-\sum _{i=1}^{n}m_{i}\;\ldots {\text{ cross-product scalar multiplication}}\\&=-\sum _{i=1}^{n}m_{i}+-\sum _{i=1}^{n}m_{i}\;\ldots {\text{ summation distributivity}}\\{\boldsymbol {\tau }}&=-\sum _{i=1}^{n}m_{i}+{\boldsymbol {\omega }}\times -\sum _{i=1}^{n}m_{i}\;\ldots \;{\boldsymbol {\omega }}{\text{ is not characteristic of particle }}P_{i}\end{aligned}}}
33155:
29435:{\displaystyle {\begin{aligned}&\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\\={}&\left(\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\times \mathbf {\hat {k}} \right)\cdot \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\\={}&\left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\cdot \left(-\Delta \mathbf {r} _{i}\times \mathbf {\hat {k}} \right)\\={}&-\mathbf {\hat {k}} \cdot \left(\Delta \mathbf {r} _{i}\times \Delta \mathbf {r} _{i}\times \mathbf {\hat {k}} \right)\\={}&-\mathbf {\hat {k}} \cdot \left^{2}\mathbf {\hat {k}} .\end{aligned}}}
14015:{\displaystyle {\begin{aligned}E_{\text{K}}&={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left(\left{\boldsymbol {\omega }}\right)\cdot \left(\left{\boldsymbol {\omega }}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\\&={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}^{\mathsf {T}}\left^{\mathsf {T}}\left{\boldsymbol {\omega }}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\\&={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \left(-\sum _{i=1}^{n}m_{i}\left^{2}\right){\boldsymbol {\omega }}+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }.\end{aligned}}}
9039:
32265:
858:
8477:
19297:
33150:{\displaystyle \mathbf {I} ={\begin{bmatrix}I_{11}&I_{12}&I_{13}\\I_{21}&I_{22}&I_{23}\\I_{31}&I_{32}&I_{33}\end{bmatrix}}={\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{bmatrix}}={\begin{bmatrix}\sum _{k=1}^{N}m_{k}\left(y_{k}^{2}+z_{k}^{2}\right)&-\sum _{k=1}^{N}m_{k}x_{k}y_{k}&-\sum _{k=1}^{N}m_{k}x_{k}z_{k}\\-\sum _{k=1}^{N}m_{k}x_{k}y_{k}&\sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+z_{k}^{2}\right)&-\sum _{k=1}^{N}m_{k}y_{k}z_{k}\\-\sum _{k=1}^{N}m_{k}x_{k}z_{k}&-\sum _{k=1}^{N}m_{k}y_{k}z_{k}&\sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+y_{k}^{2}\right)\end{bmatrix}}.}
18177:
12001:
35496:
7638:
9034:{\displaystyle {\begin{aligned}E_{\text{K}}&={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i},\\&={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right)\cdot \left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right),\\&={\frac {1}{2}}\omega ^{2}\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\mathbf {\hat {t}} _{i}\cdot \mathbf {\hat {t}} _{i}\right)+\omega \mathbf {V} \cdot \left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {t}} _{i}\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {V} \cdot \mathbf {V} .\end{aligned}}}
936:
11611:
29936:
13219:
8056:
47:
35125:
7302:
28812:
19292:{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}\\&=\sum _{i=1}^{n}m_{i}\\&=\sum _{i=1}^{n}m_{i}\\&=\sum _{i=1}^{n}m_{i}-\Delta \mathbf {r} _{i}(\Delta \mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\;\ldots \;{\boldsymbol {\omega }}(\Delta \mathbf {r} _{i}\cdot \Delta \mathbf {r} _{i})-{\boldsymbol {\omega }}(\Delta \mathbf {r} _{i}\cdot \Delta \mathbf {r} _{i})=0\\&=\sum _{i=1}^{n}m_{i}-{\boldsymbol {\omega }}(\Delta \mathbf {r} _{i}\cdot \Delta \mathbf {r} _{i})\}]\;\ldots {\text{ addition associativity}}\\\end{aligned}}}
10594:
1254:
28521:
29518:
12892:
3964:
25351:
2479:
7659:
2032:
1242:
5970:
28528:
2692:
11996:{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \mathbf {v} _{i}\\&=\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {R} }\right)\\&=\left(-\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \left(\Delta \mathbf {r} _{i}\times {\boldsymbol {\omega }}\right)\right)+\left(\sum _{i=1}^{n}m_{i}\,\Delta \mathbf {r} _{i}\times \mathbf {V} _{\mathbf {R} }\right),\end{aligned}}}
4976:
10224:
15676:
10217:
35491:{\displaystyle I=mr^{2}=m\left(\mathbf {x} -\left(\mathbf {x} \cdot \mathbf {\hat {n}} \right)\mathbf {\hat {n}} \right)\cdot \left(\mathbf {x} -\left(\mathbf {x} \cdot \mathbf {\hat {n}} \right)\mathbf {\hat {n}} \right)=m\left(\mathbf {x} ^{2}-2\mathbf {x} \left(\mathbf {x} \cdot \mathbf {\hat {n}} \right)\mathbf {\hat {n}} +\left(\mathbf {x} \cdot \mathbf {\hat {n}} \right)^{2}\mathbf {\hat {n}} ^{2}\right)=m\left(\mathbf {x} ^{2}-\left(\mathbf {x} \cdot \mathbf {\hat {n}} \right)^{2}\right).}
28201:
7633:{\displaystyle {\begin{aligned}\mathbf {\hat {e}} _{i}&={\frac {\Delta \mathbf {r} _{i}}{\Delta r_{i}}},\quad \mathbf {\hat {k}} ={\frac {\boldsymbol {\omega }}{\omega }},\quad \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i},\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} =\omega \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {V} =\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \end{aligned}}}
31127:
6650:
24966:
2263:
35725:
1820:
12829:
871:
8467:
916:
29931:{\displaystyle {\begin{aligned}I_{L}&=\sum _{i=1}^{N}m_{i}\left|\Delta \mathbf {r} _{i}^{\perp }\right|^{2}\\&=-\sum _{i=1}^{N}m_{i}\mathbf {\hat {k}} \cdot \left^{2}\mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \left(-\sum _{i=1}^{N}m_{i}\left^{2}\right)\mathbf {\hat {k}} \\&=\mathbf {\hat {k}} \cdot \mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} =\mathbf {\hat {k}} ^{\mathsf {T}}\mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} ,\end{aligned}}}
13214:{\displaystyle E_{\text{K}}={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+\left(\sum _{i=1}^{n}m_{i}\mathbf {V} _{\mathbf {C} }\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\mathbf {V} _{\mathbf {C} }\cdot \mathbf {V} _{\mathbf {C} }\right).}
5354:
15152:
31526:
9948:
8051:{\displaystyle {\begin{aligned}\mathbf {L} &=\sum _{i=1}^{n}m_{i}\Delta \mathbf {r} _{i}\times \mathbf {v} _{i}\\&=\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\times \left(\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \right)\\&=\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\right)\omega \mathbf {\hat {k}} +\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\right)\times \mathbf {V} .\end{aligned}}}
37127:
15133:
28807:{\displaystyle \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\equiv \left(\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\times \mathbf {\hat {k}} \right)\cdot \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right),}
30727:
10589:{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\times \left(\alpha \Delta r_{i}\mathbf {\hat {t}} _{i}-\omega ^{2}\Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} \right)\\&=\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}^{2}\right)\alpha \mathbf {\hat {k}} +\left(\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}\right)\times \mathbf {A} ,\end{aligned}}}
7053:
6224:
35509:
12586:
28516:{\displaystyle {\begin{aligned}\left|\Delta \mathbf {r} _{i}^{\perp }\right|^{2}&=\left(-\left^{2}\Delta \mathbf {r} _{i}\right)\cdot \left(-\left^{2}\Delta \mathbf {r} _{i}\right)\\&=\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \Delta \mathbf {r} _{i}\right)\right)\end{aligned}}}
27116:
1257:
1256:
1261:
1260:
1255:
27763:
9232:
5078:
1262:
25346:{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=-\sum _{i=1}^{n}m_{i}+{\boldsymbol {\omega }}\times -\sum _{i=1}^{n}m_{i}\Delta \mathbf {r} _{i}\times (\Delta \mathbf {r} _{i}\times {\boldsymbol {\omega }})]\\&=\left(-\sum _{i=1}^{n}m_{i}^{2}\right){\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(-\sum _{i=1}^{n}m_{i}^{2}\right){\boldsymbol {\omega }}\end{aligned}}}
14621:
2474:{\displaystyle {\begin{aligned}\mathbf {L} &=\mathbf {r} \times \mathbf {p} =\mathbf {r} \times \left(m{\boldsymbol {\omega }}\times \mathbf {r} \right)\\&=m\left(\left(\mathbf {r} \cdot \mathbf {r} \right){\boldsymbol {\omega }}-\left(\mathbf {r} \cdot {\boldsymbol {\omega }}\right)\mathbf {r} \right)\\&=mr^{2}{\boldsymbol {\omega }}=I\omega \mathbf {\hat {k}} ,\end{aligned}}}
31138:
2027:{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\mathbf {r} \times \mathbf {F} =\mathbf {r} \times (m{\boldsymbol {\alpha }}\times \mathbf {r} )\\&=m\left(\left(\mathbf {r} \cdot \mathbf {r} \right){\boldsymbol {\alpha }}-\left(\mathbf {r} \cdot {\boldsymbol {\alpha }}\right)\mathbf {r} \right)\\&=mr^{2}{\boldsymbol {\alpha }}=I\alpha \mathbf {\hat {k}} ,\end{aligned}}}
14971:
36035:
945:
4399:
11560:
6882:
1259:
15671:{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}(\mathbf {r_{i}} -\mathbf {R} )\times (m_{i}\mathbf {a} _{i})\\&=\sum _{i=1}^{n}\Delta \mathbf {r} _{i}\times (m_{i}\mathbf {a} _{i})\\&=\sum _{i=1}^{n}m_{i}\;\ldots {\text{ cross-product scalar multiplication}}\\&=\sum _{i=1}^{n}m_{i}\\&=\sum _{i=1}^{n}m_{i}\\\end{aligned}}}
10212:{\displaystyle {\begin{aligned}\mathbf {A} _{i}&=\alpha \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}-\omega \mathbf {\hat {k}} \times \omega \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} \\&=\alpha \Delta r_{i}\mathbf {\hat {t}} _{i}-\omega ^{2}\Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {A} .\end{aligned}}}
28164:
26891:
31122:{\displaystyle {\begin{aligned}I_{xx}\ &{\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{N}m_{k}\left(y_{k}^{2}+z_{k}^{2}\right),\\I_{yy}\ &{\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+z_{k}^{2}\right),\\I_{zz}\ &{\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{N}m_{k}\left(x_{k}^{2}+y_{k}^{2}\right),\end{aligned}}}
37900:
14964:
27586:
9501:
6645:{\displaystyle {\begin{aligned}I_{C,{\text{sphere}}}&=\int _{-R}^{R}{\tfrac {1}{2}}\pi \rho r(z)^{4}\,dz=\int _{-R}^{R}{\tfrac {1}{2}}\pi \rho \left(R^{2}-z^{2}\right)^{2}\,dz\\&={\tfrac {1}{2}}\pi \rho \left_{-R}^{R}\\&=\pi \rho \left(1-{\tfrac {2}{3}}+{\tfrac {1}{5}}\right)R^{5}\\&={\tfrac {2}{5}}mR^{2},\end{aligned}}}
14477:
35720:{\displaystyle I=m\left(\mathbf {x} ^{\textsf {T}}\mathbf {x} -\mathbf {\hat {n}} ^{\textsf {T}}\mathbf {x} \mathbf {x} ^{\textsf {T}}\mathbf {\hat {n}} \right)=m\cdot \mathbf {\hat {n}} ^{\textsf {T}}\left(\mathbf {x} ^{\textsf {T}}\mathbf {x} \cdot \mathbf {E_{3}} -\mathbf {x} \mathbf {x} ^{\textsf {T}}\right)\mathbf {\hat {n}} ,}
16229:
26391:
12824:{\displaystyle E_{\text{K}}={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {C} }\right)\cdot \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {C} }\right),}
9752:
8251:
35748:
5722:
25431:
14127:
4157:
11361:
5349:{\displaystyle I_{C,{\text{rod}}}=\iiint _{Q}\rho \,x^{2}\,dV=\int _{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}\rho \,x^{2}s\,dx=\left.\rho s{\frac {x^{3}}{3}}\right|_{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}={\frac {\rho s}{3}}\left({\frac {\ell ^{3}}{8}}+{\frac {\ell ^{3}}{8}}\right)={\frac {m\ell ^{2}}{12}},}
1092:(inertia) plays in linear kinetics—both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, and will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by
37534:
31521:{\displaystyle {\begin{aligned}I_{xy}=I_{yx}\ &{\stackrel {\mathrm {def} }{=}}\ -\sum _{k=1}^{N}m_{k}x_{k}y_{k},\\I_{xz}=I_{zx}\ &{\stackrel {\mathrm {def} }{=}}\ -\sum _{k=1}^{N}m_{k}x_{k}z_{k},\\I_{yz}=I_{zy}\ &{\stackrel {\mathrm {def} }{=}}\ -\sum _{k=1}^{N}m_{k}y_{k}z_{k}.\end{aligned}}}
30408:
5976:
As one more example, consider the moment of inertia of a solid sphere of constant density about an axis through its center of mass. This is determined by summing the moments of inertia of the thin discs that can form the sphere whose centers are along the axis chosen for consideration. If the surface
4054:
is the perpendicular distance to the specified axis. To see how moment of inertia arises in the study of the movement of an extended body, it is convenient to consider a rigid assembly of point masses. (This equation can be used for axes that are not principal axes provided that it is understood that
1221:
is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal, horizontal and vertical axes determine how steering forces on the control surfaces of its wings, elevators and rudder(s) affect the plane's motions
1162:
is the mass. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that
27994:
10883:
The scalar moments of inertia appear as elements in a matrix when a system of particles is assembled into a rigid body that moves in three-dimensional space. This inertia matrix appears in the calculation of the angular momentum, kinetic energy and resultant torque of the rigid system of particles.
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For a given amount of angular momentum, a decrease in the moment of inertia results in an increase in the angular velocity. Figure skaters can change their moment of inertia by pulling in their arms. Thus, the angular velocity achieved by a skater with outstretched arms results in a greater angular
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for a rigid body as a physical parameter that combines its shape and mass. There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations
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There are some CAD and CAE applications such as SolidWorks, Unigraphics NX/Siemens NX and MSC Adams that use an alternate convention for the products of inertia. According to this convention, the minus sign is removed from the product of inertia formulas and instead inserted in the inertia matrix:
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Theoria motus corporum solidorum seu rigidorum: Ex primis nostrae cognitionis principiis stabilita et ad omnes motus, qui in huiusmodi corpora cadere possunt, accommodata [The theory of motion of solid or rigid bodies: established from first principles of our knowledge and appropriate for all
15128:{\displaystyle \Delta \mathbf {r} _{i}\times \left({\boldsymbol {\omega }}\times \left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\right)+{\boldsymbol {\omega }}\times \left(\left({\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}\right)\times \Delta \mathbf {r} _{i}\right)=0,}
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The natural frequency of oscillation of a compound pendulum is obtained from the ratio of the torque imposed by gravity on the mass of the pendulum to the resistance to acceleration defined by the moment of inertia. Comparison of this natural frequency to that of a simple pendulum consisting of a
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The use of the inertia matrix in Newton's second law assumes its components are computed relative to axes parallel to the inertial frame and not relative to a body-fixed reference frame. This means that as the body moves the components of the inertia matrix change with time. In contrast, the
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7048:{\displaystyle {\begin{aligned}\Delta \mathbf {r} _{i}&=\mathbf {r} _{i}-\mathbf {R} ,\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {V} ={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} ,\end{aligned}}}
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Mathematically, the moment of inertia of a simple pendulum is the ratio of the torque due to gravity about the pivot of a pendulum to its angular acceleration about that pivot point. For a simple pendulum this is found to be the product of the mass of the particle
27111:{\displaystyle I_{L}=\mathbf {\hat {k}} \cdot \left(-\sum _{i=1}^{N}m_{i}\left^{2}\right)\mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} =\mathbf {\hat {k}} ^{\mathsf {T}}\mathbf {I} _{\mathbf {R} }\mathbf {\hat {k}} ,}
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27758:{\displaystyle \Delta \mathbf {r} _{i}^{\perp }=\Delta \mathbf {r} _{i}-\left(\mathbf {\hat {k}} \cdot \Delta \mathbf {r} _{i}\right)\mathbf {\hat {k}} =\left(\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}\right)\Delta \mathbf {r} _{i},}
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14616:{\displaystyle \mathbf {a} _{i}={\boldsymbol {\alpha }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+{\boldsymbol {\omega }}\times \left({\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)\right)+\mathbf {A} _{\mathbf {R} }.}
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constructed from a thin disc mounted at the end of a thin rod that oscillates around a pivot at the other end of the rod, begins with the calculation of the moment of inertia of the thin rod and thin disc about their respective centers of mass.
36030:{\displaystyle I=m{\begin{bmatrix}n_{1}&n_{2}&n_{3}\end{bmatrix}}{\begin{bmatrix}y^{2}+z^{2}&-xy&-xz\\-yx&x^{2}+z^{2}&-yz\\-zx&-zy&x^{2}+y^{2}\end{bmatrix}}{\begin{bmatrix}n_{1}\\n_{2}\\n_{3}\end{bmatrix}}.}
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The moment of inertia of a continuous body rotating about a specified axis is calculated in the same way, except with infinitely many point particles. Thus the limits of summation are removed, and the sum is written as follows:
4394:{\displaystyle E_{\text{K}}=\sum _{i=1}^{N}{\frac {1}{2}}\,m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}=\sum _{i=1}^{N}{\frac {1}{2}}\,m_{i}\left(\omega r_{i}\right)^{2}={\frac {1}{2}}\,\omega ^{2}\sum _{i=1}^{N}m_{i}r_{i}^{2}.}
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11555:{\displaystyle {\begin{aligned}\mathbf {b} \times \mathbf {y} &\equiv \left\mathbf {y} \\\left&\equiv {\begin{bmatrix}0&-b_{z}&b_{y}\\b_{z}&0&-b_{x}\\-b_{y}&b_{x}&0\end{bmatrix}}.\end{aligned}}}
3490:. A trifilar pendulum is a platform supported by three wires designed to oscillate in torsion around its vertical centroidal axis. The period of oscillation of the trifilar pendulum yields the moment of inertia of the system.
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Thus, moment of inertia is a physical property that combines the mass and distribution of the particles around the rotation axis. Notice that rotation about different axes of the same body yield different moments of inertia.
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38144:(Definition 7. 422. A body's moment of inertia with respect to any axis is the sum of all of the products, which arise, if the individual elements of the body are multiplied by the square of their distances from the axis.)
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is a convenient way to summarize all moments of inertia of an object with one quantity. It may be calculated with respect to any point in space, although for practical purposes the center of mass is most commonly used.
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is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot. Its moment of inertia is the sum of the moments of inertia of each of the particles that it is composed of. The
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is the distance from the pivot point to the center of mass of the object. Measuring this frequency of oscillation over small angular displacements provides an effective way of measuring moment of inertia of a body.
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In that situation this moment of inertia only describes how a torque applied along that axis causes a rotation about that axis. But, torques not aligned along a principal axis will also cause rotations about other
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28159:{\displaystyle -\left^{2}\equiv \left|\mathbf {\hat {k}} \right|^{2}\left(\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}\right)=\mathbf {E} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}},}
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to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second
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which usually correspond approximately to rotations about the three principal axes. If the vehicle has bilateral symmetry then one of the principal axes will correspond exactly to the transverse (pitch) axis.
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Video of rotating chair experiment, illustrating moment of inertia. When the spinning professor pulls his arms, his moment of inertia decreases; to conserve angular momentum, his angular velocity increases.
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For the same object, different axes of rotation will have different moments of inertia about those axes. In general, the moments of inertia are not equal unless the object is symmetric about all axes. The
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38142:"Definitio 7. 422. Momentum inertiae corporis respectu eujuspiam axis est summa omnium productorum, quae oriuntur, si singula corporis elementa per quadrata distantiarum suarum ab axe multiplicentur."
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apparatus, from 1897 scientific journal. The portable gravimeter developed in 1890 by Thomas C. Mendenhall provided the most accurate relative measurements of the local gravitational field of the Earth.
37895:{\displaystyle \mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {x} =\|\mathbf {x} \|^{2}\mathbf {n} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {n} =\|\mathbf {x} \|^{2}I_{\mathbf {n} }=1.}
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14959:{\displaystyle {\boldsymbol {\tau }}=\left(-\sum _{i=1}^{n}m_{i}\left^{2}\right){\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(-\sum _{i=1}^{n}m_{i}\left^{2}\right){\boldsymbol {\omega }}}
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9496:{\displaystyle {\begin{aligned}\mathbf {F} &=\sum _{i=1}^{n}m_{i}\mathbf {A} _{i},\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}\Delta \mathbf {r} _{i}\times m_{i}\mathbf {A} _{i},\end{aligned}}}
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37081:, the rigid body is a symmetric top. If a rigid body has at least two symmetry axes that are not parallel or perpendicular to each other, it is a spherical top, for example, a cube or any other
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does in linear motion. A body's moment of inertia about a particular axis depends both on the mass and its distribution relative to the axis, increasing with mass & distance from the axis.
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16224:{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}\end{aligned}}}
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26386:{\displaystyle \mathbf {I} _{\mathbf {R} }=-\left(\sum _{i=1}^{n}m_{i}^{2}\right)+\left(\sum _{i=1}^{n}m_{i}\right)+\left(\sum _{i=1}^{n}m_{i}\right)-\left(\sum _{i=1}^{n}m_{i}\right)^{2}.}
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9747:{\displaystyle \mathbf {A} _{i}={\boldsymbol {\alpha }}\times \Delta \mathbf {r} _{i}+{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {A} .}
8246:{\displaystyle {\begin{aligned}\Delta r_{i}\mathbf {\hat {e}} _{i}&=\mathbf {r} _{i}-\mathbf {C} ,\\\sum _{i=1}^{n}m_{i}\,\Delta r_{i}\mathbf {\hat {e}} _{i}&=0,\end{aligned}}}
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Next, one calculates the eigenvectors for the two matrices. The matrix whose eigenvectors are parallel to the principal axes corresponds to the inertia convention that has been used.
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This 1906 rotary shear uses the moment of inertia of two flywheels to store kinetic energy which when released is used to cut metal stock (International
Library of Technology, 1906).
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The cylinders with higher moment of inertia roll down a slope with a smaller acceleration, as more of their potential energy needs to be converted into the rotational kinetic energy.
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6754:. The definition of the polar moment of inertia can be obtained by considering momentum, kinetic energy and Newton's laws for the planar movement of a rigid system of particles.
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These quantities can be generalized to an object with distributed mass, described by a mass density function, in a similar fashion to the scalar moment of inertia. One then has
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38793:"A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares"
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The moment of inertia of a complex system such as a vehicle or airplane around its vertical axis can be measured by suspending the system from three points to form a trifilar
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perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors
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5717:{\displaystyle I_{C,{\text{disc}}}=\iiint _{Q}\rho \,r^{2}\,dV=\int _{0}^{2\pi }\int _{0}^{R}\rho r^{2}sr\,dr\,d\theta =2\pi \rho s{\frac {R^{4}}{4}}={\frac {1}{2}}mR^{2},}
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3502:. These calculations are commonly used in civil engineering for structural design of beams and columns. Cross-sectional areas calculated for vertical moment of the x-axis
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25426:{\displaystyle {\boldsymbol {\tau }}=\mathbf {I} _{\mathbf {C} }{\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }},}
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14122:{\displaystyle E_{\text{K}}={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }}+{\frac {1}{2}}M\mathbf {V} _{\mathbf {C} }^{2}.}
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is the length of the pendulum. Notice that the parallel axis theorem is used to shift the moment of inertia from the center of mass to the pivot point of the pendulum.
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is either at rest or moving at a constant velocity but not accelerated, or the origin of the fixed (world) coordinate reference system is placed at the center of mass
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Notice that the distance to the center of oscillation of the seconds pendulum must be adjusted to accommodate different values for the local acceleration of gravity.
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The inertia matrix of a body depends on the choice of the reference point. There is a useful relationship between the inertia matrix relative to the center of mass
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37529:{\displaystyle \left({\frac {x}{1/{\sqrt {I_{1}}}}}\right)^{2}+\left({\frac {y}{1/{\sqrt {I_{2}}}}}\right)^{2}+\left({\frac {z}{1/{\sqrt {I_{3}}}}}\right)^{2}=1,}
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so that it swings freely in a plane perpendicular to the direction of the desired moment of inertia, then measure its natural frequency or period of oscillation (
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30403:{\displaystyle I_{ij}\ {\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{N}m_{k}\left(\left\|\mathbf {r} _{k}\right\|^{2}\delta _{ij}-x_{i}^{(k)}x_{j}^{(k)}\right)}
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4843:. The moment of inertia of a flat surface is similar with the mass density being replaced by its areal mass density with the integral evaluated over its area.
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of a beam's cross-section are often confused. The moment of inertia of a body with the shape of the cross-section is the second moment of this area about the
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If the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning
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is how mass combines with the shape of a body to define rotational inertia. The moment of inertia of an arbitrarily shaped body is the sum of the values
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3461:{\displaystyle L={\frac {g}{\omega _{\text{n}}^{2}}}\approx {\frac {9.81\ \mathrm {m/s^{2}} }{(3.14\ \mathrm {rad/s} )^{2}}}\approx 0.99\ \mathrm {m} .}
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The inertia tensor can be used in the same way as the inertia matrix to compute the scalar moment of inertia about an arbitrary axis in the direction
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37022:(although it need not be spherical) and any axis can be considered a principal axis, meaning that the moment of inertia is the same about any axis.
30215:{\displaystyle \mathbf {I} ={\begin{bmatrix}I_{11}&I_{12}&I_{13}\\I_{21}&I_{22}&I_{23}\\I_{31}&I_{32}&I_{33}\end{bmatrix}}.}
31785:{\displaystyle \mathbf {I} =\iiint _{V}\rho (x,y,z)\left(\|\mathbf {r} \|^{2}\mathbf {E} _{3}-\mathbf {r} \otimes \mathbf {r} \right)\,dx\,dy\,dz,}
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that appears in planar movement. However, to make this to work out correctly a minus sign is needed. This minus sign can be absorbed into the term
12281:{\displaystyle \mathbf {L} =\left(-\sum _{i=1}^{n}m_{i}\left^{2}\right){\boldsymbol {\omega }}=\mathbf {I} _{\mathbf {C} }{\boldsymbol {\omega }},}
2608:{\displaystyle E_{\text{K}}={\frac {1}{2}}m\mathbf {v} \cdot \mathbf {v} ={\frac {1}{2}}\left(mr^{2}\right)\omega ^{2}={\frac {1}{2}}I\omega ^{2}.}
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and there is no unique choice for the two corresponding principal axes. If all three principal moments are the same, the rigid body is called a
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formulas for standard body shapes provides a way to obtain the moment of inertia of a complex body as an assembly of simpler shaped bodies. The
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The inertia matrix appears in the application of Newton's second law to a rigid assembly of particles. The resultant torque on this system is,
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The moment of inertia of the compound pendulum is now obtained by adding the moment of inertia of the rod and the disc around the pivot point
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This shows that the inertia matrix can be used to calculate the moment of inertia of a body around any specified rotation axis in the body.
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26646:: By using the skew symmetric matrix of position vectors relative to the reference point, the inertia matrix of each particle has the form
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This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses
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be the displacement vector of the body. The inertia tensor of the translated body respect to its original center of mass is given by:
6828:, are assembled into a rigid body, then the momentum of the system can be written in terms of positions relative to a reference point
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5927:{\displaystyle I_{P}=I_{C,{\text{rod}}}+M_{\text{rod}}\left({\frac {L}{2}}\right)^{2}+I_{C,{\text{disc}}}+M_{\text{disc}}(L+R)^{2},}
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To improve their maneuverability, combat aircraft are designed to minimize moments of inertia, while civil aircraft often are not.
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Gracey, William, The experimental determination of the moments of inertia of airplanes by a simplified compound-pendulum method,
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For multiple particles, we need only recall that the moment of inertia is additive in order to see that this formula is correct.
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1746:, of the string and mass around this axis. Since the mass is constrained to a circle the tangential acceleration of the mass is
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is used in the names of types of rigid bodies. When all principal moments of inertia are distinct, the principal axes through
36863:{\displaystyle {\boldsymbol {\Lambda }}={\begin{bmatrix}I_{1}&0&0\\0&I_{2}&0\\0&0&I_{3}\end{bmatrix}}.}
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velocity when the arms are pulled in, because of the reduced moment of inertia. A figure skater is not, however, a rigid body.
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For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along
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14283:{\displaystyle {\boldsymbol {\tau }}=\sum _{i=1}^{n}\left(\mathbf {r_{i}} -\mathbf {R} \right)\times m_{i}\mathbf {a} _{i},}
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can be defined, dependent on a particular axis of rotation, with such a value that its moment of inertia around the axis is
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This defines the relative position vector and the velocity vector for the rigid system of the particles moving in a plane.
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Spinning figure skaters can reduce their moment of inertia by pulling in their arms, allowing them to spin faster due to
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The components of tensors of degree two can be assembled into a matrix. For the inertia tensor this matrix is given by,
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Similarly, the kinetic energy of the pendulum mass is defined by the velocity of the pendulum around the pivot to yield
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The principal axes are often aligned with the object's symmetry axes. If a rigid body has an axis of symmetry of order
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For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a
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is a unit vector. Then the relationship presented above, between the inertia matrix and the scalar moment of inertia
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The moment of inertia matrix in body-frame coordinates is a quadratic form that defines a surface in the body called
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32033:{\displaystyle \mathbf {I} =\iiint _{V}\rho (\mathbf {r} )^{\textsf {T}}\,dV=-\iiint _{Q}\rho (\mathbf {r} )^{2}\,dV}
9181:{\displaystyle E_{\text{K}}={\frac {1}{2}}I_{\mathbf {C} }\omega ^{2}+{\frac {1}{2}}M\mathbf {V} \cdot \mathbf {V} .}
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Four objects with identical masses and radii racing down a plane while rolling without slipping. From back to front:
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Measured in the body frame, the inertia matrix is a constant real symmetric matrix. A real symmetric matrix has the
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37632:{\displaystyle a={\frac {1}{\sqrt {I_{1}}}},\quad b={\frac {1}{\sqrt {I_{2}}}},\quad c={\frac {1}{\sqrt {I_{3}}}}.}
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36569:{\displaystyle \mathbf {I} _{\mathbf {C} }=\mathbf {A} \mathbf {I} _{\mathbf {C} }^{B}\mathbf {A} ^{\mathsf {T}}.}
11169:{\displaystyle \mathbf {v} _{i}={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} _{\mathbf {R} }}
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is constrained to move parallel to a fixed plane, then the rotation of a body in the system occurs around an axis
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where the dot and the cross products have been interchanged. Exchanging products, and simplifying by noting that
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parallel to this plane. In this case, the moment of inertia of the mass in this system is a scalar known as the
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36751:{\displaystyle \mathbf {I} _{\mathbf {C} }^{B}=\mathbf {Q} {\boldsymbol {\Lambda }}\mathbf {Q} ^{\mathsf {T}},}
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35086:{\displaystyle \left|\mathbf {x} -\left(\mathbf {x} \cdot \mathbf {\hat {n}} \right)\mathbf {\hat {n}} \right|}
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To relate this scalar moment of inertia to the inertia matrix of the body, introduce the skew-symmetric matrix
26722:
8446:
about an axis perpendicular to the movement of the rigid system and through the center of mass is known as the
1600:
1493:
1416:
1234:
is defined as the product of mass of section and the square of the distance between the reference axis and the
486:
226:
37201:
36673:
40234:
40166:
38961:
36984:
28817:
27525:
14455:
14433:
11179:
9788:
9635:
9613:
7058:
4765:
is a vector perpendicular to the axis of rotation and extending from a point on the rotation axis to a point
3474:
is a compound pendulum that uses this property to measure the local acceleration of gravity, and is called a
2237:
2072:
1727:
1082:
3272:
101:
40259:
6201:
Therefore, the moment of inertia of the sphere is the sum of the moments of inertia of the discs along the
1344:
888:
875:
636:
559:
481:
324:
26446:. And the last term is the total mass of the system multiplied by the square of the skew-symmetric matrix
40497:
35096:
34990:
29489:
28849:
28169:
27840:
27557:
27390:
26840:
12420:
is the symmetric inertia matrix of the rigid system of particles measured relative to the center of mass
9759:
6724:
5482:
2039:
555:
356:
38603:
27288:
26424:
relative to the center of mass. The second and third terms are zero by definition of the center of mass
12477:
10921:
9266:
6780:
2718:
1191:
334:
40308:
40128:
38913:
An introductory lesson on moment of inertia: keeping a vertical pole not falling down (Java simulation)
38016:
37220:
be the inertia matrix relative to the center of mass aligned with the principal axes, then the surface
36501:
in the inertial frame. Then, the inertia matrix of the body measured in the inertial frame is given by
5959:
5727:
4970:
1638:
This can be shown as follows: The force of gravity on the mass of a simple pendulum generates a torque
1546:
775:
664:
590:
450:
383:
135:
37729:
36056:
34699:
without knowing which inertia convention that has been used, it can be determined if one also has the
29941:
27339:
27121:
26398:
25526:
25436:
14293:
14132:
12557:
12528:
12291:
12006:
11201:
11001:
10972:
10795:
9846:
9506:
9193:
9068:
8420:
8258:
7200:
7149:
6853:
5966:
is used to shift the reference point of the individual bodies to the reference point of the assembly.
2893:
Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot point
40627:
39980:
39590:
39395:
39249:
36397:, and define the orientation of the body frame relative to the inertial frame by the rotation matrix
36286:
32124:
is taken with the corresponding elements in the component tensors. A product of inertia term such as
2066:
1170:
introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a
922:
use the moment of inertia of a long rod for balance as they walk the rope. Samuel Dixon crossing the
529:
38759:
L. W. Tsai, Robot
Analysis: The mechanics of serial and parallel manipulators, John-Wiley, NY, 1999.
1198:
single point of mass provides a mathematical formulation for moment of inertia of an extended body.
40564:
40482:
40436:
40143:
39362:
38031:
12035:
5359:
1214:
yield a 3 × 3 matrix of moments of inertia, called the inertia matrix or inertia tensor.
1026:
value, matters. For bodies free to rotate in three dimensions, their moments can be described by a
805:
649:
38125:
38095:
37929:
37907:
37758:
37707:
37644:
36875:
36651:
36581:
36482:
36460:
36400:
36266:
36089:
34968:
32045:
30698:
30660:
29970:
29467:
27768:
27419:
27387:
as a reference point and compute the moment of inertia around a line L defined by the unit vector
27368:
27150:
26869:
26621:
26599:
26577:
26477:
26427:
25824:
25802:
25780:
25687:
25555:
25501:
25479:
21278:
15740:
15718:
14628:
14380:
12423:
11589:
11567:
11230:
11030:
10773:
9824:
9591:
9569:
9321:
9046:
8066:
7178:
7127:
7102:
7080:
6831:
4746:
2094:
1701:
1679:
935:
40534:
40221:
40138:
40108:
39373:
38917:
38837:
37115:
34703:. With the principal axes method, one makes inertia matrices from the following two assumptions:
26649:
14650:
755:
519:
39533:
39433:
38429:
26046:{\displaystyle \mathbf {I} _{\mathbf {R} }=-\sum _{i=1}^{n}m_{i}^{2}=-\sum _{i=1}^{n}m_{i}^{2}.}
25770:{\displaystyle \mathbf {R} =(\mathbf {R} -\mathbf {C} )+\mathbf {C} =\mathbf {d} +\mathbf {C} ,}
11564:
The inertia matrix is constructed by considering the angular momentum, with the reference point
9239:
on the engine. The large moment of inertia of the flywheel smooths the operation of the tractor.
3067:
A simple pendulum that has the same natural frequency as a compound pendulum defines the length
1479:. An arbitrary object's moment of inertia thus depends on the spatial distribution of its mass.
795:
40492:
40348:
40303:
38459:
Physics for
Scientists and Engineers, Vol. 1: Mechanics, Oscillations and Waves, Thermodynamics
37195:
36980:
7656:
The angular momentum vector for the planar movement of a rigid system of particles is given by
5055:
about a perpendicular axis through its center of mass is determined by integration. Align the
4683:
2065:
is a unit vector perpendicular to the plane of the pendulum. (The second to last step uses the
1042:
1007:
800:
617:
38922:
38897:
36234:
2620:
2159:
2116:
1290:
40617:
40574:
40529:
40009:
39954:
39508:
39378:
39116:
25471:
10888:
5963:
4850:
4768:
4708:
3499:
3088:
1363:
810:
785:
471:
289:
46:
31:
38680:
37058:
33252:
33222:
32127:
31583:
31533:
26692:
4406:
4007:
3572:) are the linear measures, except for circles, which are effectively half-breadth derived,
3535:
3505:
2656:
1095:
40549:
40477:
40363:
40229:
40191:
40123:
39830:
39755:
39383:
39155:
38918:
Tutorial on finding moments of inertia, with problems and solutions on various basic shapes
38061:
37097:
36951:
36924:
36897:
30631:
30040:
27892:
27498:
26813:
14353:
14322:
11264:
10740:
9542:
5438:
5075:-axis with the rod and locate the origin its center of mass at the center of the rod, then
5038:
5018:
4873:-axis perpendicular to the cross-section, weighted by its density. This is also called the
4688:
4108:
4081:
3092:
2745:
1721:
1355:
1329:
1066:
988:
830:
790:
698:
694:
686:
676:
466:
459:
215:
38616:
3996:
The time for each object to reach the finishing line depends on their moment of inertia. (
3032:{\displaystyle I_{P}={\frac {mgr}{\omega _{\text{n}}^{2}}}={\frac {mgrt^{2}}{4\pi ^{2}}},}
825:
8:
40426:
40249:
40239:
40088:
40073:
40029:
39561:
39290:
39238:
38958:
10730:{\displaystyle \mathbf {\hat {e}} _{i}\times \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} }
9938:{\displaystyle \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i}}
7292:{\displaystyle \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i}}
5456:
3471:
1337:
1241:
1210:
605:
546:
524:
269:
264:
259:
159:
38118:
38089:
5455:
about an axis through its center and perpendicular to its face (parallel to its axis of
40559:
40416:
40269:
40083:
40019:
38422:
38341:
38167:
37173:
37153:
37133:
37028:
36289:
that represents a body's rotation. The inertia tensor of the rotated body is given by:
36207:
34948:
33202:
33182:
33162:
32242:
32222:
31633:
31613:
31563:
30498:
30478:
30458:
30438:
30418:
30020:
29447:
27869:
27268:
14161:
12457:
10901:
10792:
as the reference point and define the moment of inertia relative to the center of mass
9246:
6760:
6204:
6091:
6071:
6051:
5937:
5772:
5462:
5418:
5398:
5058:
4998:
4944:
4924:
4900:
4880:
4856:
4826:
4806:
4135:
4061:
4037:
3575:
3314:
3183:{\displaystyle \omega _{\text{n}}={\sqrt {\frac {g}{L}}}={\sqrt {\frac {mgr}{I_{P}}}},}
3098:
3070:
3042:
2916:
2896:
2872:
2852:
2832:
1580:
1560:
1534:
1167:
1145:
1125:
735:
476:
351:
319:
279:
37104:
A practical example of this mathematical phenomenon is the routine automotive task of
27172:
26777:
26449:
13295:
12067:
40622:
40554:
40323:
40298:
40113:
40024:
40004:
39580:
39334:
39093:
38902:
38868:
38841:
38727:
38686:
38634:
38583:
38547:
38522:
38497:
38463:
38433:
38400:
38320:
38290:
38230:
38202:
38174:
38129:
38065:
38026:
6718:
4921:
are calculated using the second moment of the cross-sectional area around either the
4587:
3963:
2712:
2704:
1171:
745:
702:
659:
654:
595:
361:
254:
33159:
It is common in rigid body mechanics to use notation that explicitly identifies the
12450:
The kinetic energy of a rigid system of particles can be formulated in terms of the
40569:
40244:
40211:
40196:
40078:
39947:
39667:
39637:
39603:
39322:
39278:
39233:
38807:
38770:
38057:
37089:
36338:{\displaystyle \mathbf {I} =\mathbf {R} \mathbf {I_{0}} \mathbf {R} ^{\textsf {T}}}
30718:
26837:, of a body about a specified axis whose direction is specified by the unit vector
9065:
of the system so the second term becomes zero, and introduce the moment of inertia
8451:
4918:
3266:
2257:
2193:
1282:
1275:
1202:
1070:
1062:
1027:
1023:
1016:
1012:
919:
840:
820:
765:
760:
706:
681:
536:
394:
339:
314:
38792:
10657:{\displaystyle \mathbf {\hat {e}} _{i}\times \mathbf {\hat {e}} _{i}=\mathbf {0} }
8474:
The kinetic energy of a rigid system of particles moving in the plane is given by
7124:
For planar movement the angular velocity vector is directed along the unit vector
3943:{\displaystyle I_{xx}=I_{yy}={\frac {1}{4}}{\pi }r^{4}={\frac {1}{64}}{\pi }d^{4}}
3059:
is the period (duration) of oscillation (usually averaged over multiple periods).
2691:
54:
have large moments of inertia to smooth out changes in rates of rotational motion.
40597:
40539:
40487:
40431:
40411:
40313:
40201:
40068:
40039:
39912:
39872:
38862:
37263:{\displaystyle \mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Lambda }}\mathbf {x} =1,}
37047:
35737:
31836:
30688:
30621:{\displaystyle \mathbf {r} _{k}=\left(x_{1}^{(k)},x_{2}^{(k)},x_{3}^{(k)}\right)}
17190:
16234:
15136:
10869:{\displaystyle {\boldsymbol {\tau }}=I_{\mathbf {C} }\alpha \mathbf {\hat {k}} .}
9315:
5969:
1398:
1035:
835:
780:
730:
725:
644:
27:
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
39675:
39611:
36987:. The principal axis with the highest moment of inertia is sometimes called the
32212:{\displaystyle I_{12}=\mathbf {e} _{1}\cdot \mathbf {I} \cdot \mathbf {e} _{2},}
25552:
obtained for a rigid system of particles measured relative to a reference point
13292:, the second term in this equation is zero. Introduce the skew-symmetric matrix
1698:
is the distance vector from the torque axis to the pendulum center of mass, and
1343:
If the shape of the body does not change, then its moment of inertia appears in
40579:
40544:
40441:
40274:
40264:
40254:
40176:
40148:
40133:
40118:
40034:
39856:
39820:
39745:
39526:
39413:
39303:
39204:
38927:
38011:
37082:
37007:
36083:
12451:
12058:
10768:
8061:
4975:
1333:
1206:
1183:
1078:
1003:
862:
770:
671:
388:
40524:
38811:
36353:
components of the inertia matrix measured in a body-fixed frame are constant.
29967:
is the moment of inertia matrix of the system relative to the reference point
27787:
is the identity matrix, so as to avoid confusion with the inertia matrix, and
27147:
is the moment of inertia matrix of the system relative to the reference point
26567:{\displaystyle \mathbf {I} _{\mathbf {R} }=\mathbf {I} _{\mathbf {C} }-M^{2},}
11261:
by combining the first operand and the operator into a skew-symmetric matrix,
4673:{\displaystyle I_{P}=\iiint _{Q}\rho (x,y,z)\left\|\mathbf {r} \right\|^{2}dV}
40606:
40516:
40421:
40333:
40206:
39730:
39598:
39312:
38021:
36252:
31825:
15140:
11027:
relative to a fixed reference frame. For a (possibly moving) reference point
1046:
923:
750:
577:
38747:
36361:
Let the body frame inertia matrix relative to the center of mass be denoted
28525:
The simplification of this equation uses the triple scalar product identity
7146:
which is perpendicular to the plane of movement. Introduce the unit vectors
4403:
This shows that the moment of inertia of the body is the sum of each of the
40584:
40388:
40373:
40338:
40186:
40171:
39805:
39791:
39691:
39627:
39352:
39135:
39046:
38912:
37105:
36999:
36894:
define the directions of the principal axes of the body, and the constants
25675:{\displaystyle \mathbf {I} _{\mathbf {R} }=-\sum _{i=1}^{n}m_{i}\left^{2}.}
12413:{\displaystyle \mathbf {I} _{\mathbf {C} }=-\sum _{i=1}^{n}m_{i}\left^{2},}
3311:
for the pendulum. In this case, the distance to the center of oscillation,
815:
740:
429:
309:
10878:
3256:{\displaystyle L={\frac {g}{\omega _{\text{n}}^{2}}}={\frac {I_{P}}{mr}}.}
1676:
around the axis perpendicular to the plane of the pendulum movement. Here
40472:
40446:
40368:
40057:
39996:
39173:
38907:
38681:
Ferdinand P. Beer; E. Russell
Johnston, Jr.; Phillip J. Cornwell (2010).
32121:
12454:
and a matrix of mass moments of inertia of the system. Let the system of
37536:
to see that the semi-principal diameters of this ellipsoid are given by
37114:
as asymmetric, symmetric, or spherical tops, and the structure of their
35975:
35816:
33808:
33675:
32411:
32282:
27984:{\displaystyle \left\mathbf {y} =\mathbf {\hat {k}} \times \mathbf {y} }
25356:
Thus, the resultant torque on the rigid system of particles is given by
14350:
of a rigid body yields the formula for the acceleration of the particle
9539:
of a rigid body yields the formula for the acceleration of the particle
8408:{\displaystyle \mathbf {L} =I_{\mathbf {C} }\omega \mathbf {\hat {k}} .}
40353:
37996:{\displaystyle \|\mathbf {x} \|={\frac {1}{\sqrt {I_{\mathbf {n} }}}}.}
34578:
32111:{\displaystyle I_{n}=\mathbf {n} \cdot \mathbf {I} \cdot \mathbf {n} ,}
18174:
The final result can then be substituted to the main proof as follows:
14347:
10737:
is the unit vector perpendicular to the plane for all of the particles
9536:
8466:
4004:
The moment of inertia about an axis of a body is calculated by summing
3475:
2696:
2227:{\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} }
1777:{\displaystyle \mathbf {a} ={\boldsymbol {\alpha }}\times \mathbf {r} }
980:
915:
600:
274:
38368:
37663:
on this ellipsoid be defined in terms of its magnitude and direction,
33982:
32567:
14024:
Thus, the kinetic energy of the rigid system of particles is given by
983:
is defined relative to a rotational axis. It is the ratio between the
40328:
40279:
39266:
39193:
39181:
37363:
37126:
35501:
27255:{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {R} }
12886:
is the position vector of a particle relative to the center of mass.
12879:{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {C} }
12150:{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {C} }
11097:{\displaystyle \Delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {R} }
2709:
2260:
of the mass about the pivot point. This angular momentum is given by
1669:{\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} }
1454:
Thus, the moment of inertia of the pendulum depends on both the mass
622:
38898:
Angular momentum and rigid-body rotation in two and three dimensions
37014:. If two principal moments are the same, the rigid body is called a
27830:{\displaystyle \mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {T}}}
40358:
40343:
39570:
39469:
39223:
39082:
39036:
24963:
Finally, the result is used to complete the main proof as follows:
9236:
6709:
3487:
1235:
1218:
541:
424:
399:
51:
38124:(in Latin). Rostock and Greifswald (Germany): A. F. Röse. p.
11259:
cross product can be equivalently written as matrix multiplication
8351:{\displaystyle I_{\mathbf {C} }=\sum _{i}m_{i}\,\Delta r_{i}^{2},}
40052:
40014:
39896:
37093:
27488:{\displaystyle \mathbf {L} (t)=\mathbf {R} +t\mathbf {\hat {k}} }
9610:
of the reference particle as well as the angular velocity vector
1458:
of a body and its geometry, or shape, as defined by the distance
1088:
The moment of inertia plays the role in rotational kinetics that
1006:
the moment of inertia is simply the mass times the square of the
514:
367:
284:
72:
30:
For the quantity also known as the "area moment of inertia", see
38578:
Uicker, John J.; Pennock, Gordon R.; Shigley, Joseph E. (2010).
14647:
as the reference point, and introduce the skew-symmetric matrix
8454:
with respect to the orthogonal distance from an axis (or pole).
8255:
and define the moment of inertia relative to the center of mass
2822:{\displaystyle \omega _{\text{n}}={\sqrt {\frac {mgr}{I_{P}}}},}
1720:
is the net force on the mass. Associated with this torque is an
991:
about that axis. It plays the same role in rotational motion as
40378:
39970:
39810:
39779:
39735:
39713:
39167:
39066:
39031:
36347:
30065:
30004:
14430:
of the reference point, as well as the angular velocity vector
5479:-axis with the axis of the disc and define a volume element as
4803:
in the solid, and the integration is evaluated over the volume
1348:
1163:
depends on the dimensions, shape and total mass of the object.
1058:
984:
573:
419:
329:
38923:
Notes on mechanics of manipulation: the angular inertia tensor
34926:{\displaystyle (I_{12}=-I_{xy},I_{13}=-I_{xz},I_{23}=-I_{yz})}
39844:
39769:
39703:
39307:
39213:
39208:
39145:
39120:
13285:{\displaystyle \sum _{i=1}^{n}m_{i}\Delta \mathbf {r} _{i}=0}
9231:
3091:
of the compound pendulum. This point also corresponds to the
1401:, this definition yields a formula for the moment of inertia
1179:
1038:
and torques around the axes act independently of each other.
409:
404:
346:
38286:
34987:
from the axis of rotation passing through the origin in the
31891:{\displaystyle \mathbf {x} =\mathbf {r} \times \mathbf {x} }
4849:: The moment of inertia of a body moving in a plane and the
2196:
of a simple pendulum, which is calculated from the velocity
1471:
each multiplied by the square of its perpendicular distance
39926:
39886:
39459:
39125:
39056:
39021:
38903:
Lecture notes on rigid-body rotation and moments of inertia
38707:
Analysis and Design of
Elastic Beams: Computational Methods
38633:(7 ed.). New Jersey: Prentice Hall. pp. 192–196.
34807:{\displaystyle (I_{12}=I_{xy},I_{13}=I_{xz},I_{23}=I_{yz})}
34692:{\displaystyle (I_{xx},I_{yy},I_{zz},I_{xy},I_{xz},I_{yz})}
32219:
and can be interpreted as the moment of inertia around the
10821:, then the equation for the resultant torque simplifies to
5196:
2742:) of a compound pendulum depends on its moment of inertia,
1089:
992:
414:
377:
39939:
37055:
about the given axis, that axis is a principal axis. When
27495:. The perpendicular vector from this line to the particle
15759:. And distributing the cross product over the sum, we get
37697:{\displaystyle \mathbf {x} =\|\mathbf {x} \|\mathbf {n} }
35743:
This leads to a tensor formula for the moment of inertia
25520:. This relationship is called the parallel axis theorem.
14158:
is the inertia matrix relative to the center of mass and
3594:
207:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
40598:
Easy to use and Free Moment of
Inertia Calculator online
38517:
Halliday, David; Resnick, Robert; Walker, Jearl (2005).
37130:
An ellipsoid with the semi-principal diameters labelled
27837:
is the outer product matrix formed from the unit vector
944:
37010:
are uniquely specified and the rigid body is called an
31843:
is a region of space completely containing the object.
30655:
from the point about which the tensor is calculated and
10879:
Motion in space of a rigid body, and the inertia matrix
1074:
36778:
36450:{\displaystyle \mathbf {x} =\mathbf {A} \mathbf {y} ,}
35766:
34940:
31580:-axis when the objects are rotated around the x-axis,
30088:
27207:
is the skew symmetric matrix obtained from the vector
26805:
25462:
is the inertia matrix relative to the center of mass.
24625:
24181:
23588:
22980:
22348:
22077:
21909:
21787:
21619:
21446:
14761:{\displaystyle (\mathbf {r} _{i}-\mathbf {C} )\times }
11436:
6658:
6611:
6574:
6559:
6491:
6456:
6416:
6341:
6276:
39536:
38516:
37954:
37932:
37910:
37783:
37761:
37732:
37710:
37669:
37647:
37542:
37372:
37276:
37226:
37204:
37176:
37156:
37136:
37061:
37031:
37002:
is an example of a rotating rigid body, and the word
36954:
36927:
36900:
36878:
36764:
36698:
36676:
36654:
36606:
36584:
36507:
36485:
36463:
36425:
36403:
36367:
36295:
36269:
36237:
36210:
36114:
36092:
36059:
35751:
35512:
35128:
35122:
is unit vector. The moment of inertia on the axis is
35099:
35022:
34993:
34971:
34951:
34823:
34713:
34589:
33294:
33255:
33225:
33205:
33185:
33165:
32268:
32245:
32225:
32157:
32130:
32070:
32048:
31904:
31855:
31846:
Alternatively it can also be written in terms of the
31800:
31659:
31636:
31616:
31586:
31566:
31536:
31141:
30730:
30724:
The diagonal elements are more succinctly written as
30701:
30663:
30634:
30523:
30501:
30481:
30461:
30441:
30421:
30230:
30074:
30043:
30023:
29973:
29944:
29521:
29492:
29470:
29450:
28881:
28852:
28820:
28531:
28204:
28198:
The magnitude squared of the perpendicular vector is
28172:
27997:
27932:
27895:
27872:
27843:
27793:
27771:
27589:
27560:
27528:
27501:
27444:
27422:
27393:
27371:
27342:
27291:
27271:
27213:
27175:
27153:
27124:
26894:
26872:
26843:
26816:
26780:
26774:, if desired, by using the skew-symmetry property of
26725:
26695:
26652:
26624:
26602:
26580:
26505:
26480:
26452:
26430:
26401:
26061:
25849:
25827:
25805:
25783:
25712:
25690:
25580:
25558:
25529:
25504:
25482:
25439:
25362:
24969:
21303:
21281:
19304:
18180:
17199:
16243:
15765:
15743:
15721:
15684:
15155:
14974:
14774:
14728:
14653:
14631:
14480:
14458:
14436:
14405:
14383:
14356:
14325:
14296:
14192:
14164:
14135:
14030:
13336:
13298:
13229:
12895:
12837:
12589:
12560:
12531:
12480:
12460:
12426:
12323:
12294:
12163:
12108:
12070:
12038:
12009:
11614:
11592:
11570:
11364:
11297:
11267:
11233:
11204:
11182:
11110:
11055:
11033:
11004:
10975:
10924:
10904:
10827:
10798:
10776:
10743:
10670:
10604:
10227:
9951:
9878:
9849:
9827:
9791:
9762:
9660:
9638:
9616:
9594:
9572:
9545:
9509:
9346:
9324:
9269:
9249:
9196:
9100:
9071:
9049:
8480:
8423:
8366:
8360:
then the equation for angular momentum simplifies to
8290:
8261:
8091:
8069:
7662:
7305:
7232:
7203:
7181:
7152:
7130:
7105:
7083:
7061:
6885:
6856:
6834:
6783:
6763:
6727:
6227:
6207:
6114:
6094:
6074:
6054:
5983:
5940:
5795:
5775:
5730:
5531:
5485:
5465:
5441:
5421:
5401:
5362:
5081:
5061:
5041:
5021:
5001:
4947:
4927:
4903:
4883:
4859:
4829:
4809:
4771:
4749:
4711:
4691:
4596:
4525:
4439:
4409:
4160:
4138:
4111:
4084:
4064:
4055:
this does not fully describe the moment of inertia.)
4040:
4010:
3854:
3797:
3740:
3685:
3617:
3578:
3538:
3508:
3337:
3317:
3275:
3196:
3121:
3101:
3073:
3045:
2939:
2919:
2899:
2875:
2855:
2835:
2775:
2748:
2721:
2659:
2623:
2490:
2481:
using a similar derivation to the previous equation.
2266:
2240:
2202:
2162:
2119:
2097:
2075:
2042:
1823:
1790:
1752:
1730:
1704:
1682:
1644:
1603:
1583:
1563:
1496:
1419:
1368:
1295:
1148:
1128:
1098:
171:
104:
38577:
37366:
in the body frame. Write this equation in the form,
36479:
in the body fixed coordinate frame have coordinates
34579:
Determine inertia convention (Principal axes method)
27265:
This is derived as follows. Let a rigid assembly of
4505:{\displaystyle I_{P}=\sum _{i=1}^{N}m_{i}r_{i}^{2}.}
1030:
3-by-3 matrix, with a set of mutually perpendicular
38771:"Physically Based Modeling - Rigid Body Simulation"
37355:{\displaystyle I_{1}x^{2}+I_{2}y^{2}+I_{3}z^{2}=1,}
25843:. Use this equation to compute the inertia matrix,
39551:
38421:
38166:
37995:
37940:
37918:
37894:
37769:
37747:
37718:
37696:
37655:
37631:
37528:
37354:
37262:
37212:
37182:
37162:
37142:
37073:
37037:
36967:
36940:
36913:
36886:
36862:
36750:
36684:
36662:
36628:
36592:
36568:
36493:
36471:
36449:
36411:
36389:
36337:
36277:
36243:
36216:
36196:
36100:
36082:be the inertia tensor of a body calculated at its
36074:
36049:Parallel axis theorem § Tensor generalization
36029:
35719:
35490:
35114:
35085:
35008:
34979:
34957:
34925:
34806:
34691:
34568:
33271:
33241:
33211:
33191:
33171:
33149:
32251:
32231:
32211:
32143:
32110:
32056:
32032:
31890:
31816:
31784:
31642:
31622:
31602:
31572:
31552:
31520:
31121:
30709:
30679:
30647:
30620:
30507:
30487:
30467:
30447:
30427:
30402:
30214:
30056:
30029:
29981:
29959:
29930:
29507:
29478:
29456:
29434:
28867:
28838:
28806:
28515:
28187:
28158:
27983:
27918:
27878:
27858:
27829:
27779:
27757:
27575:
27546:
27514:
27487:
27430:
27408:
27379:
27357:
27328:
27277:
27254:
27199:
27161:
27139:
27110:
26880:
26858:
26829:
26794:
26766:
26711:
26681:
26632:
26610:
26588:
26566:
26488:
26466:
26438:
26416:
26385:
26045:
25835:
25813:
25791:
25769:
25698:
25674:
25566:
25544:
25512:
25490:
25454:
25425:
25345:
24953:
21289:
21265:
19291:
18164:
17179:
16223:
15751:
15729:
15707:
15670:
15127:
14958:
14760:
14714:
14639:
14615:
14466:
14444:
14422:
14391:
14369:
14338:
14311:
14282:
14170:
14150:
14121:
14014:
13322:
13284:
13213:
12878:
12823:
12575:
12546:
12517:
12466:
12434:
12412:
12309:
12280:
12149:
12094:
12049:
12024:
11995:
11600:
11578:
11554:
11350:
11283:
11241:
11219:
11190:
11168:
11096:
11041:
11019:
10990:
10961:
10910:
10868:
10813:
10784:
10756:
10729:
10656:
10588:
10221:This yields the resultant torque on the system as
10211:
9937:
9864:
9835:
9813:
9777:
9746:
9646:
9624:
9602:
9580:
9558:
9524:
9495:
9332:
9306:
9255:
9211:
9180:
9086:
9057:
9033:
8438:
8407:
8350:
8276:
8245:
8077:
8050:
7632:
7291:
7218:
7189:
7167:
7138:
7113:
7091:
7069:
7047:
6871:
6842:
6820:
6769:
6742:
6693:
6644:
6213:
6191:
6100:
6080:
6060:
6038:
5946:
5926:
5781:
5758:
5716:
5517:
5471:
5447:
5427:
5407:
5380:
5348:
5067:
5047:
5027:
5007:
4953:
4933:
4909:
4889:
4865:
4835:
4815:
4795:
4757:
4735:
4697:
4672:
4586:Another expression replaces the summation with an
4576:
4504:
4425:
4393:
4144:
4124:
4097:
4070:
4046:
4026:
3942:
3838:
3781:
3726:
3669:
3584:
3554:
3524:
3460:
3323:
3303:
3255:
3182:
3107:
3079:
3051:
3031:
2925:
2905:
2881:
2861:
2841:
2821:
2761:
2734:
2675:
2645:
2607:
2473:
2248:
2226:
2184:
2141:
2105:
2083:
2057:
2026:
1809:
1776:
1738:
1712:
1690:
1668:
1628:
1589:
1569:
1521:
1444:
1386:
1317:
1154:
1134:
1114:
1077:units and pound-foot-second squared (lbf·ft·s) in
206:
123:
35708:
35622:
35595:
35556:
35463:
35400:
35376:
35347:
35330:
35266:
35249:
35203:
35186:
35106:
35072:
35055:
35000:
31131:while the off-diagonal elements, also called the
29915:
29881:
29865:
29836:
29814:
29723:
29708:
29661:
29499:
29419:
29372:
29339:
29283:
29250:
29186:
29135:
29110:
29067:
29047:
28986:
28966:
28918:
28898:
28859:
28772:
28747:
28704:
28684:
28631:
28611:
28563:
28543:
28475:
28455:
28407:
28387:
28330:
28272:
28179:
28139:
28126:
28090:
28077:
28042:
28012:
27967:
27943:
27906:
27850:
27813:
27800:
27718:
27705:
27677:
27642:
27567:
27479:
27400:
27099:
27065:
27049:
27020:
27005:
26914:
26850:
25498:and the inertia matrix relative to another point
10857:
10721:
10700:
10678:
10634:
10612:
10554:
10488:
10384:
10339:
10296:
10182:
10137:
10084:
10055:
10037:
10013:
9984:
9923:
9907:
9886:
9799:
9769:
8940:
8852:
8830:
8710:
8653:
8396:
8214:
8116:
8016:
7950:
7846:
7802:
7606:
7559:
7530:
7449:
7433:
7412:
7382:
7317:
7277:
7261:
7240:
6734:
6192:{\displaystyle r(z)^{2}=x^{2}+y^{2}=R^{2}-z^{2}.}
4877:, and is the sum of the second moments about the
3670:{\displaystyle I_{xx}=I_{yy}={\frac {b^{4}}{12}}}
2458:
2049:
2011:
1065:. The amount of torque needed to cause any given
40604:
38957:
38220:
38218:
36042:
31817:{\displaystyle \mathbf {r} \otimes \mathbf {r} }
25706:be the center of mass of the rigid system, then
19574: cross-product distributivity over addition
11351:{\displaystyle \mathbf {b} =(b_{x},b_{y},b_{z})}
1142:is the distance of the point from the axis, and
37045:, meaning it is symmetrical under rotations of
36629:{\displaystyle \mathbf {I} _{\mathbf {C} }^{B}}
36390:{\displaystyle \mathbf {I} _{\mathbf {C} }^{B}}
36197:{\displaystyle \mathbf {I} =\mathbf {I} _{0}+m}
34817:The alternate inertia convention has been used
33282:
3498:Moment of inertia of area is also known as the
38790:
38628:
38319:. Silly Beagle Productions. pp. 142–143.
38164:
38049:
34707:The standard inertia convention has been used
31630:-axis when the objects are rotated around the
9043:Let the reference point be the center of mass
4058:Consider the kinetic energy of an assembly of
3481:
1188:Theoria motus corporum solidorum seu rigidorum
1057:When a body is free to rotate around an axis,
1045:, simply "inertia" is often used to refer to "
39955:
38943:
38215:
38169:Classical dynamics of particles & systems
38160:
38158:
38156:
38154:
38152:
38150:
36258:
27554:by removing the component that projects onto
21246: is not characteristic of particle
15708:{\displaystyle \mathbf {A} _{\mathbf {R} }=0}
4577:{\displaystyle I_{P}=\sum _{i}m_{i}r_{i}^{2}}
2683:for all of the elements of mass in the body.
2234:of the pendulum mass around the pivot, where
896:
38831:
38717:
38715:
38444:ice skater conservation of angular momentum.
38396:New Understanding Physics for Advanced Level
38264:
38192:
38190:
38121:motions which can occur in such bodies.]
37963:
37955:
37865:
37856:
37820:
37811:
37686:
37678:
36348:Inertia matrix in different reference frames
33279:, for the components of the inertia tensor.
31716:
31707:
29444:Thus, the moment of inertia around the line
26055:Distribute over the cross product to obtain
21852: cross-product as matrix multiplication
21018:
20968:
20808:
20758:
20638:
20585:
20465:
20415:
20307:
20213:
20045:
19951:
19773:
19679:
19507:
19413:
19270:
19123:
18913:
18766:
18658:
18605:
11586:of the body chosen to be the center of mass
3493:
2153:of this single mass around the pivot point.
38494:Kinematics and Dynamics of Planar Machinery
38308:
38306:
14423:{\displaystyle \mathbf {A} _{\mathbf {R} }}
12889:This equation expands to yield three terms
12102:obtained from the relative position vector
11198:is the angular velocity of the system, and
9235:A 1920s John Deere tractor with the spoked
6694:{\textstyle m={\frac {4}{3}}\pi R^{3}\rho }
5459:) is determined by integration. Align the
3839:{\displaystyle I_{xx}={\frac {bh^{3}}{36}}}
3782:{\displaystyle I_{yy}={\frac {hb^{3}}{12}}}
3727:{\displaystyle I_{xx}={\frac {bh^{3}}{12}}}
39962:
39948:
38950:
38936:
38392:
38386:
38147:
24926:
24922:
21846:
21238:
21234:
21024:
20814:
20644:
20471:
20102:
19840:
19568:
19276:
18923:
18919:
18149:
18007:
17927:
17764:
17627:
17480:
17025:
16866:
16649:
15404:
7077:is the angular velocity of the system and
3953:
1810:{\displaystyle \mathbf {F} =m\mathbf {a} }
903:
889:
45:
38721:
38712:
38582:(4th ed.). Oxford University Press.
38573:
38571:
38569:
38567:
38565:
38563:
38487:
38485:
38483:
38481:
38479:
38312:
38224:
38187:
36329:
35691:
35649:
35630:
35583:
35564:
35535:
32239:-axis when the object rotates around the
32023:
31968:
31951:
31772:
31765:
31758:
31610:denotes the moment of inertia around the
31560:denotes the moment of inertia around the
26767:{\displaystyle m\left^{\mathsf {T}}\left}
26499:The result is the parallel axis theorem,
24113:
24042:
23990:
23934:
23882:
23811:
23755:
23703:
23651:
23520:
23482:
23430:
23392:
23336:
23298:
23246:
23194:
23152:
23100:
23062:
23010:
22909:
22876:
22815:
22782:
22717:
22684:
22623:
22590:
22525:
22492:
22431:
22398:
22275:
22242:
22205:
22172:
22135:
22102:
20820: cross-product scalar multiplication
19846: cross-product scalar multiplication
18013: cross-product scalar multiplication
17933: cross-product scalar multiplication
16872: cross-product scalar multiplication
15410: cross-product scalar multiplication
11948:
11852:
11734:
11662:
10533:
10455:
10275:
9532:denotes the trajectory of each particle.
8919:
8804:
8689:
8632:
8326:
8193:
7995:
7917:
7825:
7781:
7585:
6398:
6312:
5977:of the sphere is defined by the equation
5646:
5639:
5577:
5566:
5508:
5501:
5184:
5170:
5127:
5116:
4331:
4277:
4205:
1015:of mass with respect to distance from an
40319:Covariance and contravariance of vectors
38683:Vector mechanics for engineers: Dynamics
38629:Morrow, H. W.; Kokernak, Robert (2011).
38303:
38110:
37213:{\displaystyle {\boldsymbol {\Lambda }}}
37125:
36685:{\displaystyle {\boldsymbol {\Lambda }}}
25465:
13223:Since the center of mass is defined by
9230:
8465:
6708:
6039:{\displaystyle x^{2}+y^{2}+z^{2}=R^{2},}
5968:
4974:
3962:
3062:
2690:
1274:is also defined as the ratio of the net
1252:
1240:
914:
38750:, Vol 1. 2nd Ed., Pergamon Press, 1969.
38676:
38674:
38672:
38670:
38668:
38666:
38664:
38662:
38660:
38510:
38419:
38413:
38339:
38333:
38056:(2 ed.). Oxford University Press.
38050:Escudier, Marcel; Atkins, Tony (2019).
28839:{\displaystyle \Delta \mathbf {r} _{i}}
27547:{\displaystyle \Delta \mathbf {r} _{i}}
26866:and passes through the body at a point
25416:
25394:
25386:
25364:
25335:
25257:
25249:
25158:
25077:
25063:
24975:
21240:
21224:
21140:
21126:
21038:
21011:
20961:
20910:
20801:
20751:
20740:
20631:
20578:
20567:
20440:
20408:
20379:
20300:
20217:
20206:
20177:
20038:
19955:
19944:
19915:
19830:
19822:
19766:
19683:
19672:
19643:
19522:
19514:
19500:
19417:
19406:
19377:
19310:
19227:
19213:
19130:
19116:
19087:
18972:
18925:
18906:
18820:
18773:
18759:
18730:
18651:
18598:
18569:
18490:
18440:
18411:
18314:
18303:
18253:
18186:
18139:
18089:
18053:
18042:
17979:
17947:
17896:
17867:
17822:
17793:
17685:
17656:
17614:
17606:
17538:
17509:
17449:
17438:
17391:
17362:
17293:
17279:
17237:
17226:
17127:
17113:
17081:
17070:
16964:
16950:
16918:
16907:
16838:
16806:
16747:
16733:
16701:
16690:
16624:
16589:
16536:
16522:
16490:
16479:
16440:
16390:
16337:
16323:
16291:
16280:
16186:
16175:
16125:
16058:
16020:
15970:
15903:
15771:
15161:
15066:
15048:
15012:
14999:
14952:
14868:
14860:
14776:
14551:
14538:
14497:
14467:{\displaystyle {\boldsymbol {\alpha }}}
14460:
14445:{\displaystyle {\boldsymbol {\omega }}}
14438:
14194:
14077:
14055:
13916:
13832:
13712:
13640:
13478:
13435:
13091:
12997:
12961:
12774:
12721:
12271:
12249:
11895:
11759:
11191:{\displaystyle {\boldsymbol {\omega }}}
11184:
11127:
10893:Euler's equations (rigid body dynamics)
10889:Precession § Classical (Newtonian)
10829:
10233:
9814:{\displaystyle \mathbf {\hat {e}} _{i}}
9711:
9703:
9677:
9647:{\displaystyle {\boldsymbol {\alpha }}}
9640:
9625:{\displaystyle {\boldsymbol {\omega }}}
9618:
9414:
7488:
7393:
7070:{\displaystyle {\boldsymbol {\omega }}}
7063:
7008:
6959:
2439:
2396:
2375:
2316:
2249:{\displaystyle {\boldsymbol {\omega }}}
2242:
2212:
2084:{\displaystyle {\boldsymbol {\alpha }}}
2077:
1992:
1949:
1928:
1871:
1829:
1762:
1739:{\displaystyle {\boldsymbol {\alpha }}}
1732:
1646:
1318:{\displaystyle I={\frac {L}{\omega }}.}
14:
40605:
38784:
38560:
38541:
38476:
38455:
38449:
38278:
38276:
38260:
38258:
38256:
38254:
38252:
38250:
38248:
38246:
38083:
38081:
38053:A Dictionary of Mechanical Engineering
37837:
37792:
37235:
37112:Rotating molecules are also classified
36739:
36648:into the product of a rotation matrix
36557:
29889:
28147:
28098:
27821:
27726:
27073:
26745:
26596:is the vector from the center of mass
25799:is the vector from the center of mass
13680:
13646:
4034:for every particle in the body, where
3595:Sectional areas moment calculated thus
3304:{\displaystyle \pi \ \mathrm {rad/s} }
2869:is local acceleration of gravity, and
1201:The moment of inertia also appears in
124:{\displaystyle I={\frac {L}{\omega }}}
39943:
38931:
38860:
38685:(9th ed.). Boston: McGraw-Hill.
38366:
38360:
38265:Kane, T. R.; Levinson, D. A. (1985).
38196:
38116:
26395:The first term is the inertia matrix
20650: cross-product anticommutativity
16655: cross-product anticommutativity
11291:, constructed from the components of
11252:
9654:of the rigid system of particles as,
4705:gives the mass density at each point
3087:from the pivot to a point called the
2686:
1190:in 1765, and it is incorporated into
585:Newton's law of universal gravitation
38825:
38753:
38657:
38521:(7th ed.). Hoboken, NJ: Wiley.
38491:
38282:
38087:
38062:10.1093/acref/9780198832102.001.0001
24934: is not characteristic of
14319:is the acceleration of the particle
10887:For analysis of a spinning top, see
9243:Newton's laws for a rigid system of
3532:and horizontal moment of the y-axis
1482:In general, given an object of mass
1217:The moment of inertia of a rotating
40462:
38273:
38243:
38078:
36872:The columns of the rotation matrix
35115:{\displaystyle \mathbf {\hat {n}} }
35009:{\displaystyle \mathbf {\hat {n}} }
34941:Derivation of the tensor components
34700:
29508:{\displaystyle \mathbf {\hat {k}} }
28868:{\displaystyle \mathbf {\hat {k}} }
28188:{\displaystyle \mathbf {\hat {k}} }
27859:{\displaystyle \mathbf {\hat {k}} }
27576:{\displaystyle \mathbf {\hat {k}} }
27409:{\displaystyle \mathbf {\hat {k}} }
26859:{\displaystyle \mathbf {\hat {k}} }
26806:Scalar moment of inertia in a plane
14181:
12445:
12057:) sum to zero by the definition of
9778:{\displaystyle \mathbf {\hat {k}} }
7651:
6743:{\displaystyle \mathbf {\hat {k}} }
5518:{\displaystyle dV=sr\,dr\,d\theta }
2058:{\displaystyle \mathbf {\hat {k}} }
1332:pull in their outstretched arms or
1031:
566:Mechanics of planar particle motion
174:
24:
40182:Tensors in curvilinear coordinates
38864:Mechanics of Robotics Manipulation
38631:Statics and Strengths of Materials
36231:is the 3 × 3 identity matrix, and
33593:
33590:
33587:
33472:
33469:
33466:
33351:
33348:
33345:
31446:
31443:
31440:
31322:
31319:
31316:
31198:
31195:
31192:
31027:
31024:
31021:
30899:
30896:
30893:
30771:
30768:
30765:
30263:
30260:
30257:
29777:
29676:
29580:
29387:
29315:
29297:
29226:
29195:
29144:
29076:
28995:
28927:
28821:
28781:
28713:
28640:
28572:
28484:
28416:
28346:
28288:
28215:
27737:
27651:
27613:
27590:
27529:
27329:{\displaystyle P_{i},i=1,\dots ,n}
27214:
27179:
26968:
25306:
25220:
25139:
25118:
25044:
25023:
24889:
24811:
24790:
24721:
24585:
24558:
24531:
24512:
24491:
24472:
24449:
24430:
24401:
24374:
24347:
24328:
24305:
24286:
24265:
24246:
24217:
24190:
24086:
24059:
24023:
24004:
23971:
23952:
23915:
23896:
23855:
23828:
23792:
23773:
23736:
23717:
23684:
23665:
23624:
23597:
23496:
23463:
23444:
23406:
23373:
23354:
23312:
23279:
23260:
23227:
23208:
23170:
23133:
23114:
23076:
23043:
23024:
22986:
22890:
22857:
22832:
22796:
22763:
22738:
22698:
22665:
22640:
22604:
22571:
22546:
22506:
22473:
22448:
22412:
22379:
22354:
22256:
22223:
22186:
22153:
22116:
22083:
22040:
22019:
21993:
21964:
21941:
21920:
21750:
21729:
21703:
21674:
21651:
21630:
21577:
21556:
21530:
21501:
21478:
21457:
21366:
21345:
21205:
21184:
21107:
21086:
20992:
20971:
20891:
20870:
20782:
20761:
20721:
20700:
20612:
20588:
20548:
20524:
20447:
20418:
20386:
20357:
20281:
20263:
20242:
20224:
20184:
20155:
20072:
20054:
20019:
20001:
19980:
19962:
19922:
19893:
19800:
19782:
19747:
19729:
19708:
19690:
19650:
19621:
19547:
19529:
19481:
19463:
19442:
19424:
19384:
19355:
19252:
19234:
19194:
19176:
19155:
19137:
19094:
19065:
18997:
18979:
18950:
18932:
18887:
18869:
18845:
18827:
18798:
18780:
18737:
18708:
18632:
18614:
18576:
18547:
18471:
18453:
18418:
18389:
18321:
18281:
18260:
18231:
18120:
18102:
18060:
18020:
17986:
17960:
17903:
17877:
17829:
17800:
17731:
17713:
17692:
17663:
17584:
17566:
17545:
17516:
17456:
17419:
17398:
17369:
17321:
17300:
17244:
17204:
17193:can then be continued as follows:
17155:
17134:
17088:
17048:
16992:
16971:
16925:
16885:
16845:
16813:
16775:
16754:
16708:
16668:
16631:
16596:
16564:
16543:
16497:
16457:
16421:
16397:
16365:
16344:
16298:
16258:
16193:
16153:
16132:
16103:
15998:
15977:
15948:
15857:
15816:
15588:
15457:
15371:
15281:
15096:
15073:
15019:
14975:
14968:The calculation uses the identity
14920:
14828:
14659:
13884:
13691:
13658:
13457:
13414:
13302:
13261:
13098:
13004:
12968:
12838:
12781:
12728:
12518:{\displaystyle P_{i},i=1,\dots ,n}
12381:
12217:
12109:
12074:
11949:
11876:
11853:
11766:
11735:
11663:
11134:
11104:and the (absolute) velocities are
11056:
10962:{\displaystyle P_{i},i=1,\dots ,n}
10534:
10456:
10364:
10319:
10276:
10162:
10117:
10064:
9993:
9718:
9684:
9446:
9307:{\displaystyle P_{i},i=1,\dots ,n}
9094:so the kinetic energy is given by
8920:
8805:
8690:
8633:
8327:
8194:
8096:
7996:
7918:
7826:
7782:
7710:
7586:
7539:
7495:
7356:
7339:
7015:
6890:
6821:{\displaystyle P_{i},i=1,\dots ,n}
3451:
3424:
3416:
3413:
3410:
3388:
3379:
3297:
3289:
3286:
3283:
2735:{\displaystyle \omega _{\text{n}}}
1551:
1289:
25:
40639:
38891:
38768:
38746:L. D. Landau and E. M. Lifshitz,
38580:Theory of Machines and Mechanisms
38267:Dynamics, Theory and Applications
38165:Marion, JB; Thornton, ST (1995).
36979:. This result was first shown by
36639:
36600:changes as the body moves, while
29995:
29515:is obtained from the calculation
8461:
6068:of the disc at the cross-section
5759:{\displaystyle m=\pi R^{2}\rho s}
1409:of the pendulum and its distance
1362:
1361:around a principal axis, that is
1288:around a principal axis, that is
38726:(2nd ed.). Addison-Wesley.
38227:Fundamentals of Applied Dynamics
38201:(3rd ed.). Addison-Wesley.
37981:
37959:
37934:
37912:
37880:
37860:
37849:
37844:
37831:
37815:
37804:
37799:
37786:
37763:
37755:around an axis in the direction
37748:{\displaystyle I_{\mathbf {n} }}
37739:
37712:
37690:
37682:
37671:
37649:
37247:
37242:
37229:
37206:
36880:
36766:
36733:
36727:
36722:
36707:
36701:
36678:
36656:
36615:
36609:
36586:
36551:
36538:
36532:
36526:
36516:
36510:
36487:
36465:
36440:
36435:
36427:
36405:
36376:
36370:
36323:
36315:
36311:
36305:
36297:
36271:
36187:
36179:
36165:
36156:
36148:
36125:
36116:
36094:
36075:{\displaystyle \mathbf {I} _{0}}
36062:
35705:
35685:
35679:
35669:
35665:
35656:
35643:
35619:
35592:
35577:
35571:
35553:
35542:
35529:
35460:
35450:
35430:
35397:
35373:
35363:
35344:
35327:
35317:
35307:
35290:
35263:
35246:
35236:
35223:
35200:
35183:
35173:
35160:
35103:
35069:
35052:
35042:
35029:
34997:
34973:
33663:
32270:
32196:
32187:
32173:
32101:
32093:
32085:
32050:
32009:
31998:
31961:
31941:
31930:
31906:
31884:
31876:
31868:
31860:
31810:
31802:
31749:
31741:
31727:
31711:
31661:
30703:
30628:is the vector to the point mass
30526:
30317:
30076:
29975:
29960:{\displaystyle \mathbf {I_{R}} }
29951:
29947:
29912:
29903:
29897:
29878:
29862:
29853:
29847:
29833:
29811:
29782:
29720:
29705:
29681:
29658:
29585:
29496:
29472:
29416:
29392:
29369:
29336:
29320:
29302:
29280:
29247:
29231:
29200:
29183:
29149:
29132:
29107:
29081:
29064:
29044:
29000:
28983:
28963:
28932:
28915:
28895:
28856:
28826:
28786:
28769:
28744:
28718:
28701:
28681:
28645:
28628:
28608:
28577:
28560:
28540:
28489:
28472:
28452:
28421:
28404:
28384:
28351:
28327:
28293:
28269:
28220:
28176:
28136:
28123:
28113:
28087:
28074:
28064:
28039:
28009:
27977:
27964:
27954:
27940:
27903:
27847:
27810:
27797:
27773:
27742:
27715:
27702:
27692:
27674:
27656:
27639:
27618:
27595:
27564:
27534:
27476:
27463:
27446:
27424:
27397:
27373:
27358:{\displaystyle \mathbf {r} _{i}}
27345:
27248:
27234:
27219:
27184:
27155:
27140:{\displaystyle \mathbf {I_{R}} }
27131:
27127:
27096:
27087:
27081:
27062:
27046:
27037:
27031:
27017:
27002:
26973:
26911:
26874:
26847:
26785:
26756:
26735:
26665:
26626:
26604:
26582:
26547:
26531:
26525:
26514:
26508:
26482:
26457:
26432:
26417:{\displaystyle \mathbf {I_{C}} }
26408:
26404:
26366:
26306:
26292:
26244:
26230:
26214:
26200:
26137:
26123:
26070:
26064:
26026:
26013:
25999:
25933:
25925:
25906:
25858:
25852:
25829:
25807:
25785:
25760:
25752:
25744:
25733:
25725:
25714:
25692:
25654:
25640:
25589:
25583:
25560:
25545:{\displaystyle \mathbf {I_{R}} }
25536:
25532:
25506:
25484:
25455:{\displaystyle \mathbf {I_{C}} }
25446:
25442:
25409:
25403:
25379:
25373:
25144:
25123:
25049:
25028:
24928:
24918:
24830:
24816:
24795:
24745:
21385:
21371:
21350:
21283:
21210:
21189:
21112:
21091:
20997:
20976:
20896:
20875:
20787:
20766:
20726:
20705:
20617:
20593:
20553:
20529:
20452:
20423:
20391:
20362:
20286:
20268:
20247:
20229:
20189:
20160:
20077:
20059:
20024:
20006:
19985:
19967:
19927:
19898:
19805:
19787:
19752:
19734:
19713:
19695:
19655:
19626:
19552:
19534:
19486:
19468:
19447:
19429:
19389:
19360:
19257:
19239:
19199:
19181:
19160:
19142:
19099:
19070:
19002:
18984:
18955:
18937:
18892:
18874:
18850:
18832:
18803:
18785:
18742:
18713:
18637:
18619:
18581:
18552:
18476:
18458:
18423:
18394:
18326:
18286:
18265:
18236:
18125:
18107:
18065:
18025:
17991:
17965:
17908:
17882:
17834:
17805:
17736:
17718:
17697:
17668:
17589:
17571:
17550:
17521:
17461:
17424:
17403:
17374:
17326:
17305:
17249:
17209:
17160:
17139:
17093:
17053:
16997:
16976:
16930:
16890:
16850:
16818:
16780:
16759:
16713:
16673:
16636:
16601:
16569:
16548:
16502:
16462:
16426:
16402:
16370:
16349:
16303:
16263:
16198:
16158:
16137:
16108:
16029:
16003:
15982:
15953:
15877:
15862:
15836:
15821:
15745:
15723:
15693:
15687:
15634:
15611:
15593:
15532:
15526:
15503:
15480:
15462:
15391:
15376:
15314:
15286:
15237:
15212:
15202:
15198:
15101:
15078:
15024:
14980:
14925:
14833:
14748:
14734:
14703:
14689:
14664:
14633:
14604:
14598:
14579:
14565:
14525:
14511:
14483:
14452:and angular acceleration vector
14414:
14408:
14385:
14312:{\displaystyle \mathbf {a} _{i}}
14299:
14267:
14243:
14233:
14229:
14151:{\displaystyle \mathbf {I_{C}} }
14142:
14138:
14105:
14099:
14070:
14064:
13999:
13993:
13982:
13976:
13889:
13805:
13799:
13788:
13782:
13696:
13663:
13571:
13565:
13554:
13548:
13462:
13419:
13307:
13266:
13197:
13191:
13180:
13174:
13103:
13076:
13070:
13009:
12973:
12872:
12858:
12843:
12807:
12801:
12786:
12754:
12748:
12733:
12661:
12646:
12576:{\displaystyle \mathbf {v} _{i}}
12563:
12547:{\displaystyle \mathbf {r} _{i}}
12534:
12428:
12386:
12332:
12326:
12310:{\displaystyle \mathbf {I_{C}} }
12301:
12297:
12264:
12258:
12222:
12165:
12143:
12129:
12114:
12079:
12064:Then, the skew-symmetric matrix
12043:
12025:{\displaystyle \mathbf {V_{R}} }
12016:
12012:
11975:
11969:
11954:
11881:
11858:
11792:
11786:
11771:
11740:
11683:
11668:
11620:
11594:
11572:
11416:
11403:
11394:
11378:
11370:
11299:
11273:
11235:
11220:{\displaystyle \mathbf {V_{R}} }
11211:
11207:
11160:
11154:
11139:
11113:
11090:
11076:
11061:
11035:
11020:{\displaystyle \mathbf {v} _{i}}
11007:
10991:{\displaystyle \mathbf {r} _{i}}
10978:
10854:
10842:
10814:{\displaystyle I_{\mathbf {C} }}
10805:
10778:
10718:
10697:
10675:
10650:
10631:
10609:
10575:
10551:
10485:
10400:
10381:
10336:
10293:
10198:
10179:
10134:
10100:
10081:
10052:
10034:
10010:
9981:
9958:
9920:
9904:
9883:
9865:{\displaystyle \mathbf {r} _{i}}
9852:
9829:
9796:
9766:
9737:
9723:
9689:
9663:
9632:and angular acceleration vector
9596:
9574:
9525:{\displaystyle \mathbf {r} _{i}}
9512:
9476:
9451:
9396:
9352:
9326:
9318:and torque at a reference point
9226:
9212:{\displaystyle I_{\mathbf {C} }}
9203:
9171:
9163:
9130:
9087:{\displaystyle I_{\mathbf {C} }}
9078:
9051:
9020:
9012:
8937:
8876:
8849:
8827:
8726:
8707:
8669:
8650:
8560:
8545:
8439:{\displaystyle I_{\mathbf {C} }}
8430:
8393:
8381:
8368:
8297:
8277:{\displaystyle I_{\mathbf {C} }}
8268:
8211:
8151:
8137:
8113:
8071:
8037:
8013:
7947:
7862:
7843:
7799:
7730:
7715:
7668:
7622:
7603:
7575:
7556:
7527:
7514:
7500:
7470:
7446:
7430:
7409:
7379:
7344:
7314:
7274:
7258:
7237:
7219:{\displaystyle \mathbf {r} _{i}}
7206:
7183:
7168:{\displaystyle \mathbf {e} _{i}}
7155:
7132:
7107:
7085:
7034:
7020:
7000:
6987:
6973:
6941:
6928:
6914:
6895:
6872:{\displaystyle \mathbf {v} _{i}}
6859:
6836:
6731:
4751:
4650:
4233:
4218:
3115:is determined from the formula,
2526:
2518:
2455:
2406:
2388:
2365:
2357:
2324:
2300:
2292:
2284:
2272:
2220:
2204:
2099:
2046:
2008:
1959:
1941:
1918:
1910:
1879:
1857:
1849:
1841:
1803:
1792:
1770:
1754:
1706:
1684:
1662:
1654:
1577:with the square of its distance
1247:conservation of angular momentum
943:
934:
870:
869:
856:
189:
38974:Linear/translational quantities
38854:
38762:
38740:
38699:
38647:
38622:
38609:
38596:
38535:
37904:Thus, the magnitude of a point
37600:
37571:
37096:is often described in terms of
32151:is obtained by the computation
30012:
18155: dot-product commutativity
14722:to represent the cross product
9314:, can be written in terms of a
7404:
7375:
2067:vector triple product expansion
1387:{\displaystyle \tau =I\alpha .}
1340:during a dive, to spin faster.
1052:
38346:Wolfram Demonstrations Project
38289:. AuthorHouse. p. 10.10.
38043:
36191:
36160:
36144:
36141:
34920:
34824:
34801:
34714:
34686:
34590:
32014:
32005:
32002:
31994:
31965:
31957:
31946:
31937:
31934:
31926:
31864:
31856:
31699:
31681:
30695:Note that, by the definition,
30608:
30602:
30584:
30578:
30560:
30554:
30390:
30384:
30369:
30363:
30327:
30312:
30224:Its components are defined as
27456:
27450:
27194:
27176:
26810:The scalar moment of inertia,
26789:
26781:
26552:
26543:
26461:
26453:
26371:
26362:
26310:
26287:
26248:
26240:
26234:
26226:
26218:
26195:
26142:
26118:
26031:
25989:
25943:
25901:
25737:
25721:
25320:
25303:
25234:
25217:
25165:
25162:
25136:
25070:
25067:
25041:
25020:
24903:
24886:
24837:
24834:
24808:
24787:
24735:
24718:
24609:
24555:
24425:
24371:
24241:
24187:
24110:
24056:
23879:
23825:
23648:
23594:
22920:
22851:
22826:
22757:
22728:
22659:
22634:
22565:
22536:
22467:
22442:
22373:
21392:
21389:
21363:
21342:
21231:
21228:
21202:
21181:
21133:
21130:
21104:
21083:
21030: summation distributivity
21021:
21015:
20989:
20957:
20917:
20914:
20888:
20867:
20811:
20805:
20779:
20744:
20718:
20697:
20641:
20635:
20609:
20571:
20545:
20521:
20468:
20462:
20436:
20401:
20375:
20354:
20310:
20304:
20278:
20257:
20221:
20199:
20173:
20152:
20099:
20096:
20090:
20087:
20051:
20042:
20016:
19995:
19959:
19937:
19911:
19890:
19837:
19834:
19818:
19815:
19779:
19770:
19744:
19723:
19687:
19665:
19639:
19618:
19565:
19562:
19526:
19504:
19478:
19457:
19421:
19399:
19373:
19352:
19273:
19267:
19231:
19220:
19217:
19191:
19170:
19134:
19126:
19109:
19083:
19062:
19012:
18976:
18965:
18929:
18916:
18910:
18884:
18863:
18860:
18824:
18813:
18777:
18769:
18752:
18726:
18705:
18661:
18655:
18629:
18591:
18565:
18544:
18500:
18497:
18494:
18468:
18450:
18433:
18407:
18386:
18342:
18339:
18336:
18310:
18299:
18275:
18249:
18228:
18146:
18143:
18117:
18099:
18078:
18075:
18049:
18038:
18004:
18001:
17975:
17957:
17924:
17921:
17918:
17892:
17874:
17863:
17847:
17844:
17818:
17815:
17789:
17786:
17761:
17758:
17755:
17749:
17728:
17707:
17681:
17678:
17652:
17649:
17624:
17621:
17618:
17602:
17581:
17560:
17534:
17531:
17505:
17502:
17477:
17474:
17471:
17445:
17434:
17413:
17387:
17384:
17358:
17355:
17339:
17336:
17315:
17289:
17286:
17275:
17262:
17259:
17233:
17222:
17170:
17149:
17123:
17120:
17106:
17103:
17077:
17066:
17022:
17016:
17007:
16986:
16960:
16957:
16943:
16940:
16914:
16903:
16863:
16860:
16834:
16828:
16802:
16799:
16790:
16769:
16743:
16740:
16726:
16723:
16697:
16686:
16646:
16620:
16611:
16585:
16579:
16558:
16532:
16529:
16515:
16512:
16486:
16475:
16444:
16418:
16412:
16386:
16380:
16359:
16333:
16330:
16316:
16313:
16287:
16276:
16214:
16211:
16208:
16182:
16171:
16147:
16121:
16100:
16050:
16047:
16016:
15992:
15966:
15945:
15895:
15813:
15661:
15658:
15606:
15585:
15541:
15538:
15475:
15454:
15401:
15368:
15324:
15299:
15247:
15222:
15216:
15193:
14752:
14729:
13330:so the kinetic energy becomes
13317:
13299:
12525:be located at the coordinates
12089:
12071:
11345:
11306:
10969:be located at the coordinates
6303:
6296:
6125:
6118:
6048:then the square of the radius
5912:
5899:
4790:
4772:
4730:
4712:
4654:
4646:
4641:
4623:
3429:
3400:
1883:
1864:
1061:must be applied to change its
13:
1:
40235:Exterior covariant derivative
40167:Tensor (intrinsic definition)
38979:Angular/rotational quantities
38546:. Boca Raton, FL: CRC Press.
38428:. Light and Matter. pp.
38399:. Nelson Thomas. p. 64.
38373:The Physics of Everyday Stuff
38037:
36356:
36043:Inertia tensor of translation
15143:as shown in the proof below:
12583:, then the kinetic energy is
12050:{\displaystyle =\mathbf {C} }
11049:, the relative positions are
6704:
5381:{\displaystyle m=\rho s\ell }
4964:
4961:-axis depending on the load.
4847:Note on second moment of area
3958:
2695:Pendulums used in Mendenhall
2617:This shows that the quantity
2069:with the perpendicularity of
1817:the torque equation becomes:
1225:
492:Koopman–von Neumann mechanics
40260:Raising and lowering indices
38908:The moment of inertia tensor
38617:NACA Technical Note No. 1629
37948:on the inertia ellipsoid is
37941:{\displaystyle \mathbf {n} }
37919:{\displaystyle \mathbf {x} }
37770:{\displaystyle \mathbf {n} }
37719:{\displaystyle \mathbf {n} }
37656:{\displaystyle \mathbf {x} }
37121:
37118:is different for each type.
36977:principal moments of inertia
36887:{\displaystyle \mathbf {Q} }
36663:{\displaystyle \mathbf {Q} }
36593:{\displaystyle \mathbf {A} }
36494:{\displaystyle \mathbf {x} }
36472:{\displaystyle \mathbf {y} }
36412:{\displaystyle \mathbf {A} }
36278:{\displaystyle \mathbf {R} }
36101:{\displaystyle \mathbf {R} }
34980:{\displaystyle \mathbf {x} }
34583:If one has the inertia data
33283:Alternate inertia convention
32057:{\displaystyle \mathbf {n} }
30710:{\displaystyle \mathbf {I} }
30680:{\displaystyle \delta _{ij}}
29982:{\displaystyle \mathbf {R} }
29479:{\displaystyle \mathbf {R} }
27991:, then we have the identity
27780:{\displaystyle \mathbf {E} }
27431:{\displaystyle \mathbf {R} }
27416:through the reference point
27380:{\displaystyle \mathbf {R} }
27162:{\displaystyle \mathbf {R} }
26881:{\displaystyle \mathbf {R} }
26633:{\displaystyle \mathbf {R} }
26611:{\displaystyle \mathbf {C} }
26589:{\displaystyle \mathbf {d} }
26489:{\displaystyle \mathbf {d} }
26439:{\displaystyle \mathbf {C} }
25836:{\displaystyle \mathbf {R} }
25814:{\displaystyle \mathbf {C} }
25792:{\displaystyle \mathbf {d} }
25699:{\displaystyle \mathbf {C} }
25567:{\displaystyle \mathbf {R} }
25523:Consider the inertia matrix
25513:{\displaystyle \mathbf {R} }
25491:{\displaystyle \mathbf {C} }
21290:{\displaystyle \mathbf {u} }
19282: addition associativity
15752:{\displaystyle \mathbf {C} }
15730:{\displaystyle \mathbf {R} }
14640:{\displaystyle \mathbf {C} }
14392:{\displaystyle \mathbf {R} }
12435:{\displaystyle \mathbf {C} }
11601:{\displaystyle \mathbf {C} }
11579:{\displaystyle \mathbf {R} }
11242:{\displaystyle \mathbf {R} }
11042:{\displaystyle \mathbf {R} }
10785:{\displaystyle \mathbf {C} }
9836:{\displaystyle \mathbf {R} }
9603:{\displaystyle \mathbf {A} }
9581:{\displaystyle \mathbf {R} }
9333:{\displaystyle \mathbf {R} }
9058:{\displaystyle \mathbf {C} }
8078:{\displaystyle \mathbf {C} }
7190:{\displaystyle \mathbf {R} }
7139:{\displaystyle \mathbf {k} }
7114:{\displaystyle \mathbf {R} }
7092:{\displaystyle \mathbf {V} }
6843:{\displaystyle \mathbf {R} }
4995:with constant cross-section
4758:{\displaystyle \mathbf {r} }
2106:{\displaystyle \mathbf {r} }
1713:{\displaystyle \mathbf {F} }
1691:{\displaystyle \mathbf {r} }
560:Non-inertial reference frame
7:
40498:Gluon field strength tensor
39969:
38778:Pixar Graphics Technologies
38005:
35500:Rewrite the equation using
26682:{\displaystyle -m\left^{2}}
21275:Notice that for any vector
20477: vector triple product
17633: scalar triple product
17486: vector triple product
14715:{\displaystyle \left=\left}
12003:where the terms containing
6701:is the mass of the sphere.
5391:The moment of inertia of a
4991:The moment of inertia of a
4982:The moment of inertia of a
3482:Measuring moment of inertia
2849:is the mass of the object,
1540:
1002:(additive) property: for a
487:Appell's equation of motion
357:Inertial frame of reference
10:
40644:
40309:Cartan formalism (physics)
40129:Penrose graphical notation
39552:{\displaystyle {\dot {m}}}
38976:
38861:Mason, Matthew T. (2001).
38492:Paul, Burton (June 1979).
38462:. Macmillan. p. 304.
38420:Crowell, Benjamin (2003).
38017:List of moments of inertia
36985:Sylvester's law of inertia
36259:Inertia tensor of rotation
36046:
30455:is equal to 1, 2 or 3 for
26689:, which is similar to the
25469:
16237:is used on the last term:
10886:
8450:. Specifically, it is the
8085:as the reference point so
6850:, and absolute velocities
5960:list of moments of inertia
4971:List of moments of inertia
4968:
4105:that lie at the distances
1547:List of moments of inertia
1544:
1049:" or "moment of inertia".
987:applied and the resulting
29:
40515:
40455:
40404:
40397:
40289:
40220:
40157:
40101:
40048:
39995:
39988:
39981:Glossary of tensor theory
39977:
39250:specific angular momentum
38978:
38973:
38968:
38812:10.1080/14786445208647087
38606:, Happresearch.com, 2016.
38604:Trifilar Pendulum for MOI
38340:Wolfram, Stephen (2014).
38316:Honors Physics Essentials
38173:(4th ed.). Thomson.
31848:angular momentum operator
14377:in terms of the position
12157:, can be used to define,
9821:from the reference point
9566:in terms of the position
7646:Note on the cross product
7175:from the reference point
4917:-axes. The stresses in a
3494:Moment of inertia of area
1629:{\displaystyle I=mr^{2}.}
1522:{\displaystyle I=mk^{2},}
1462:to the axis of rotation.
1445:{\displaystyle I=mr^{2}.}
1413:from the pivot point as,
1336:curl their bodies into a
1034:for which this matrix is
963:, otherwise known as the
134:
91:
81:
71:
59:
44:
39:
40565:Gregorio Ricci-Curbastro
40437:Riemann curvature tensor
40144:Van der Waerden notation
38456:Tipler, Paul A. (1999).
38393:Breithaupt, Jim (2000).
38269:. New York: McGraw-Hill.
38117:Euler, Leonhard (1765).
38091:The Science of Mechanics
38032:Moment of inertia factor
36244:{\displaystyle \otimes }
30064:, the moment of inertia
20108: self cross-product
17770: self cross-product
17031: self cross-product
14474:of the rigid system as,
4875:polar moment of the area
3331:, can be computed to be
2646:{\displaystyle I=mr^{2}}
2185:{\displaystyle I=mr^{2}}
2142:{\displaystyle I=mr^{2}}
1222:in roll, pitch and yaw.
1178:("momentum inertiae" in
650:Rotating reference frame
482:Hamilton–Jacobi equation
18:Moment of inertia tensor
40535:Elwin Bruno Christoffel
40468:Angular momentum tensor
40139:Tetrad (index notation)
40109:Abstract index notation
39374:rotational acceleration
38838:Oxford University Press
38791:Sylvester, J J (1852).
38602:C. Couch and J. Mayes,
38519:Fundamentals of physics
38313:Fullerton, Dan (2011).
26618:to the reference point
25821:to the reference point
21297:, the following holds:
17189:The result of applying
15678:In the last statement,
14625:Use the center of mass
9221:polar moment of inertia
8448:polar moment of inertia
6752:polar moment of inertia
5388:is the mass of the rod.
4796:{\displaystyle (x,y,z)}
4736:{\displaystyle (x,y,z)}
3954:Motion in a fixed plane
1211:Newton's laws of motion
1069:(the rate of change in
969:angular/rotational mass
591:Newton's laws of motion
451:Newton's laws of motion
40349:Levi-Civita connection
39553:
38800:Philosophical Magazine
38722:Goldstein, H. (1980).
38369:"Figure Skating Spins"
38367:Hokin, Samuel (2014).
38225:Tenenbaum, RA (2004).
37997:
37942:
37920:
37896:
37771:
37749:
37720:
37698:
37657:
37633:
37530:
37356:
37264:
37214:
37191:
37184:
37164:
37144:
37075:
37074:{\displaystyle m>2}
37039:
36981:J. J. Sylvester (1852)
36969:
36942:
36915:
36888:
36864:
36752:
36686:
36670:and a diagonal matrix
36664:
36630:
36594:
36570:
36495:
36473:
36451:
36413:
36391:
36339:
36279:
36245:
36218:
36198:
36102:
36076:
36031:
35721:
35492:
35116:
35087:
35010:
34981:
34959:
34927:
34808:
34693:
34570:
34497:
34444:
34388:
34330:
34251:
34198:
34140:
34084:
34005:
33624:
33503:
33382:
33273:
33272:{\displaystyle I_{xy}}
33243:
33242:{\displaystyle I_{xx}}
33213:
33193:
33173:
33151:
33082:
33029:
32973:
32915:
32836:
32783:
32725:
32669:
32590:
32253:
32233:
32213:
32145:
32144:{\displaystyle I_{12}}
32112:
32058:
32034:
31892:
31818:
31786:
31644:
31624:
31604:
31603:{\displaystyle I_{xy}}
31574:
31554:
31553:{\displaystyle I_{xx}}
31522:
31480:
31356:
31232:
31123:
31058:
30930:
30802:
30711:
30681:
30649:
30622:
30509:
30489:
30469:
30449:
30429:
30404:
30294:
30216:
30058:
30031:
30017:For a rigid object of
29983:
29961:
29932:
29760:
29644:
29563:
29509:
29480:
29458:
29436:
28869:
28840:
28808:
28517:
28189:
28160:
27985:
27920:
27880:
27860:
27831:
27781:
27759:
27577:
27548:
27516:
27489:
27432:
27410:
27381:
27359:
27330:
27279:
27256:
27201:
27163:
27141:
27112:
26951:
26882:
26860:
26831:
26796:
26768:
26713:
26712:{\displaystyle mr^{2}}
26683:
26644:Note on the minus sign
26634:
26612:
26590:
26568:
26490:
26468:
26440:
26418:
26387:
26346:
26276:
26184:
26107:
26047:
25978:
25890:
25837:
25815:
25793:
25771:
25700:
25676:
25621:
25568:
25546:
25514:
25492:
25456:
25427:
25347:
25292:
25206:
25107:
25009:
24955:
24875:
24776:
24707:
24165:
23572:
22964:
22332:
21888:
21425:
21331:
21291:
21267:
21170:
21072:
20946:
20856:
20686:
20510:
20343:
20141:
19879:
19607:
19341:
19293:
19051:
18694:
18533:
18375:
18217:
18166:
17181:
16225:
16089:
15934:
15802:
15753:
15731:
15709:
15672:
15574:
15443:
15357:
15280:
15192:
15129:
14960:
14903:
14811:
14762:
14716:
14641:
14617:
14468:
14446:
14424:
14393:
14371:
14340:
14313:
14284:
14221:
14172:
14152:
14123:
14016:
13958:
13867:
13764:
13622:
13530:
13393:
13324:
13286:
13250:
13215:
13161:
13057:
12944:
12880:
12825:
12704:
12633:
12577:
12548:
12519:
12468:
12436:
12414:
12364:
12311:
12282:
12200:
12151:
12096:
12051:
12026:
11997:
11937:
11841:
11723:
11651:
11602:
11580:
11556:
11352:
11285:
11243:
11221:
11192:
11170:
11098:
11043:
11021:
10992:
10963:
10912:
10870:
10815:
10786:
10758:
10731:
10658:
10590:
10522:
10444:
10264:
10213:
9939:
9866:
9837:
9815:
9779:
9748:
9648:
9626:
9604:
9582:
9560:
9526:
9497:
9445:
9383:
9334:
9308:
9257:
9240:
9213:
9190:The moment of inertia
9182:
9088:
9059:
9035:
8995:
8908:
8793:
8613:
8532:
8471:
8440:
8417:The moment of inertia
8409:
8352:
8278:
8247:
8182:
8079:
8052:
7984:
7906:
7770:
7699:
7634:
7293:
7226:, and the unit vector
7220:
7191:
7169:
7140:
7115:
7093:
7071:
7049:
6873:
6844:
6822:
6771:
6744:
6714:
6695:
6646:
6215:
6193:
6102:
6082:
6062:
6040:
5973:
5948:
5928:
5783:
5760:
5718:
5519:
5473:
5449:
5429:
5409:
5395:of constant thickness
5382:
5350:
5069:
5049:
5029:
5009:
4979:
4955:
4935:
4911:
4891:
4867:
4837:
4817:
4797:
4759:
4737:
4699:
4674:
4578:
4506:
4473:
4427:
4426:{\displaystyle mr^{2}}
4395:
4362:
4266:
4194:
4146:
4126:
4099:
4072:
4048:
4028:
4027:{\displaystyle mr^{2}}
4001:
3944:
3840:
3783:
3728:
3671:
3586:
3556:
3555:{\displaystyle I_{yy}}
3526:
3525:{\displaystyle I_{xx}}
3462:
3325:
3305:
3257:
3184:
3109:
3081:
3053:
3033:
2927:
2907:
2883:
2863:
2843:
2823:
2763:
2736:
2700:
2677:
2676:{\displaystyle mr^{2}}
2647:
2609:
2475:
2250:
2228:
2186:
2143:
2107:
2085:
2059:
2028:
1811:
1778:
1740:
1714:
1692:
1670:
1630:
1597:to the pivot, that is
1591:
1571:
1523:
1486:, an effective radius
1446:
1388:
1345:Newton's law of motion
1319:
1270:The moment of inertia
1267:
1250:
1156:
1136:
1116:
1115:{\displaystyle mr^{2}}
1043:mechanical engineering
1008:perpendicular distance
975:, or most accurately,
965:mass moment of inertia
927:
618:Simple harmonic motion
531:Euler's laws of motion
325:D'Alembert's principle
208:
125:
40613:Mechanical quantities
40575:Jan Arnoldus Schouten
40530:Augustin-Louis Cauchy
40010:Differential geometry
39554:
38834:Undergraduate algebra
38832:Norman, C.W. (1986).
38542:French, A.P. (1971).
38342:"Spinning Ice Skater"
37998:
37943:
37921:
37897:
37772:
37750:
37721:
37699:
37658:
37634:
37531:
37357:
37265:
37215:
37185:
37165:
37145:
37129:
37076:
37040:
36970:
36968:{\displaystyle I_{3}}
36943:
36941:{\displaystyle I_{2}}
36916:
36914:{\displaystyle I_{1}}
36889:
36865:
36753:
36687:
36665:
36631:
36595:
36571:
36496:
36474:
36452:
36414:
36392:
36340:
36280:
36246:
36219:
36199:
36103:
36077:
36032:
35722:
35493:
35117:
35088:
35011:
34982:
34960:
34928:
34809:
34694:
34571:
34477:
34424:
34368:
34310:
34231:
34178:
34120:
34064:
33985:
33604:
33483:
33362:
33274:
33244:
33214:
33194:
33174:
33152:
33062:
33009:
32953:
32895:
32816:
32763:
32705:
32649:
32570:
32254:
32234:
32214:
32146:
32113:
32059:
32035:
31893:
31819:
31787:
31645:
31625:
31605:
31575:
31555:
31523:
31460:
31336:
31212:
31124:
31038:
30910:
30782:
30712:
30682:
30650:
30648:{\displaystyle m_{k}}
30623:
30510:
30490:
30470:
30450:
30430:
30405:
30274:
30217:
30059:
30057:{\displaystyle m_{k}}
30032:
29984:
29962:
29933:
29740:
29624:
29543:
29510:
29481:
29459:
29437:
28870:
28841:
28809:
28518:
28190:
28161:
27986:
27921:
27919:{\displaystyle \left}
27881:
27861:
27832:
27782:
27760:
27578:
27549:
27517:
27515:{\displaystyle P_{i}}
27490:
27433:
27411:
27382:
27360:
27331:
27280:
27257:
27202:
27164:
27142:
27113:
26931:
26883:
26861:
26832:
26830:{\displaystyle I_{L}}
26797:
26769:
26714:
26684:
26635:
26613:
26591:
26569:
26491:
26469:
26441:
26419:
26388:
26326:
26256:
26164:
26087:
26048:
25958:
25870:
25838:
25816:
25794:
25772:
25701:
25677:
25601:
25569:
25547:
25515:
25493:
25472:Parallel axis theorem
25466:Parallel axis theorem
25457:
25428:
25348:
25272:
25186:
25087:
24989:
24956:
24855:
24756:
24687:
24145:
23552:
22944:
22312:
21868:
21405:
21311:
21292:
21268:
21150:
21052:
20926:
20836:
20666:
20490:
20323:
20121:
19859:
19587:
19321:
19294:
19031:
18674:
18513:
18355:
18197:
18167:
17182:
16226:
16069:
15914:
15782:
15754:
15732:
15710:
15673:
15554:
15423:
15337:
15260:
15172:
15130:
14961:
14883:
14791:
14763:
14717:
14642:
14618:
14469:
14447:
14425:
14394:
14372:
14370:{\displaystyle P_{i}}
14341:
14339:{\displaystyle P_{i}}
14314:
14285:
14201:
14173:
14153:
14124:
14017:
13938:
13847:
13744:
13602:
13510:
13373:
13325:
13287:
13230:
13216:
13141:
13037:
12924:
12881:
12826:
12684:
12613:
12578:
12549:
12520:
12469:
12437:
12415:
12344:
12312:
12283:
12180:
12152:
12097:
12052:
12027:
11998:
11917:
11821:
11703:
11631:
11603:
11581:
11557:
11353:
11286:
11284:{\displaystyle \left}
11244:
11222:
11193:
11171:
11099:
11044:
11022:
10993:
10964:
10913:
10871:
10816:
10787:
10759:
10757:{\displaystyle P_{i}}
10732:
10659:
10591:
10502:
10424:
10244:
10214:
9940:
9872:and the unit vectors
9867:
9838:
9816:
9780:
9749:
9649:
9627:
9605:
9583:
9561:
9559:{\displaystyle P_{i}}
9527:
9498:
9425:
9363:
9335:
9309:
9258:
9234:
9214:
9183:
9089:
9060:
9036:
8975:
8888:
8773:
8593:
8512:
8469:
8452:second moment of mass
8441:
8410:
8353:
8279:
8248:
8162:
8080:
8053:
7964:
7886:
7750:
7679:
7635:
7294:
7221:
7192:
7170:
7141:
7116:
7094:
7072:
7050:
6874:
6845:
6823:
6772:
6745:
6712:
6696:
6647:
6216:
6194:
6103:
6083:
6063:
6041:
5972:
5964:parallel axis theorem
5949:
5929:
5784:
5761:
5719:
5520:
5474:
5450:
5448:{\displaystyle \rho }
5430:
5410:
5383:
5351:
5070:
5050:
5048:{\displaystyle \ell }
5030:
5028:{\displaystyle \rho }
5010:
4978:
4956:
4936:
4912:
4892:
4868:
4851:second moment of area
4838:
4818:
4798:
4760:
4738:
4700:
4698:{\displaystyle \rho }
4675:
4579:
4507:
4453:
4428:
4396:
4342:
4246:
4174:
4147:
4132:from the pivot point
4127:
4125:{\displaystyle r_{i}}
4100:
4098:{\displaystyle m_{i}}
4073:
4049:
4029:
3986:cylindrical ring, and
3966:
3945:
3841:
3784:
3729:
3672:
3587:
3557:
3527:
3500:second moment of area
3463:
3326:
3306:
3258:
3185:
3110:
3089:center of oscillation
3082:
3063:Center of oscillation
3054:
3034:
2928:
2908:
2884:
2864:
2844:
2824:
2764:
2762:{\displaystyle I_{P}}
2737:
2694:
2678:
2648:
2610:
2476:
2251:
2229:
2187:
2144:
2108:
2086:
2060:
2029:
1812:
1779:
1741:
1715:
1693:
1671:
1631:
1592:
1572:
1524:
1447:
1405:in terms of the mass
1389:
1320:
1265:
1244:
1157:
1137:
1117:
973:second moment of mass
918:
472:Hamiltonian mechanics
290:Statistical mechanics
209:
126:
32:Second moment of area
40550:Carl Friedrich Gauss
40483:stress–energy tensor
40478:Cauchy stress tensor
40230:Covariant derivative
40192:Antisymmetric tensor
40124:Multi-index notation
39534:
39384:angular acceleration
39156:angular displacement
38840:. pp. 360–361.
38544:Vibrations and waves
38088:Mach, Ernst (1919).
37952:
37930:
37908:
37781:
37759:
37730:
37708:
37667:
37645:
37540:
37370:
37274:
37224:
37202:
37174:
37154:
37134:
37098:yaw, pitch, and roll
37059:
37029:
36952:
36925:
36898:
36876:
36762:
36696:
36674:
36652:
36604:
36582:
36505:
36483:
36461:
36423:
36401:
36365:
36293:
36267:
36235:
36224:is the body's mass,
36208:
36112:
36090:
36057:
35749:
35510:
35126:
35097:
35020:
34991:
34969:
34949:
34821:
34711:
34587:
33292:
33253:
33223:
33203:
33183:
33163:
32266:
32243:
32223:
32155:
32128:
32068:
32046:
31902:
31853:
31798:
31657:
31634:
31614:
31584:
31564:
31534:
31139:
30728:
30699:
30661:
30632:
30521:
30499:
30479:
30459:
30439:
30419:
30228:
30072:
30041:
30021:
29971:
29942:
29519:
29490:
29468:
29448:
28879:
28850:
28818:
28529:
28202:
28170:
27995:
27930:
27893:
27870:
27841:
27791:
27769:
27587:
27558:
27526:
27499:
27442:
27420:
27391:
27369:
27340:
27289:
27269:
27211:
27173:
27151:
27122:
26892:
26870:
26841:
26814:
26778:
26723:
26693:
26650:
26622:
26600:
26578:
26503:
26478:
26450:
26428:
26399:
26059:
25847:
25825:
25803:
25781:
25710:
25688:
25578:
25556:
25527:
25502:
25480:
25437:
25360:
24967:
21301:
21279:
19302:
18178:
17197:
16241:
16233:Then, the following
15763:
15741:
15719:
15682:
15153:
14972:
14772:
14726:
14651:
14629:
14478:
14456:
14434:
14403:
14381:
14354:
14323:
14294:
14190:
14162:
14133:
14028:
13334:
13296:
13227:
12893:
12835:
12587:
12558:
12529:
12478:
12458:
12424:
12321:
12292:
12161:
12106:
12068:
12036:
12007:
11612:
11590:
11568:
11362:
11295:
11265:
11231:
11202:
11180:
11108:
11053:
11031:
11002:
10973:
10922:
10902:
10825:
10796:
10774:
10741:
10668:
10602:
10225:
9949:
9876:
9847:
9825:
9789:
9760:
9658:
9636:
9614:
9592:
9570:
9543:
9507:
9344:
9322:
9267:
9247:
9194:
9098:
9069:
9047:
8478:
8421:
8364:
8288:
8259:
8089:
8067:
7660:
7303:
7230:
7201:
7179:
7150:
7128:
7103:
7081:
7059:
6883:
6854:
6832:
6781:
6761:
6725:
6656:
6225:
6205:
6112:
6092:
6072:
6052:
5981:
5938:
5793:
5773:
5728:
5529:
5483:
5463:
5439:
5419:
5399:
5360:
5079:
5059:
5039:
5019:
4999:
4945:
4925:
4901:
4881:
4857:
4827:
4807:
4769:
4747:
4709:
4689:
4594:
4523:
4437:
4407:
4158:
4136:
4109:
4082:
4062:
4038:
4008:
3852:
3795:
3738:
3683:
3615:
3576:
3536:
3506:
3335:
3315:
3273:
3194:
3119:
3099:
3093:center of percussion
3071:
3043:
2937:
2917:
2897:
2873:
2853:
2833:
2773:
2746:
2719:
2657:
2621:
2488:
2264:
2238:
2200:
2192:also appears in the
2160:
2117:
2095:
2073:
2040:
1821:
1788:
1750:
1728:
1722:angular acceleration
1702:
1680:
1642:
1601:
1581:
1561:
1494:
1417:
1366:
1356:angular acceleration
1293:
1182:) was introduced by
1146:
1126:
1096:
1067:angular acceleration
989:angular acceleration
695:Angular acceleration
687:Rotational frequency
467:Lagrangian mechanics
460:Analytical mechanics
216:Second law of motion
169:
102:
40427:Nonmetricity tensor
40282:(2nd-order tensors)
40250:Hodge star operator
40240:Exterior derivative
40089:Transport phenomena
40074:Continuum mechanics
40030:Multilinear algebra
39474:weighted position:
39291:rotational velocity
39239:kinematic viscosity
38959:Classical mechanics
38724:Classical Mechanics
38709:, John Wiley, 2002.
38283:Winn, Will (2010).
37196:Poinsot's ellipsoid
36983:, and is a form of
36717:
36625:
36548:
36386:
34545:
34527:
34299:
34281:
34053:
34035:
33130:
33112:
32884:
32866:
32638:
32620:
31133:products of inertia
31106:
31088:
30978:
30960:
30850:
30832:
30612:
30588:
30564:
30394:
30373:
29599:
28234:
27609:
27336:, have coordinates
24608:
24581:
24424:
24397:
24240:
24213:
24109:
24082:
23878:
23851:
23647:
23620:
23519:
23429:
23335:
23193:
23099:
23009:
14178:is the total mass.
14115:
11227:is the velocity of
10473:
8822:
8344:
7935:
7099:is the velocity of
6530:
6339:
6274:
5619:
5604:
5457:rotational symmetry
5252:
5166:
4573:
4498:
4387:
3363:
3222:
2980:
1347:as the ratio of an
1281:of a system to its
547:Harmonic oscillator
525:Equations of motion
160:Classical mechanics
154:Part of a series on
40560:Tullio Levi-Civita
40503:Metric tensor (GR)
40417:Levi-Civita symbol
40270:Tensor contraction
40084:General relativity
40020:Euclidean geometry
39549:
38705:Walter D. Pilkey,
38348:. Mathematica, Inc
38197:Symon, KR (1971).
37993:
37938:
37916:
37892:
37767:
37745:
37716:
37694:
37653:
37629:
37526:
37352:
37260:
37210:
37192:
37180:
37160:
37140:
37116:rotational spectra
37071:
37035:
36965:
36938:
36911:
36884:
36860:
36851:
36748:
36699:
36682:
36660:
36646:eigendecomposition
36636:remains constant.
36626:
36607:
36590:
36566:
36530:
36491:
36469:
36447:
36409:
36387:
36368:
36335:
36275:
36241:
36214:
36194:
36098:
36072:
36027:
36018:
35964:
35805:
35717:
35488:
35112:
35083:
35006:
34977:
34955:
34923:
34804:
34689:
34566:
34564:
34553:
34531:
34513:
34285:
34267:
34039:
34021:
33968:
33790:
33269:
33239:
33209:
33189:
33169:
33147:
33138:
33116:
33098:
32870:
32852:
32624:
32606:
32553:
32397:
32249:
32229:
32209:
32141:
32108:
32054:
32030:
31888:
31814:
31782:
31650:-axis, and so on.
31640:
31620:
31600:
31570:
31550:
31518:
31516:
31119:
31117:
31092:
31074:
30964:
30946:
30836:
30818:
30707:
30677:
30645:
30618:
30592:
30568:
30544:
30505:
30485:
30465:
30445:
30425:
30400:
30374:
30353:
30212:
30203:
30054:
30027:
30003:moment of inertia
29979:
29957:
29928:
29926:
29583:
29505:
29476:
29454:
29432:
29430:
28865:
28836:
28804:
28513:
28511:
28218:
28195:is a unit vector.
28185:
28156:
27981:
27916:
27876:
27856:
27827:
27777:
27755:
27593:
27573:
27544:
27512:
27485:
27428:
27406:
27377:
27355:
27326:
27275:
27252:
27197:
27159:
27137:
27108:
26878:
26856:
26827:
26792:
26764:
26709:
26679:
26630:
26608:
26586:
26564:
26486:
26464:
26436:
26414:
26383:
26043:
25833:
25811:
25789:
25767:
25696:
25672:
25564:
25542:
25510:
25488:
25452:
25423:
25343:
25341:
24951:
24949:
24668:
24614:
24588:
24561:
24404:
24377:
24220:
24193:
24126:
24089:
24062:
23858:
23831:
23627:
23600:
23533:
23499:
23409:
23315:
23173:
23079:
22989:
22925:
22288:
22066:
21830:
21776:
21603:
21287:
21263:
21261:
19289:
19287:
18162:
18160:
17177:
17175:
16221:
16219:
15749:
15727:
15705:
15668:
15666:
15148:
15135:obtained from the
15125:
14956:
14758:
14712:
14637:
14613:
14464:
14442:
14420:
14389:
14367:
14336:
14309:
14280:
14168:
14148:
14119:
14097:
14012:
14010:
13320:
13282:
13211:
12876:
12821:
12573:
12544:
12515:
12464:
12432:
12410:
12307:
12278:
12147:
12092:
12047:
12022:
11993:
11991:
11598:
11576:
11552:
11550:
11539:
11348:
11281:
11239:
11217:
11188:
11166:
11094:
11039:
11017:
10988:
10959:
10908:
10898:Let the system of
10866:
10811:
10782:
10754:
10727:
10654:
10586:
10584:
10459:
10209:
10207:
9935:
9862:
9833:
9811:
9775:
9744:
9644:
9622:
9600:
9578:
9556:
9522:
9493:
9491:
9330:
9304:
9253:
9241:
9209:
9178:
9084:
9055:
9031:
9029:
8808:
8472:
8436:
8405:
8348:
8330:
8315:
8274:
8243:
8241:
8075:
8048:
8046:
7921:
7630:
7628:
7289:
7216:
7187:
7165:
7136:
7111:
7089:
7067:
7045:
7043:
6869:
6840:
6818:
6767:
6740:
6715:
6691:
6642:
6640:
6620:
6583:
6568:
6500:
6465:
6433:
6425:
6350:
6322:
6285:
6257:
6211:
6189:
6098:
6078:
6058:
6036:
5974:
5944:
5924:
5779:
5756:
5714:
5605:
5587:
5515:
5469:
5445:
5425:
5405:
5378:
5346:
5194:
5137:
5065:
5045:
5025:
5005:
4980:
4951:
4931:
4907:
4887:
4863:
4833:
4813:
4793:
4755:
4733:
4695:
4670:
4574:
4559:
4548:
4502:
4484:
4423:
4391:
4373:
4142:
4122:
4095:
4068:
4044:
4024:
4002:
3940:
3836:
3779:
3724:
3667:
3582:
3552:
3522:
3458:
3349:
3321:
3301:
3253:
3208:
3180:
3105:
3077:
3049:
3029:
2966:
2923:
2903:
2879:
2859:
2839:
2819:
2759:
2732:
2701:
2687:Compound pendulums
2673:
2643:
2605:
2471:
2469:
2246:
2224:
2182:
2139:
2103:
2081:
2055:
2024:
2022:
1807:
1774:
1736:
1710:
1688:
1666:
1626:
1587:
1567:
1535:radius of gyration
1519:
1442:
1384:
1315:
1268:
1251:
1192:Euler's second law
1168:Christiaan Huygens
1152:
1132:
1112:
977:rotational inertia
928:
863:Physics portal
477:Routhian mechanics
352:Frame of reference
204:
121:
40593:
40592:
40555:Hermann Grassmann
40511:
40510:
40463:Moment of inertia
40324:Differential form
40299:Affine connection
40114:Einstein notation
40097:
40096:
40025:Exterior calculus
40005:Coordinate system
39937:
39936:
39932:
39931:
39546:
39499:moment of inertia
38874:978-0-262-13396-8
38496:. Prentice Hall.
38424:Conservation Laws
38135:978-1-4297-4281-8
38071:978-0-19-883210-2
38027:Rotational energy
37988:
37987:
37926:in the direction
37624:
37623:
37595:
37594:
37566:
37565:
37505:
37502:
37456:
37453:
37407:
37404:
37183:{\displaystyle c}
37163:{\displaystyle b}
37143:{\displaystyle a}
37038:{\displaystyle m}
36331:
36217:{\displaystyle m}
35711:
35693:
35651:
35632:
35625:
35598:
35585:
35566:
35559:
35537:
35466:
35403:
35379:
35350:
35333:
35269:
35252:
35206:
35189:
35109:
35075:
35058:
35003:
34965:of a particle at
34958:{\displaystyle r}
33603:
33598:
33572:
33482:
33477:
33451:
33361:
33356:
33330:
33212:{\displaystyle z}
33192:{\displaystyle y}
33172:{\displaystyle x}
32252:{\displaystyle y}
32232:{\displaystyle x}
31953:
31643:{\displaystyle x}
31623:{\displaystyle y}
31573:{\displaystyle x}
31456:
31451:
31425:
31332:
31327:
31301:
31208:
31203:
31177:
31037:
31032:
31006:
30909:
30904:
30878:
30781:
30776:
30750:
30508:{\displaystyle z}
30488:{\displaystyle y}
30468:{\displaystyle x}
30448:{\displaystyle j}
30428:{\displaystyle i}
30273:
30268:
30246:
30030:{\displaystyle N}
29918:
29884:
29868:
29839:
29817:
29726:
29711:
29664:
29502:
29486:in the direction
29457:{\displaystyle L}
29422:
29375:
29342:
29286:
29253:
29189:
29138:
29113:
29070:
29050:
28989:
28969:
28921:
28901:
28862:
28775:
28750:
28707:
28687:
28634:
28614:
28566:
28546:
28478:
28458:
28410:
28390:
28333:
28275:
28182:
28142:
28129:
28093:
28080:
28045:
28015:
27970:
27946:
27909:
27879:{\displaystyle L}
27853:
27816:
27803:
27721:
27708:
27680:
27645:
27570:
27522:is obtained from
27482:
27403:
27278:{\displaystyle n}
27102:
27068:
27052:
27023:
27008:
26917:
26853:
26474:constructed from
24935:
21853:
21247:
21031:
20821:
20651:
20478:
20109:
19847:
19575:
19283:
18156:
18014:
17934:
17771:
17634:
17487:
17032:
16873:
16656:
16037:
15885:
15844:
15642:
15619:
15511:
15488:
15411:
15146:
14399:and acceleration
14171:{\displaystyle M}
14092:
14052:
14038:
13931:
13829:
13737:
13595:
13503:
13366:
13348:
13134:
12917:
12903:
12682:
12611:
12597:
12467:{\displaystyle n}
10911:{\displaystyle n}
10860:
10724:
10703:
10681:
10637:
10615:
10557:
10491:
10387:
10342:
10299:
10185:
10140:
10087:
10058:
10040:
10016:
9987:
9926:
9910:
9889:
9802:
9772:
9588:and acceleration
9256:{\displaystyle n}
9157:
9122:
9108:
8968:
8943:
8855:
8833:
8756:
8713:
8656:
8591:
8510:
8492:
8399:
8306:
8217:
8119:
8019:
7953:
7849:
7805:
7609:
7562:
7533:
7452:
7436:
7415:
7399:
7385:
7370:
7320:
7280:
7264:
7243:
6770:{\displaystyle n}
6737:
6719:mechanical system
6673:
6619:
6582:
6567:
6499:
6464:
6424:
6349:
6284:
6246:
6214:{\displaystyle z}
6101:{\displaystyle z}
6081:{\displaystyle z}
6061:{\displaystyle r}
5947:{\displaystyle L}
5896:
5882:
5853:
5837:
5823:
5782:{\displaystyle P}
5696:
5683:
5546:
5472:{\displaystyle z}
5428:{\displaystyle R}
5408:{\displaystyle s}
5341:
5311:
5291:
5269:
5250:
5239:
5220:
5164:
5153:
5096:
5068:{\displaystyle x}
5008:{\displaystyle s}
4984:compound pendulum
4954:{\displaystyle y}
4934:{\displaystyle x}
4910:{\displaystyle y}
4890:{\displaystyle x}
4866:{\displaystyle z}
4836:{\displaystyle Q}
4816:{\displaystyle V}
4539:
4329:
4275:
4203:
4168:
4145:{\displaystyle P}
4071:{\displaystyle N}
4047:{\displaystyle r}
3923:
3895:
3834:
3777:
3722:
3665:
3585:{\displaystyle r}
3449:
3439:
3408:
3377:
3364:
3356:
3324:{\displaystyle L}
3281:
3248:
3223:
3215:
3175:
3174:
3145:
3144:
3129:
3108:{\displaystyle L}
3080:{\displaystyle L}
3052:{\displaystyle t}
3024:
2981:
2973:
2926:{\displaystyle t}
2906:{\displaystyle P}
2882:{\displaystyle r}
2862:{\displaystyle g}
2842:{\displaystyle m}
2814:
2813:
2783:
2729:
2705:compound pendulum
2587:
2541:
2512:
2498:
2461:
2151:moment of inertia
2052:
2014:
1590:{\displaystyle r}
1570:{\displaystyle m}
1537:around the axis.
1354:on a body to the
1310:
1263:
1232:moment of inertia
1176:moment of inertia
1172:compound pendulum
1155:{\displaystyle m}
1135:{\displaystyle r}
961:moment of inertia
920:Tightrope walkers
913:
912:
660:Centrifugal force
655:Centripetal force
611:Euler's equations
596:Relative velocity
372:Moment of inertia
202:
176:
149:
148:
119:
40:Moment of inertia
16:(Redirected from
40635:
40628:Moment (physics)
40570:Bernhard Riemann
40402:
40401:
40245:Exterior product
40212:Two-point tensor
40197:Symmetric tensor
40079:Electromagnetism
39993:
39992:
39964:
39957:
39950:
39941:
39940:
39924:
39920:
39908:
39904:
39884:
39880:
39868:
39864:
39842:
39838:
39828:
39818:
39803:
39799:
39789:
39767:
39763:
39753:
39743:
39728:
39724:
39711:
39687:
39681:
39673:
39662:
39656:
39645:
39638:angular momentum
39623:
39617:
39609:
39594:
39588:
39578:
39565:
39559:
39558:
39556:
39555:
39550:
39548:
39547:
39539:
39512:
39506:
39490:
39467:
39448:
39443:
39421:
39399:
39393:
39376:
39360:
39338:
39332:
39323:angular velocity
39320:
39301:
39300:
39287:
39286:
39279:rotational speed
39275:
39274:
39257:
39246:
39231:
39221:
39202:
39201:
39184:
39179:
39165:
39153:
39133:
39114:
39113:
39107:
39101:
39090:
39064:
39044:
39029:
39013:
39006:
38971:
38970:
38952:
38945:
38938:
38929:
38928:
38886:
38885:
38883:
38881:
38858:
38852:
38851:
38829:
38823:
38822:
38820:
38818:
38797:
38788:
38782:
38781:
38775:
38766:
38760:
38757:
38751:
38744:
38738:
38737:
38719:
38710:
38703:
38697:
38696:
38678:
38655:
38651:
38645:
38644:
38626:
38620:
38613:
38607:
38600:
38594:
38593:
38575:
38558:
38557:
38539:
38533:
38532:
38514:
38508:
38507:
38489:
38474:
38473:
38453:
38447:
38446:
38427:
38417:
38411:
38410:
38390:
38384:
38383:
38381:
38379:
38364:
38358:
38357:
38355:
38353:
38337:
38331:
38330:
38310:
38301:
38300:
38280:
38271:
38270:
38262:
38241:
38240:
38222:
38213:
38212:
38194:
38185:
38184:
38172:
38162:
38145:
38140:From page 166:
38139:
38114:
38108:
38107:
38105:
38103:
38085:
38076:
38075:
38047:
38002:
38000:
37999:
37994:
37989:
37986:
37985:
37984:
37974:
37970:
37962:
37947:
37945:
37944:
37939:
37937:
37925:
37923:
37922:
37917:
37915:
37901:
37899:
37898:
37893:
37885:
37884:
37883:
37873:
37872:
37863:
37852:
37847:
37842:
37841:
37840:
37834:
37828:
37827:
37818:
37807:
37802:
37797:
37796:
37795:
37789:
37776:
37774:
37773:
37768:
37766:
37754:
37752:
37751:
37746:
37744:
37743:
37742:
37725:
37723:
37722:
37717:
37715:
37703:
37701:
37700:
37695:
37693:
37685:
37674:
37662:
37660:
37659:
37654:
37652:
37638:
37636:
37635:
37630:
37625:
37622:
37621:
37612:
37608:
37596:
37593:
37592:
37583:
37579:
37567:
37564:
37563:
37554:
37550:
37535:
37533:
37532:
37527:
37516:
37515:
37510:
37506:
37504:
37503:
37501:
37500:
37491:
37489:
37477:
37467:
37466:
37461:
37457:
37455:
37454:
37452:
37451:
37442:
37440:
37428:
37418:
37417:
37412:
37408:
37406:
37405:
37403:
37402:
37393:
37391:
37379:
37361:
37359:
37358:
37353:
37342:
37341:
37332:
37331:
37319:
37318:
37309:
37308:
37296:
37295:
37286:
37285:
37269:
37267:
37266:
37261:
37250:
37245:
37240:
37239:
37238:
37232:
37219:
37217:
37216:
37211:
37209:
37189:
37187:
37186:
37181:
37169:
37167:
37166:
37161:
37149:
37147:
37146:
37141:
37106:balancing a tire
37080:
37078:
37077:
37072:
37054:
37044:
37042:
37041:
37036:
36974:
36972:
36971:
36966:
36964:
36963:
36947:
36945:
36944:
36939:
36937:
36936:
36920:
36918:
36917:
36912:
36910:
36909:
36893:
36891:
36890:
36885:
36883:
36869:
36867:
36866:
36861:
36856:
36855:
36848:
36847:
36819:
36818:
36790:
36789:
36769:
36757:
36755:
36754:
36749:
36744:
36743:
36742:
36736:
36730:
36725:
36716:
36711:
36710:
36704:
36691:
36689:
36688:
36683:
36681:
36669:
36667:
36666:
36661:
36659:
36635:
36633:
36632:
36627:
36624:
36619:
36618:
36612:
36599:
36597:
36596:
36591:
36589:
36575:
36573:
36572:
36567:
36562:
36561:
36560:
36554:
36547:
36542:
36541:
36535:
36529:
36521:
36520:
36519:
36513:
36500:
36498:
36497:
36492:
36490:
36478:
36476:
36475:
36470:
36468:
36456:
36454:
36453:
36448:
36443:
36438:
36430:
36418:
36416:
36415:
36410:
36408:
36396:
36394:
36393:
36388:
36385:
36380:
36379:
36373:
36344:
36342:
36341:
36336:
36334:
36333:
36332:
36326:
36320:
36319:
36318:
36308:
36300:
36284:
36282:
36281:
36276:
36274:
36250:
36248:
36247:
36242:
36223:
36221:
36220:
36215:
36203:
36201:
36200:
36195:
36190:
36182:
36174:
36173:
36168:
36159:
36151:
36134:
36133:
36128:
36119:
36107:
36105:
36104:
36099:
36097:
36081:
36079:
36078:
36073:
36071:
36070:
36065:
36036:
36034:
36033:
36028:
36023:
36022:
36015:
36014:
36001:
36000:
35987:
35986:
35969:
35968:
35961:
35960:
35948:
35947:
35901:
35900:
35888:
35887:
35841:
35840:
35828:
35827:
35810:
35809:
35802:
35801:
35790:
35789:
35778:
35777:
35726:
35724:
35723:
35718:
35713:
35712:
35704:
35701:
35697:
35696:
35695:
35694:
35688:
35682:
35674:
35673:
35672:
35659:
35654:
35653:
35652:
35646:
35635:
35634:
35633:
35627:
35626:
35618:
35605:
35601:
35600:
35599:
35591:
35588:
35587:
35586:
35580:
35574:
35569:
35568:
35567:
35561:
35560:
35552:
35545:
35540:
35539:
35538:
35532:
35502:matrix transpose
35497:
35495:
35494:
35489:
35484:
35480:
35479:
35478:
35473:
35469:
35468:
35467:
35459:
35453:
35439:
35438:
35433:
35416:
35412:
35411:
35410:
35405:
35404:
35396:
35392:
35391:
35386:
35382:
35381:
35380:
35372:
35366:
35352:
35351:
35343:
35340:
35336:
35335:
35334:
35326:
35320:
35310:
35299:
35298:
35293:
35276:
35272:
35271:
35270:
35262:
35259:
35255:
35254:
35253:
35245:
35239:
35226:
35213:
35209:
35208:
35207:
35199:
35196:
35192:
35191:
35190:
35182:
35176:
35163:
35147:
35146:
35121:
35119:
35118:
35113:
35111:
35110:
35102:
35092:
35090:
35089:
35084:
35082:
35078:
35077:
35076:
35068:
35065:
35061:
35060:
35059:
35051:
35045:
35032:
35015:
35013:
35012:
35007:
35005:
35004:
34996:
34986:
34984:
34983:
34978:
34976:
34964:
34962:
34961:
34956:
34932:
34930:
34929:
34924:
34919:
34918:
34900:
34899:
34887:
34886:
34868:
34867:
34855:
34854:
34836:
34835:
34813:
34811:
34810:
34805:
34800:
34799:
34784:
34783:
34771:
34770:
34755:
34754:
34742:
34741:
34726:
34725:
34698:
34696:
34695:
34690:
34685:
34684:
34669:
34668:
34653:
34652:
34637:
34636:
34621:
34620:
34605:
34604:
34575:
34573:
34572:
34567:
34565:
34558:
34557:
34550:
34546:
34544:
34539:
34526:
34521:
34507:
34506:
34496:
34491:
34474:
34473:
34464:
34463:
34454:
34453:
34443:
34438:
34418:
34417:
34408:
34407:
34398:
34397:
34387:
34382:
34360:
34359:
34350:
34349:
34340:
34339:
34329:
34324:
34304:
34300:
34298:
34293:
34280:
34275:
34261:
34260:
34250:
34245:
34228:
34227:
34218:
34217:
34208:
34207:
34197:
34192:
34170:
34169:
34160:
34159:
34150:
34149:
34139:
34134:
34114:
34113:
34104:
34103:
34094:
34093:
34083:
34078:
34058:
34054:
34052:
34047:
34034:
34029:
34015:
34014:
34004:
33999:
33973:
33972:
33965:
33964:
33950:
33949:
33932:
33931:
33912:
33911:
33894:
33893:
33879:
33878:
33859:
33858:
33841:
33840:
33823:
33822:
33795:
33794:
33787:
33786:
33775:
33774:
33763:
33762:
33749:
33748:
33737:
33736:
33725:
33724:
33711:
33710:
33699:
33698:
33687:
33686:
33666:
33654:
33653:
33644:
33643:
33634:
33633:
33623:
33618:
33601:
33600:
33599:
33597:
33596:
33584:
33579:
33570:
33569:
33568:
33553:
33552:
33533:
33532:
33523:
33522:
33513:
33512:
33502:
33497:
33480:
33479:
33478:
33476:
33475:
33463:
33458:
33449:
33448:
33447:
33432:
33431:
33412:
33411:
33402:
33401:
33392:
33391:
33381:
33376:
33359:
33358:
33357:
33355:
33354:
33342:
33337:
33328:
33327:
33326:
33311:
33310:
33278:
33276:
33275:
33270:
33268:
33267:
33248:
33246:
33245:
33240:
33238:
33237:
33218:
33216:
33215:
33210:
33198:
33196:
33195:
33190:
33178:
33176:
33175:
33170:
33156:
33154:
33153:
33148:
33143:
33142:
33135:
33131:
33129:
33124:
33111:
33106:
33092:
33091:
33081:
33076:
33059:
33058:
33049:
33048:
33039:
33038:
33028:
33023:
33003:
33002:
32993:
32992:
32983:
32982:
32972:
32967:
32945:
32944:
32935:
32934:
32925:
32924:
32914:
32909:
32889:
32885:
32883:
32878:
32865:
32860:
32846:
32845:
32835:
32830:
32813:
32812:
32803:
32802:
32793:
32792:
32782:
32777:
32755:
32754:
32745:
32744:
32735:
32734:
32724:
32719:
32699:
32698:
32689:
32688:
32679:
32678:
32668:
32663:
32643:
32639:
32637:
32632:
32619:
32614:
32600:
32599:
32589:
32584:
32558:
32557:
32550:
32549:
32535:
32534:
32520:
32519:
32503:
32502:
32488:
32487:
32473:
32472:
32456:
32455:
32441:
32440:
32426:
32425:
32402:
32401:
32394:
32393:
32382:
32381:
32370:
32369:
32356:
32355:
32344:
32343:
32332:
32331:
32318:
32317:
32306:
32305:
32294:
32293:
32273:
32258:
32256:
32255:
32250:
32238:
32236:
32235:
32230:
32218:
32216:
32215:
32210:
32205:
32204:
32199:
32190:
32182:
32181:
32176:
32167:
32166:
32150:
32148:
32147:
32142:
32140:
32139:
32117:
32115:
32114:
32109:
32104:
32096:
32088:
32080:
32079:
32063:
32061:
32060:
32055:
32053:
32039:
32037:
32036:
32031:
32022:
32021:
32012:
32001:
31990:
31989:
31964:
31956:
31955:
31954:
31944:
31933:
31922:
31921:
31909:
31897:
31895:
31894:
31889:
31887:
31879:
31871:
31863:
31823:
31821:
31820:
31815:
31813:
31805:
31791:
31789:
31788:
31783:
31757:
31753:
31752:
31744:
31736:
31735:
31730:
31724:
31723:
31714:
31677:
31676:
31664:
31649:
31647:
31646:
31641:
31629:
31627:
31626:
31621:
31609:
31607:
31606:
31601:
31599:
31598:
31579:
31577:
31576:
31571:
31559:
31557:
31556:
31551:
31549:
31548:
31527:
31525:
31524:
31519:
31517:
31510:
31509:
31500:
31499:
31490:
31489:
31479:
31474:
31454:
31453:
31452:
31450:
31449:
31437:
31432:
31423:
31422:
31421:
31406:
31405:
31386:
31385:
31376:
31375:
31366:
31365:
31355:
31350:
31330:
31329:
31328:
31326:
31325:
31313:
31308:
31299:
31298:
31297:
31282:
31281:
31262:
31261:
31252:
31251:
31242:
31241:
31231:
31226:
31206:
31205:
31204:
31202:
31201:
31189:
31184:
31175:
31174:
31173:
31158:
31157:
31128:
31126:
31125:
31120:
31118:
31111:
31107:
31105:
31100:
31087:
31082:
31068:
31067:
31057:
31052:
31035:
31034:
31033:
31031:
31030:
31018:
31013:
31004:
31003:
31002:
30983:
30979:
30977:
30972:
30959:
30954:
30940:
30939:
30929:
30924:
30907:
30906:
30905:
30903:
30902:
30890:
30885:
30876:
30875:
30874:
30855:
30851:
30849:
30844:
30831:
30826:
30812:
30811:
30801:
30796:
30779:
30778:
30777:
30775:
30774:
30762:
30757:
30748:
30747:
30746:
30719:symmetric tensor
30716:
30714:
30713:
30708:
30706:
30686:
30684:
30683:
30678:
30676:
30675:
30654:
30652:
30651:
30646:
30644:
30643:
30627:
30625:
30624:
30619:
30617:
30613:
30611:
30600:
30587:
30576:
30563:
30552:
30535:
30534:
30529:
30514:
30512:
30511:
30506:
30494:
30492:
30491:
30486:
30474:
30472:
30471:
30466:
30454:
30452:
30451:
30446:
30434:
30432:
30431:
30426:
30409:
30407:
30406:
30401:
30399:
30395:
30393:
30382:
30372:
30361:
30349:
30348:
30336:
30335:
30330:
30326:
30325:
30320:
30304:
30303:
30293:
30288:
30271:
30270:
30269:
30267:
30266:
30254:
30249:
30244:
30243:
30242:
30221:
30219:
30218:
30213:
30208:
30207:
30200:
30199:
30188:
30187:
30176:
30175:
30162:
30161:
30150:
30149:
30138:
30137:
30124:
30123:
30112:
30111:
30100:
30099:
30079:
30063:
30061:
30060:
30055:
30053:
30052:
30036:
30034:
30033:
30028:
29988:
29986:
29985:
29980:
29978:
29966:
29964:
29963:
29958:
29956:
29955:
29954:
29937:
29935:
29934:
29929:
29927:
29920:
29919:
29911:
29908:
29907:
29906:
29900:
29894:
29893:
29892:
29886:
29885:
29877:
29870:
29869:
29861:
29858:
29857:
29856:
29850:
29841:
29840:
29832:
29823:
29819:
29818:
29810:
29807:
29803:
29802:
29801:
29796:
29792:
29791:
29790:
29785:
29770:
29769:
29759:
29754:
29728:
29727:
29719:
29713:
29712:
29704:
29701:
29700:
29695:
29691:
29690:
29689:
29684:
29666:
29665:
29657:
29654:
29653:
29643:
29638:
29614:
29610:
29609:
29604:
29600:
29598:
29593:
29588:
29573:
29572:
29562:
29557:
29535:
29534:
29514:
29512:
29511:
29506:
29504:
29503:
29495:
29485:
29483:
29482:
29477:
29475:
29463:
29461:
29460:
29455:
29441:
29439:
29438:
29433:
29431:
29424:
29423:
29415:
29412:
29411:
29406:
29402:
29401:
29400:
29395:
29377:
29376:
29368:
29358:
29349:
29345:
29344:
29343:
29335:
29329:
29328:
29323:
29311:
29310:
29305:
29288:
29287:
29279:
29269:
29260:
29256:
29255:
29254:
29246:
29240:
29239:
29234:
29214:
29210:
29209:
29208:
29203:
29191:
29190:
29182:
29172:
29163:
29159:
29158:
29157:
29152:
29140:
29139:
29131:
29120:
29116:
29115:
29114:
29106:
29100:
29096:
29095:
29091:
29090:
29089:
29084:
29072:
29071:
29063:
29052:
29051:
29043:
29028:
29019:
29015:
29014:
29010:
29009:
29008:
29003:
28991:
28990:
28982:
28971:
28970:
28962:
28951:
28947:
28946:
28942:
28941:
28940:
28935:
28923:
28922:
28914:
28903:
28902:
28894:
28885:
28875:are orthogonal:
28874:
28872:
28871:
28866:
28864:
28863:
28855:
28845:
28843:
28842:
28837:
28835:
28834:
28829:
28813:
28811:
28810:
28805:
28800:
28796:
28795:
28794:
28789:
28777:
28776:
28768:
28757:
28753:
28752:
28751:
28743:
28737:
28733:
28732:
28728:
28727:
28726:
28721:
28709:
28708:
28700:
28689:
28688:
28680:
28664:
28660:
28659:
28655:
28654:
28653:
28648:
28636:
28635:
28627:
28616:
28615:
28607:
28596:
28592:
28591:
28587:
28586:
28585:
28580:
28568:
28567:
28559:
28548:
28547:
28539:
28522:
28520:
28519:
28514:
28512:
28508:
28504:
28503:
28499:
28498:
28497:
28492:
28480:
28479:
28471:
28460:
28459:
28451:
28440:
28436:
28435:
28431:
28430:
28429:
28424:
28412:
28411:
28403:
28392:
28391:
28383:
28369:
28365:
28361:
28360:
28359:
28354:
28345:
28344:
28339:
28335:
28334:
28326:
28307:
28303:
28302:
28301:
28296:
28287:
28286:
28281:
28277:
28276:
28268:
28245:
28244:
28239:
28235:
28233:
28228:
28223:
28194:
28192:
28191:
28186:
28184:
28183:
28175:
28165:
28163:
28162:
28157:
28152:
28151:
28150:
28144:
28143:
28135:
28131:
28130:
28122:
28116:
28108:
28104:
28103:
28102:
28101:
28095:
28094:
28086:
28082:
28081:
28073:
28067:
28057:
28056:
28051:
28047:
28046:
28038:
28027:
28026:
28021:
28017:
28016:
28008:
27990:
27988:
27987:
27982:
27980:
27972:
27971:
27963:
27957:
27952:
27948:
27947:
27939:
27925:
27923:
27922:
27917:
27915:
27911:
27910:
27902:
27885:
27883:
27882:
27877:
27865:
27863:
27862:
27857:
27855:
27854:
27846:
27836:
27834:
27833:
27828:
27826:
27825:
27824:
27818:
27817:
27809:
27805:
27804:
27796:
27786:
27784:
27783:
27778:
27776:
27764:
27762:
27761:
27756:
27751:
27750:
27745:
27736:
27732:
27731:
27730:
27729:
27723:
27722:
27714:
27710:
27709:
27701:
27695:
27682:
27681:
27673:
27670:
27666:
27665:
27664:
27659:
27647:
27646:
27638:
27627:
27626:
27621:
27608:
27603:
27598:
27582:
27580:
27579:
27574:
27572:
27571:
27563:
27553:
27551:
27550:
27545:
27543:
27542:
27537:
27521:
27519:
27518:
27513:
27511:
27510:
27494:
27492:
27491:
27486:
27484:
27483:
27475:
27466:
27449:
27437:
27435:
27434:
27429:
27427:
27415:
27413:
27412:
27407:
27405:
27404:
27396:
27386:
27384:
27383:
27378:
27376:
27364:
27362:
27361:
27356:
27354:
27353:
27348:
27335:
27333:
27332:
27327:
27301:
27300:
27284:
27282:
27281:
27276:
27261:
27259:
27258:
27253:
27251:
27243:
27242:
27237:
27228:
27227:
27222:
27206:
27204:
27203:
27200:{\displaystyle }
27198:
27193:
27192:
27187:
27168:
27166:
27165:
27160:
27158:
27146:
27144:
27143:
27138:
27136:
27135:
27134:
27117:
27115:
27114:
27109:
27104:
27103:
27095:
27092:
27091:
27090:
27084:
27078:
27077:
27076:
27070:
27069:
27061:
27054:
27053:
27045:
27042:
27041:
27040:
27034:
27025:
27024:
27016:
27010:
27009:
27001:
26998:
26994:
26993:
26992:
26987:
26983:
26982:
26981:
26976:
26961:
26960:
26950:
26945:
26919:
26918:
26910:
26904:
26903:
26887:
26885:
26884:
26879:
26877:
26865:
26863:
26862:
26857:
26855:
26854:
26846:
26836:
26834:
26833:
26828:
26826:
26825:
26801:
26799:
26798:
26795:{\displaystyle }
26793:
26788:
26773:
26771:
26770:
26765:
26763:
26759:
26750:
26749:
26748:
26742:
26738:
26718:
26716:
26715:
26710:
26708:
26707:
26688:
26686:
26685:
26680:
26678:
26677:
26672:
26668:
26639:
26637:
26636:
26631:
26629:
26617:
26615:
26614:
26609:
26607:
26595:
26593:
26592:
26587:
26585:
26573:
26571:
26570:
26565:
26560:
26559:
26550:
26536:
26535:
26534:
26528:
26519:
26518:
26517:
26511:
26495:
26493:
26492:
26487:
26485:
26473:
26471:
26470:
26467:{\displaystyle }
26465:
26460:
26445:
26443:
26442:
26437:
26435:
26423:
26421:
26420:
26415:
26413:
26412:
26411:
26392:
26390:
26389:
26384:
26379:
26378:
26369:
26361:
26357:
26356:
26355:
26345:
26340:
26317:
26313:
26309:
26301:
26300:
26295:
26286:
26285:
26275:
26270:
26247:
26233:
26225:
26221:
26217:
26209:
26208:
26203:
26194:
26193:
26183:
26178:
26155:
26151:
26150:
26149:
26140:
26132:
26131:
26126:
26117:
26116:
26106:
26101:
26075:
26074:
26073:
26067:
26052:
26050:
26049:
26044:
26039:
26038:
26029:
26021:
26017:
26016:
26008:
26007:
26002:
25988:
25987:
25977:
25972:
25951:
25950:
25941:
25937:
25936:
25928:
25915:
25914:
25909:
25900:
25899:
25889:
25884:
25863:
25862:
25861:
25855:
25842:
25840:
25839:
25834:
25832:
25820:
25818:
25817:
25812:
25810:
25798:
25796:
25795:
25790:
25788:
25776:
25774:
25773:
25768:
25763:
25755:
25747:
25736:
25728:
25717:
25705:
25703:
25702:
25697:
25695:
25681:
25679:
25678:
25673:
25668:
25667:
25662:
25658:
25657:
25649:
25648:
25643:
25631:
25630:
25620:
25615:
25594:
25593:
25592:
25586:
25573:
25571:
25570:
25565:
25563:
25551:
25549:
25548:
25543:
25541:
25540:
25539:
25519:
25517:
25516:
25511:
25509:
25497:
25495:
25494:
25489:
25487:
25461:
25459:
25458:
25453:
25451:
25450:
25449:
25432:
25430:
25429:
25424:
25419:
25414:
25413:
25412:
25406:
25397:
25389:
25384:
25383:
25382:
25376:
25367:
25352:
25350:
25349:
25344:
25342:
25338:
25333:
25329:
25328:
25327:
25318:
25317:
25302:
25301:
25291:
25286:
25260:
25252:
25247:
25243:
25242:
25241:
25232:
25231:
25216:
25215:
25205:
25200:
25171:
25161:
25153:
25152:
25147:
25132:
25131:
25126:
25117:
25116:
25106:
25101:
25080:
25066:
25058:
25057:
25052:
25037:
25036:
25031:
25019:
25018:
25008:
25003:
24978:
24960:
24958:
24957:
24952:
24950:
24946:
24945:
24936:
24933:
24931:
24921:
24916:
24912:
24911:
24910:
24901:
24900:
24885:
24884:
24874:
24869:
24833:
24825:
24824:
24819:
24804:
24803:
24798:
24786:
24785:
24775:
24770:
24748:
24743:
24742:
24733:
24732:
24717:
24716:
24706:
24701:
24677:
24673:
24672:
24665:
24664:
24651:
24650:
24637:
24636:
24619:
24618:
24607:
24602:
24580:
24575:
24549:
24548:
24530:
24529:
24509:
24508:
24490:
24489:
24467:
24466:
24448:
24447:
24423:
24418:
24396:
24391:
24365:
24364:
24346:
24345:
24323:
24322:
24304:
24303:
24283:
24282:
24264:
24263:
24239:
24234:
24212:
24207:
24175:
24174:
24164:
24159:
24135:
24131:
24130:
24123:
24122:
24108:
24103:
24081:
24076:
24052:
24051:
24041:
24040:
24022:
24021:
24000:
23999:
23989:
23988:
23970:
23969:
23944:
23943:
23933:
23932:
23914:
23913:
23892:
23891:
23877:
23872:
23850:
23845:
23821:
23820:
23810:
23809:
23791:
23790:
23765:
23764:
23754:
23753:
23735:
23734:
23713:
23712:
23702:
23701:
23683:
23682:
23661:
23660:
23646:
23641:
23619:
23614:
23582:
23581:
23571:
23566:
23542:
23538:
23537:
23530:
23529:
23518:
23513:
23492:
23491:
23481:
23480:
23462:
23461:
23440:
23439:
23428:
23423:
23402:
23401:
23391:
23390:
23372:
23371:
23346:
23345:
23334:
23329:
23308:
23307:
23297:
23296:
23278:
23277:
23256:
23255:
23245:
23244:
23226:
23225:
23204:
23203:
23192:
23187:
23162:
23161:
23151:
23150:
23132:
23131:
23110:
23109:
23098:
23093:
23072:
23071:
23061:
23060:
23042:
23041:
23020:
23019:
23008:
23003:
22974:
22973:
22963:
22958:
22934:
22930:
22929:
22919:
22918:
22908:
22907:
22886:
22885:
22875:
22874:
22850:
22849:
22825:
22824:
22814:
22813:
22792:
22791:
22781:
22780:
22756:
22755:
22727:
22726:
22716:
22715:
22694:
22693:
22683:
22682:
22658:
22657:
22633:
22632:
22622:
22621:
22600:
22599:
22589:
22588:
22564:
22563:
22535:
22534:
22524:
22523:
22502:
22501:
22491:
22490:
22466:
22465:
22441:
22440:
22430:
22429:
22408:
22407:
22397:
22396:
22372:
22371:
22342:
22341:
22331:
22326:
22302:
22298:
22294:
22293:
22292:
22285:
22284:
22274:
22273:
22252:
22251:
22241:
22240:
22215:
22214:
22204:
22203:
22182:
22181:
22171:
22170:
22145:
22144:
22134:
22133:
22112:
22111:
22101:
22100:
22071:
22070:
22058:
22057:
22037:
22036:
22011:
22010:
21982:
21981:
21959:
21958:
21938:
21937:
21898:
21897:
21887:
21882:
21858:
21854:
21851:
21845:
21841:
21840:
21836:
21835:
21834:
21827:
21826:
21813:
21812:
21799:
21798:
21781:
21780:
21768:
21767:
21747:
21746:
21721:
21720:
21692:
21691:
21669:
21668:
21648:
21647:
21608:
21607:
21595:
21594:
21574:
21573:
21548:
21547:
21519:
21518:
21496:
21495:
21475:
21474:
21435:
21434:
21424:
21419:
21388:
21380:
21379:
21374:
21359:
21358:
21353:
21341:
21340:
21330:
21325:
21296:
21294:
21293:
21288:
21286:
21272:
21270:
21269:
21264:
21262:
21258:
21257:
21248:
21245:
21243:
21227:
21219:
21218:
21213:
21198:
21197:
21192:
21180:
21179:
21169:
21164:
21143:
21129:
21121:
21120:
21115:
21100:
21099:
21094:
21082:
21081:
21071:
21066:
21041:
21032:
21029:
21014:
21006:
21005:
21000:
20985:
20984:
20979:
20964:
20956:
20955:
20945:
20940:
20913:
20905:
20904:
20899:
20884:
20883:
20878:
20866:
20865:
20855:
20850:
20826:
20822:
20819:
20804:
20796:
20795:
20790:
20775:
20774:
20769:
20754:
20743:
20735:
20734:
20729:
20714:
20713:
20708:
20696:
20695:
20685:
20680:
20656:
20652:
20649:
20634:
20626:
20625:
20620:
20602:
20601:
20596:
20581:
20570:
20562:
20561:
20556:
20538:
20537:
20532:
20520:
20519:
20509:
20504:
20483:
20479:
20476:
20461:
20460:
20455:
20443:
20432:
20431:
20426:
20411:
20400:
20399:
20394:
20382:
20371:
20370:
20365:
20353:
20352:
20342:
20337:
20316:
20303:
20295:
20294:
20289:
20277:
20276:
20271:
20256:
20255:
20250:
20238:
20237:
20232:
20220:
20209:
20198:
20197:
20192:
20180:
20169:
20168:
20163:
20151:
20150:
20140:
20135:
20114:
20110:
20107:
20086:
20085:
20080:
20068:
20067:
20062:
20041:
20033:
20032:
20027:
20015:
20014:
20009:
19994:
19993:
19988:
19976:
19975:
19970:
19958:
19947:
19936:
19935:
19930:
19918:
19907:
19906:
19901:
19889:
19888:
19878:
19873:
19852:
19848:
19845:
19833:
19825:
19814:
19813:
19808:
19796:
19795:
19790:
19769:
19761:
19760:
19755:
19743:
19742:
19737:
19722:
19721:
19716:
19704:
19703:
19698:
19686:
19675:
19664:
19663:
19658:
19646:
19635:
19634:
19629:
19617:
19616:
19606:
19601:
19580:
19576:
19573:
19561:
19560:
19555:
19543:
19542:
19537:
19525:
19517:
19503:
19495:
19494:
19489:
19477:
19476:
19471:
19456:
19455:
19450:
19438:
19437:
19432:
19420:
19409:
19398:
19397:
19392:
19380:
19369:
19368:
19363:
19351:
19350:
19340:
19335:
19313:
19298:
19296:
19295:
19290:
19288:
19284:
19281:
19266:
19265:
19260:
19248:
19247:
19242:
19230:
19216:
19208:
19207:
19202:
19190:
19189:
19184:
19169:
19168:
19163:
19151:
19150:
19145:
19133:
19119:
19108:
19107:
19102:
19090:
19079:
19078:
19073:
19061:
19060:
19050:
19045:
19024:
19011:
19010:
19005:
18993:
18992:
18987:
18975:
18964:
18963:
18958:
18946:
18945:
18940:
18928:
18909:
18901:
18900:
18895:
18883:
18882:
18877:
18859:
18858:
18853:
18841:
18840:
18835:
18823:
18812:
18811:
18806:
18794:
18793:
18788:
18776:
18762:
18751:
18750:
18745:
18733:
18722:
18721:
18716:
18704:
18703:
18693:
18688:
18667:
18654:
18646:
18645:
18640:
18628:
18627:
18622:
18601:
18590:
18589:
18584:
18572:
18561:
18560:
18555:
18543:
18542:
18532:
18527:
18506:
18493:
18485:
18484:
18479:
18467:
18466:
18461:
18443:
18432:
18431:
18426:
18414:
18403:
18402:
18397:
18385:
18384:
18374:
18369:
18348:
18335:
18334:
18329:
18317:
18306:
18295:
18294:
18289:
18274:
18273:
18268:
18256:
18245:
18244:
18239:
18227:
18226:
18216:
18211:
18189:
18171:
18169:
18168:
18163:
18161:
18157:
18154:
18142:
18134:
18133:
18128:
18116:
18115:
18110:
18092:
18074:
18073:
18068:
18056:
18045:
18034:
18033:
18028:
18015:
18012:
18000:
17999:
17994:
17982:
17974:
17973:
17968:
17950:
17939:
17935:
17932:
17917:
17916:
17911:
17899:
17891:
17890:
17885:
17870:
17853:
17843:
17842:
17837:
17825:
17814:
17813:
17808:
17796:
17776:
17772:
17769:
17745:
17744:
17739:
17727:
17726:
17721:
17706:
17705:
17700:
17688:
17677:
17676:
17671:
17659:
17639:
17635:
17632:
17617:
17609:
17598:
17597:
17592:
17580:
17579:
17574:
17559:
17558:
17553:
17541:
17530:
17529:
17524:
17512:
17492:
17488:
17485:
17470:
17469:
17464:
17452:
17441:
17433:
17432:
17427:
17412:
17411:
17406:
17394:
17383:
17382:
17377:
17365:
17345:
17335:
17334:
17329:
17314:
17313:
17308:
17296:
17282:
17258:
17257:
17252:
17240:
17229:
17218:
17217:
17212:
17186:
17184:
17183:
17178:
17176:
17169:
17168:
17163:
17148:
17147:
17142:
17130:
17116:
17102:
17101:
17096:
17084:
17073:
17062:
17061:
17056:
17033:
17030:
17006:
17005:
17000:
16985:
16984:
16979:
16967:
16953:
16939:
16938:
16933:
16921:
16910:
16899:
16898:
16893:
16878:
16874:
16871:
16859:
16858:
16853:
16841:
16827:
16826:
16821:
16809:
16789:
16788:
16783:
16768:
16767:
16762:
16750:
16736:
16722:
16721:
16716:
16704:
16693:
16682:
16681:
16676:
16661:
16657:
16654:
16645:
16644:
16639:
16627:
16610:
16609:
16604:
16592:
16578:
16577:
16572:
16557:
16556:
16551:
16539:
16525:
16511:
16510:
16505:
16493:
16482:
16471:
16470:
16465:
16450:
16443:
16435:
16434:
16429:
16411:
16410:
16405:
16393:
16379:
16378:
16373:
16358:
16357:
16352:
16340:
16326:
16312:
16311:
16306:
16294:
16283:
16272:
16271:
16266:
16230:
16228:
16227:
16222:
16220:
16207:
16206:
16201:
16189:
16178:
16167:
16166:
16161:
16146:
16145:
16140:
16128:
16117:
16116:
16111:
16099:
16098:
16088:
16083:
16061:
16046:
16045:
16038:
16035:
16032:
16023:
16012:
16011:
16006:
15991:
15990:
15985:
15973:
15962:
15961:
15956:
15944:
15943:
15933:
15928:
15906:
15894:
15893:
15886:
15883:
15880:
15871:
15870:
15865:
15853:
15852:
15845:
15842:
15839:
15830:
15829:
15824:
15812:
15811:
15801:
15796:
15774:
15758:
15756:
15755:
15750:
15748:
15736:
15734:
15733:
15728:
15726:
15714:
15712:
15711:
15706:
15698:
15697:
15696:
15690:
15677:
15675:
15674:
15669:
15667:
15651:
15650:
15643:
15640:
15637:
15628:
15627:
15620:
15617:
15614:
15602:
15601:
15596:
15584:
15583:
15573:
15568:
15547:
15537:
15536:
15535:
15529:
15520:
15519:
15512:
15509:
15506:
15497:
15496:
15489:
15486:
15483:
15471:
15470:
15465:
15453:
15452:
15442:
15437:
15416:
15412:
15409:
15400:
15399:
15394:
15385:
15384:
15379:
15367:
15366:
15356:
15351:
15330:
15323:
15322:
15317:
15311:
15310:
15295:
15294:
15289:
15279:
15274:
15253:
15246:
15245:
15240:
15234:
15233:
15215:
15207:
15206:
15205:
15191:
15186:
15164:
15134:
15132:
15131:
15126:
15115:
15111:
15110:
15109:
15104:
15092:
15088:
15087:
15086:
15081:
15069:
15051:
15043:
15039:
15038:
15034:
15033:
15032:
15027:
15015:
15002:
14989:
14988:
14983:
14965:
14963:
14962:
14957:
14955:
14950:
14946:
14945:
14944:
14939:
14935:
14934:
14933:
14928:
14913:
14912:
14902:
14897:
14871:
14863:
14858:
14854:
14853:
14852:
14847:
14843:
14842:
14841:
14836:
14821:
14820:
14810:
14805:
14779:
14767:
14765:
14764:
14759:
14751:
14743:
14742:
14737:
14721:
14719:
14718:
14713:
14711:
14707:
14706:
14698:
14697:
14692:
14678:
14674:
14673:
14672:
14667:
14646:
14644:
14643:
14638:
14636:
14622:
14620:
14619:
14614:
14609:
14608:
14607:
14601:
14592:
14588:
14587:
14583:
14582:
14574:
14573:
14568:
14554:
14541:
14533:
14529:
14528:
14520:
14519:
14514:
14500:
14492:
14491:
14486:
14473:
14471:
14470:
14465:
14463:
14451:
14449:
14448:
14443:
14441:
14429:
14427:
14426:
14421:
14419:
14418:
14417:
14411:
14398:
14396:
14395:
14390:
14388:
14376:
14374:
14373:
14368:
14366:
14365:
14345:
14343:
14342:
14337:
14335:
14334:
14318:
14316:
14315:
14310:
14308:
14307:
14302:
14289:
14287:
14286:
14281:
14276:
14275:
14270:
14264:
14263:
14251:
14247:
14246:
14238:
14237:
14236:
14220:
14215:
14197:
14182:Resultant torque
14177:
14175:
14174:
14169:
14157:
14155:
14154:
14149:
14147:
14146:
14145:
14128:
14126:
14125:
14120:
14114:
14109:
14108:
14102:
14093:
14085:
14080:
14075:
14074:
14073:
14067:
14058:
14053:
14045:
14040:
14039:
14036:
14021:
14019:
14018:
14013:
14011:
14004:
14003:
14002:
13996:
13987:
13986:
13985:
13979:
13973:
13969:
13968:
13967:
13957:
13952:
13932:
13924:
13919:
13914:
13910:
13909:
13908:
13903:
13899:
13898:
13897:
13892:
13877:
13876:
13866:
13861:
13835:
13830:
13822:
13814:
13810:
13809:
13808:
13802:
13793:
13792:
13791:
13785:
13779:
13775:
13774:
13773:
13763:
13758:
13738:
13730:
13725:
13721:
13720:
13716:
13715:
13710:
13706:
13705:
13704:
13699:
13685:
13684:
13683:
13677:
13673:
13672:
13671:
13666:
13651:
13650:
13649:
13643:
13632:
13631:
13621:
13616:
13596:
13588:
13580:
13576:
13575:
13574:
13568:
13559:
13558:
13557:
13551:
13545:
13541:
13540:
13539:
13529:
13524:
13504:
13496:
13491:
13487:
13486:
13482:
13481:
13476:
13472:
13471:
13470:
13465:
13443:
13439:
13438:
13433:
13429:
13428:
13427:
13422:
13403:
13402:
13392:
13387:
13367:
13359:
13350:
13349:
13346:
13329:
13327:
13326:
13323:{\displaystyle }
13321:
13316:
13315:
13310:
13291:
13289:
13288:
13283:
13275:
13274:
13269:
13260:
13259:
13249:
13244:
13220:
13218:
13217:
13212:
13207:
13203:
13202:
13201:
13200:
13194:
13185:
13184:
13183:
13177:
13171:
13170:
13160:
13155:
13135:
13127:
13122:
13118:
13117:
13113:
13112:
13111:
13106:
13094:
13081:
13080:
13079:
13073:
13067:
13066:
13056:
13051:
13028:
13024:
13023:
13019:
13018:
13017:
13012:
13000:
12987:
12983:
12982:
12981:
12976:
12964:
12954:
12953:
12943:
12938:
12918:
12910:
12905:
12904:
12901:
12885:
12883:
12882:
12877:
12875:
12867:
12866:
12861:
12852:
12851:
12846:
12830:
12828:
12827:
12822:
12817:
12813:
12812:
12811:
12810:
12804:
12795:
12794:
12789:
12777:
12764:
12760:
12759:
12758:
12757:
12751:
12742:
12741:
12736:
12724:
12714:
12713:
12703:
12698:
12683:
12675:
12670:
12669:
12664:
12655:
12654:
12649:
12643:
12642:
12632:
12627:
12612:
12604:
12599:
12598:
12595:
12582:
12580:
12579:
12574:
12572:
12571:
12566:
12554:with velocities
12553:
12551:
12550:
12545:
12543:
12542:
12537:
12524:
12522:
12521:
12516:
12490:
12489:
12473:
12471:
12470:
12465:
12441:
12439:
12438:
12433:
12431:
12419:
12417:
12416:
12411:
12406:
12405:
12400:
12396:
12395:
12394:
12389:
12374:
12373:
12363:
12358:
12337:
12336:
12335:
12329:
12316:
12314:
12313:
12308:
12306:
12305:
12304:
12287:
12285:
12284:
12279:
12274:
12269:
12268:
12267:
12261:
12252:
12247:
12243:
12242:
12241:
12236:
12232:
12231:
12230:
12225:
12210:
12209:
12199:
12194:
12168:
12156:
12154:
12153:
12148:
12146:
12138:
12137:
12132:
12123:
12122:
12117:
12101:
12099:
12098:
12095:{\displaystyle }
12093:
12088:
12087:
12082:
12056:
12054:
12053:
12048:
12046:
12031:
12029:
12028:
12023:
12021:
12020:
12019:
12002:
12000:
11999:
11994:
11992:
11985:
11981:
11980:
11979:
11978:
11972:
11963:
11962:
11957:
11947:
11946:
11936:
11931:
11908:
11904:
11903:
11899:
11898:
11890:
11889:
11884:
11867:
11866:
11861:
11851:
11850:
11840:
11835:
11806:
11802:
11798:
11797:
11796:
11795:
11789:
11780:
11779:
11774:
11762:
11749:
11748:
11743:
11733:
11732:
11722:
11717:
11696:
11692:
11691:
11686:
11677:
11676:
11671:
11661:
11660:
11650:
11645:
11623:
11607:
11605:
11604:
11599:
11597:
11585:
11583:
11582:
11577:
11575:
11561:
11559:
11558:
11553:
11551:
11544:
11543:
11531:
11530:
11519:
11518:
11502:
11501:
11482:
11481:
11468:
11467:
11456:
11455:
11423:
11419:
11406:
11401:
11397:
11381:
11373:
11357:
11355:
11354:
11349:
11344:
11343:
11331:
11330:
11318:
11317:
11302:
11290:
11288:
11287:
11282:
11280:
11276:
11253:Angular momentum
11248:
11246:
11245:
11240:
11238:
11226:
11224:
11223:
11218:
11216:
11215:
11214:
11197:
11195:
11194:
11189:
11187:
11175:
11173:
11172:
11167:
11165:
11164:
11163:
11157:
11148:
11147:
11142:
11130:
11122:
11121:
11116:
11103:
11101:
11100:
11095:
11093:
11085:
11084:
11079:
11070:
11069:
11064:
11048:
11046:
11045:
11040:
11038:
11026:
11024:
11023:
11018:
11016:
11015:
11010:
10998:with velocities
10997:
10995:
10994:
10989:
10987:
10986:
10981:
10968:
10966:
10965:
10960:
10934:
10933:
10917:
10915:
10914:
10909:
10875:
10873:
10872:
10867:
10862:
10861:
10853:
10847:
10846:
10845:
10832:
10820:
10818:
10817:
10812:
10810:
10809:
10808:
10791:
10789:
10788:
10783:
10781:
10763:
10761:
10760:
10755:
10753:
10752:
10736:
10734:
10733:
10728:
10726:
10725:
10717:
10711:
10710:
10705:
10704:
10696:
10689:
10688:
10683:
10682:
10674:
10663:
10661:
10660:
10655:
10653:
10645:
10644:
10639:
10638:
10630:
10623:
10622:
10617:
10616:
10608:
10595:
10593:
10592:
10587:
10585:
10578:
10570:
10566:
10565:
10564:
10559:
10558:
10550:
10546:
10545:
10532:
10531:
10521:
10516:
10493:
10492:
10484:
10478:
10474:
10472:
10467:
10454:
10453:
10443:
10438:
10412:
10408:
10404:
10403:
10395:
10394:
10389:
10388:
10380:
10376:
10375:
10363:
10362:
10350:
10349:
10344:
10343:
10335:
10331:
10330:
10307:
10306:
10301:
10300:
10292:
10288:
10287:
10274:
10273:
10263:
10258:
10236:
10218:
10216:
10215:
10210:
10208:
10201:
10193:
10192:
10187:
10186:
10178:
10174:
10173:
10161:
10160:
10148:
10147:
10142:
10141:
10133:
10129:
10128:
10107:
10103:
10095:
10094:
10089:
10088:
10080:
10076:
10075:
10060:
10059:
10051:
10042:
10041:
10033:
10024:
10023:
10018:
10017:
10009:
10005:
10004:
9989:
9988:
9980:
9967:
9966:
9961:
9944:
9942:
9941:
9936:
9934:
9933:
9928:
9927:
9919:
9912:
9911:
9903:
9897:
9896:
9891:
9890:
9882:
9871:
9869:
9868:
9863:
9861:
9860:
9855:
9842:
9840:
9839:
9834:
9832:
9820:
9818:
9817:
9812:
9810:
9809:
9804:
9803:
9795:
9784:
9782:
9781:
9776:
9774:
9773:
9765:
9753:
9751:
9750:
9745:
9740:
9732:
9731:
9726:
9714:
9706:
9698:
9697:
9692:
9680:
9672:
9671:
9666:
9653:
9651:
9650:
9645:
9643:
9631:
9629:
9628:
9623:
9621:
9609:
9607:
9606:
9601:
9599:
9587:
9585:
9584:
9579:
9577:
9565:
9563:
9562:
9557:
9555:
9554:
9531:
9529:
9528:
9523:
9521:
9520:
9515:
9502:
9500:
9499:
9494:
9492:
9485:
9484:
9479:
9473:
9472:
9460:
9459:
9454:
9444:
9439:
9417:
9405:
9404:
9399:
9393:
9392:
9382:
9377:
9355:
9339:
9337:
9336:
9331:
9329:
9313:
9311:
9310:
9305:
9279:
9278:
9262:
9260:
9259:
9254:
9218:
9216:
9215:
9210:
9208:
9207:
9206:
9187:
9185:
9184:
9179:
9174:
9166:
9158:
9150:
9145:
9144:
9135:
9134:
9133:
9123:
9115:
9110:
9109:
9106:
9093:
9091:
9090:
9085:
9083:
9082:
9081:
9064:
9062:
9061:
9056:
9054:
9040:
9038:
9037:
9032:
9030:
9023:
9015:
9010:
9006:
9005:
9004:
8994:
8989:
8969:
8961:
8956:
8952:
8951:
8950:
8945:
8944:
8936:
8932:
8931:
8918:
8917:
8907:
8902:
8879:
8868:
8864:
8863:
8862:
8857:
8856:
8848:
8841:
8840:
8835:
8834:
8826:
8821:
8816:
8803:
8802:
8792:
8787:
8767:
8766:
8757:
8749:
8741:
8734:
8730:
8729:
8721:
8720:
8715:
8714:
8706:
8702:
8701:
8677:
8673:
8672:
8664:
8663:
8658:
8657:
8649:
8645:
8644:
8623:
8622:
8612:
8607:
8592:
8584:
8576:
8569:
8568:
8563:
8554:
8553:
8548:
8542:
8541:
8531:
8526:
8511:
8503:
8494:
8493:
8490:
8445:
8443:
8442:
8437:
8435:
8434:
8433:
8414:
8412:
8411:
8406:
8401:
8400:
8392:
8386:
8385:
8384:
8371:
8357:
8355:
8354:
8349:
8343:
8338:
8325:
8324:
8314:
8302:
8301:
8300:
8283:
8281:
8280:
8275:
8273:
8272:
8271:
8252:
8250:
8249:
8244:
8242:
8225:
8224:
8219:
8218:
8210:
8206:
8205:
8192:
8191:
8181:
8176:
8154:
8146:
8145:
8140:
8127:
8126:
8121:
8120:
8112:
8108:
8107:
8084:
8082:
8081:
8076:
8074:
8057:
8055:
8054:
8049:
8047:
8040:
8032:
8028:
8027:
8026:
8021:
8020:
8012:
8008:
8007:
7994:
7993:
7983:
7978:
7955:
7954:
7946:
7940:
7936:
7934:
7929:
7916:
7915:
7905:
7900:
7874:
7870:
7866:
7865:
7857:
7856:
7851:
7850:
7842:
7838:
7837:
7813:
7812:
7807:
7806:
7798:
7794:
7793:
7780:
7779:
7769:
7764:
7743:
7739:
7738:
7733:
7724:
7723:
7718:
7709:
7708:
7698:
7693:
7671:
7652:Angular momentum
7639:
7637:
7636:
7631:
7629:
7625:
7617:
7616:
7611:
7610:
7602:
7598:
7597:
7578:
7570:
7569:
7564:
7563:
7555:
7551:
7550:
7535:
7534:
7526:
7517:
7509:
7508:
7503:
7491:
7479:
7478:
7473:
7460:
7459:
7454:
7453:
7445:
7438:
7437:
7429:
7423:
7422:
7417:
7416:
7408:
7400:
7392:
7387:
7386:
7378:
7371:
7369:
7368:
7367:
7354:
7353:
7352:
7347:
7337:
7328:
7327:
7322:
7321:
7313:
7298:
7296:
7295:
7290:
7288:
7287:
7282:
7281:
7273:
7266:
7265:
7257:
7251:
7250:
7245:
7244:
7236:
7225:
7223:
7222:
7217:
7215:
7214:
7209:
7196:
7194:
7193:
7188:
7186:
7174:
7172:
7171:
7166:
7164:
7163:
7158:
7145:
7143:
7142:
7137:
7135:
7120:
7118:
7117:
7112:
7110:
7098:
7096:
7095:
7090:
7088:
7076:
7074:
7073:
7068:
7066:
7054:
7052:
7051:
7046:
7044:
7037:
7029:
7028:
7023:
7011:
7003:
6995:
6991:
6990:
6982:
6981:
6976:
6962:
6950:
6949:
6944:
6931:
6923:
6922:
6917:
6904:
6903:
6898:
6878:
6876:
6875:
6870:
6868:
6867:
6862:
6849:
6847:
6846:
6841:
6839:
6827:
6825:
6824:
6819:
6793:
6792:
6776:
6774:
6773:
6768:
6749:
6747:
6746:
6741:
6739:
6738:
6730:
6700:
6698:
6697:
6692:
6687:
6686:
6674:
6666:
6651:
6649:
6648:
6643:
6641:
6634:
6633:
6621:
6612:
6603:
6599:
6598:
6589:
6585:
6584:
6575:
6569:
6560:
6534:
6529:
6524:
6516:
6512:
6511:
6510:
6501:
6492:
6486:
6485:
6476:
6475:
6466:
6457:
6448:
6447:
6426:
6417:
6408:
6397:
6396:
6391:
6387:
6386:
6385:
6373:
6372:
6351:
6342:
6338:
6333:
6311:
6310:
6286:
6277:
6273:
6268:
6249:
6248:
6247:
6244:
6220:
6218:
6217:
6212:
6198:
6196:
6195:
6190:
6185:
6184:
6172:
6171:
6159:
6158:
6146:
6145:
6133:
6132:
6107:
6105:
6104:
6099:
6087:
6085:
6084:
6079:
6067:
6065:
6064:
6059:
6045:
6043:
6042:
6037:
6032:
6031:
6019:
6018:
6006:
6005:
5993:
5992:
5953:
5951:
5950:
5945:
5933:
5931:
5930:
5925:
5920:
5919:
5898:
5897:
5894:
5885:
5884:
5883:
5880:
5864:
5863:
5858:
5854:
5846:
5839:
5838:
5835:
5826:
5825:
5824:
5821:
5805:
5804:
5788:
5786:
5785:
5780:
5765:
5763:
5762:
5757:
5749:
5748:
5723:
5721:
5720:
5715:
5710:
5709:
5697:
5689:
5684:
5679:
5678:
5669:
5632:
5631:
5618:
5613:
5603:
5595:
5576:
5575:
5562:
5561:
5549:
5548:
5547:
5544:
5524:
5522:
5521:
5516:
5478:
5476:
5475:
5470:
5454:
5452:
5451:
5446:
5434:
5432:
5431:
5426:
5414:
5412:
5411:
5406:
5387:
5385:
5384:
5379:
5355:
5353:
5352:
5347:
5342:
5337:
5336:
5335:
5322:
5317:
5313:
5312:
5307:
5306:
5297:
5292:
5287:
5286:
5277:
5270:
5265:
5257:
5251:
5243:
5241:
5240:
5232:
5226:
5222:
5221:
5216:
5215:
5206:
5180:
5179:
5165:
5157:
5155:
5154:
5146:
5126:
5125:
5112:
5111:
5099:
5098:
5097:
5094:
5074:
5072:
5071:
5066:
5054:
5052:
5051:
5046:
5035:and with length
5034:
5032:
5031:
5026:
5014:
5012:
5011:
5006:
4960:
4958:
4957:
4952:
4940:
4938:
4937:
4932:
4916:
4914:
4913:
4908:
4896:
4894:
4893:
4888:
4872:
4870:
4869:
4864:
4842:
4840:
4839:
4834:
4822:
4820:
4819:
4814:
4802:
4800:
4799:
4794:
4764:
4762:
4761:
4756:
4754:
4742:
4740:
4739:
4734:
4704:
4702:
4701:
4696:
4679:
4677:
4676:
4671:
4663:
4662:
4657:
4653:
4619:
4618:
4606:
4605:
4583:
4581:
4580:
4575:
4572:
4567:
4558:
4557:
4547:
4535:
4534:
4511:
4509:
4508:
4503:
4497:
4492:
4483:
4482:
4472:
4467:
4449:
4448:
4432:
4430:
4429:
4424:
4422:
4421:
4400:
4398:
4397:
4392:
4386:
4381:
4372:
4371:
4361:
4356:
4341:
4340:
4330:
4322:
4317:
4316:
4311:
4307:
4306:
4305:
4287:
4286:
4276:
4268:
4265:
4260:
4242:
4241:
4236:
4227:
4226:
4221:
4215:
4214:
4204:
4196:
4193:
4188:
4170:
4169:
4166:
4151:
4149:
4148:
4143:
4131:
4129:
4128:
4123:
4121:
4120:
4104:
4102:
4101:
4096:
4094:
4093:
4077:
4075:
4074:
4069:
4053:
4051:
4050:
4045:
4033:
4031:
4030:
4025:
4023:
4022:
3991:
3985:
3979:
3974:spherical shell,
3973:
3949:
3947:
3946:
3941:
3939:
3938:
3929:
3924:
3916:
3911:
3910:
3901:
3896:
3888:
3883:
3882:
3867:
3866:
3845:
3843:
3842:
3837:
3835:
3830:
3829:
3828:
3815:
3810:
3809:
3788:
3786:
3785:
3780:
3778:
3773:
3772:
3771:
3758:
3753:
3752:
3733:
3731:
3730:
3725:
3723:
3718:
3717:
3716:
3703:
3698:
3697:
3676:
3674:
3673:
3668:
3666:
3661:
3660:
3651:
3646:
3645:
3630:
3629:
3607:
3606:
3602:
3591:
3589:
3588:
3583:
3561:
3559:
3558:
3553:
3551:
3550:
3531:
3529:
3528:
3523:
3521:
3520:
3472:Kater's pendulum
3467:
3465:
3464:
3459:
3454:
3447:
3440:
3438:
3437:
3436:
3427:
3423:
3406:
3398:
3397:
3396:
3395:
3386:
3375:
3370:
3365:
3362:
3357:
3354:
3345:
3330:
3328:
3327:
3322:
3310:
3308:
3307:
3302:
3300:
3296:
3279:
3267:seconds pendulum
3262:
3260:
3259:
3254:
3249:
3247:
3239:
3238:
3229:
3224:
3221:
3216:
3213:
3204:
3189:
3187:
3186:
3181:
3176:
3173:
3172:
3163:
3152:
3151:
3146:
3137:
3136:
3131:
3130:
3127:
3114:
3112:
3111:
3106:
3086:
3084:
3083:
3078:
3058:
3056:
3055:
3050:
3038:
3036:
3035:
3030:
3025:
3023:
3022:
3021:
3008:
3007:
3006:
2987:
2982:
2979:
2974:
2971:
2965:
2954:
2949:
2948:
2932:
2930:
2929:
2924:
2912:
2910:
2909:
2904:
2888:
2886:
2885:
2880:
2868:
2866:
2865:
2860:
2848:
2846:
2845:
2840:
2828:
2826:
2825:
2820:
2815:
2812:
2811:
2802:
2791:
2790:
2785:
2784:
2781:
2768:
2766:
2765:
2760:
2758:
2757:
2741:
2739:
2738:
2733:
2731:
2730:
2727:
2682:
2680:
2679:
2674:
2672:
2671:
2652:
2650:
2649:
2644:
2642:
2641:
2614:
2612:
2611:
2606:
2601:
2600:
2588:
2580:
2575:
2574:
2565:
2561:
2560:
2559:
2542:
2534:
2529:
2521:
2513:
2505:
2500:
2499:
2496:
2480:
2478:
2477:
2472:
2470:
2463:
2462:
2454:
2442:
2437:
2436:
2418:
2414:
2410:
2409:
2404:
2400:
2399:
2391:
2378:
2373:
2369:
2368:
2360:
2336:
2332:
2328:
2327:
2319:
2303:
2295:
2287:
2275:
2258:angular velocity
2255:
2253:
2252:
2247:
2245:
2233:
2231:
2230:
2225:
2223:
2215:
2207:
2194:angular momentum
2191:
2189:
2188:
2183:
2181:
2180:
2148:
2146:
2145:
2140:
2138:
2137:
2113:.) The quantity
2112:
2110:
2109:
2104:
2102:
2090:
2088:
2087:
2082:
2080:
2064:
2062:
2061:
2056:
2054:
2053:
2045:
2033:
2031:
2030:
2025:
2023:
2016:
2015:
2007:
1995:
1990:
1989:
1971:
1967:
1963:
1962:
1957:
1953:
1952:
1944:
1931:
1926:
1922:
1921:
1913:
1889:
1882:
1874:
1860:
1852:
1844:
1832:
1816:
1814:
1813:
1808:
1806:
1795:
1783:
1781:
1780:
1775:
1773:
1765:
1757:
1745:
1743:
1742:
1737:
1735:
1719:
1717:
1716:
1711:
1709:
1697:
1695:
1694:
1689:
1687:
1675:
1673:
1672:
1667:
1665:
1657:
1649:
1635:
1633:
1632:
1627:
1622:
1621:
1596:
1594:
1593:
1588:
1576:
1574:
1573:
1568:
1533:is known as the
1532:
1528:
1526:
1525:
1520:
1515:
1514:
1489:
1485:
1478:
1474:
1470:
1461:
1457:
1451:
1449:
1448:
1443:
1438:
1437:
1412:
1408:
1404:
1394:
1391:
1390:
1385:
1360:
1353:
1325:
1322:
1321:
1316:
1311:
1303:
1287:
1283:angular velocity
1280:
1276:angular momentum
1273:
1264:
1238:of the section.
1161:
1159:
1158:
1153:
1141:
1139:
1138:
1133:
1121:
1119:
1118:
1113:
1111:
1110:
1071:angular velocity
1063:angular momentum
947:
938:
905:
898:
891:
878:
873:
872:
865:
861:
860:
766:Johann Bernoulli
761:Daniel Bernoulli
682:Tangential speed
586:
562:
537:Fictitious force
532:
384:Mechanical power
374:
315:Angular momentum
213:
211:
210:
205:
203:
201:
193:
192:
183:
178:
177:
151:
150:
130:
128:
127:
122:
120:
112:
95:other quantities
93:Derivations from
49:
37:
36:
21:
40643:
40642:
40638:
40637:
40636:
40634:
40633:
40632:
40603:
40602:
40594:
40589:
40540:Albert Einstein
40507:
40488:Einstein tensor
40451:
40432:Ricci curvature
40412:Kronecker delta
40398:Notable tensors
40393:
40314:Connection form
40291:
40285:
40216:
40202:Tensor operator
40159:
40153:
40093:
40069:Computer vision
40062:
40044:
40040:Tensor calculus
39984:
39973:
39968:
39938:
39933:
39922:
39921:
39916:
39906:
39905:
39900:
39882:
39881:
39876:
39866:
39865:
39860:
39840:
39839:
39834:
39824:
39814:
39801:
39800:
39795:
39783:
39765:
39764:
39759:
39749:
39739:
39726:
39725:
39723:
39717:
39707:
39685:
39682:
39679:
39671:
39660:
39657:
39651:
39648:angular impulse
39641:
39621:
39618:
39615:
39607:
39592:
39589:
39584:
39574:
39563:
39560:
39538:
39537:
39535:
39532:
39531:
39530:
39510:
39507:
39502:
39475:
39468:
39463:
39446:
39444:
39437:
39422:
39417:
39397:
39394:
39387:
39377:
39372:
39361:
39356:
39336:
39333:
39326:
39316:
39302:
39294:
39289:
39282:
39277:
39270:
39265:
39258:
39253:
39248:
39242:
39232:
39227:
39217:
39203:
39197:
39192:
39180:
39177:
39172:
39166:
39159:
39149:
39134:
39129:
39119:
39109:
39103:
39097:
39092:
39086:
39065:
39060:
39045:
39040:
39030:
39025:
39009:
39002:
38964:
38956:
38894:
38889:
38879:
38877:
38875:
38859:
38855:
38848:
38830:
38826:
38816:
38814:
38806:(23): 138–142.
38795:
38789:
38785:
38773:
38769:David, Baraff.
38767:
38763:
38758:
38754:
38745:
38741:
38734:
38720:
38713:
38704:
38700:
38693:
38679:
38658:
38652:
38648:
38641:
38627:
38623:
38614:
38610:
38601:
38597:
38590:
38576:
38561:
38554:
38540:
38536:
38529:
38515:
38511:
38504:
38490:
38477:
38470:
38454:
38450:
38440:
38418:
38414:
38407:
38391:
38387:
38377:
38375:
38365:
38361:
38351:
38349:
38338:
38334:
38327:
38311:
38304:
38297:
38281:
38274:
38263:
38244:
38237:
38223:
38216:
38209:
38195:
38188:
38181:
38163:
38148:
38136:
38115:
38111:
38101:
38099:
38086:
38079:
38072:
38048:
38044:
38040:
38008:
37980:
37979:
37975:
37969:
37958:
37953:
37950:
37949:
37933:
37931:
37928:
37927:
37911:
37909:
37906:
37905:
37879:
37878:
37874:
37868:
37864:
37859:
37848:
37843:
37836:
37835:
37830:
37829:
37823:
37819:
37814:
37803:
37798:
37791:
37790:
37785:
37784:
37782:
37779:
37778:
37762:
37760:
37757:
37756:
37738:
37737:
37733:
37731:
37728:
37727:
37711:
37709:
37706:
37705:
37689:
37681:
37670:
37668:
37665:
37664:
37648:
37646:
37643:
37642:
37617:
37613:
37607:
37588:
37584:
37578:
37559:
37555:
37549:
37541:
37538:
37537:
37511:
37496:
37492:
37490:
37485:
37481:
37476:
37472:
37471:
37462:
37447:
37443:
37441:
37436:
37432:
37427:
37423:
37422:
37413:
37398:
37394:
37392:
37387:
37383:
37378:
37374:
37373:
37371:
37368:
37367:
37337:
37333:
37327:
37323:
37314:
37310:
37304:
37300:
37291:
37287:
37281:
37277:
37275:
37272:
37271:
37246:
37241:
37234:
37233:
37228:
37227:
37225:
37222:
37221:
37205:
37203:
37200:
37199:
37175:
37172:
37171:
37155:
37152:
37151:
37135:
37132:
37131:
37124:
37060:
37057:
37056:
37046:
37030:
37027:
37026:
36975:are called the
36959:
36955:
36953:
36950:
36949:
36932:
36928:
36926:
36923:
36922:
36905:
36901:
36899:
36896:
36895:
36879:
36877:
36874:
36873:
36850:
36849:
36843:
36839:
36837:
36832:
36826:
36825:
36820:
36814:
36810:
36808:
36802:
36801:
36796:
36791:
36785:
36781:
36774:
36773:
36765:
36763:
36760:
36759:
36738:
36737:
36732:
36731:
36726:
36721:
36712:
36706:
36705:
36700:
36697:
36694:
36693:
36677:
36675:
36672:
36671:
36655:
36653:
36650:
36649:
36642:
36620:
36614:
36613:
36608:
36605:
36602:
36601:
36585:
36583:
36580:
36579:
36556:
36555:
36550:
36549:
36543:
36537:
36536:
36531:
36525:
36515:
36514:
36509:
36508:
36506:
36503:
36502:
36486:
36484:
36481:
36480:
36464:
36462:
36459:
36458:
36439:
36434:
36426:
36424:
36421:
36420:
36404:
36402:
36399:
36398:
36381:
36375:
36374:
36369:
36366:
36363:
36362:
36359:
36350:
36328:
36327:
36322:
36321:
36314:
36310:
36309:
36304:
36296:
36294:
36291:
36290:
36270:
36268:
36265:
36264:
36261:
36236:
36233:
36232:
36230:
36209:
36206:
36205:
36186:
36178:
36169:
36164:
36163:
36155:
36147:
36129:
36124:
36123:
36115:
36113:
36110:
36109:
36093:
36091:
36088:
36087:
36066:
36061:
36060:
36058:
36055:
36054:
36051:
36045:
36017:
36016:
36010:
36006:
36003:
36002:
35996:
35992:
35989:
35988:
35982:
35978:
35971:
35970:
35963:
35962:
35956:
35952:
35943:
35939:
35937:
35926:
35914:
35913:
35902:
35896:
35892:
35883:
35879:
35877:
35865:
35864:
35853:
35842:
35836:
35832:
35823:
35819:
35812:
35811:
35804:
35803:
35797:
35793:
35791:
35785:
35781:
35779:
35773:
35769:
35762:
35761:
35750:
35747:
35746:
35738:identity matrix
35735:
35703:
35702:
35690:
35689:
35684:
35683:
35678:
35668:
35664:
35663:
35655:
35648:
35647:
35642:
35641:
35640:
35636:
35629:
35628:
35617:
35616:
35615:
35590:
35589:
35582:
35581:
35576:
35575:
35570:
35563:
35562:
35551:
35550:
35549:
35541:
35534:
35533:
35528:
35527:
35526:
35522:
35511:
35508:
35507:
35474:
35458:
35457:
35449:
35448:
35444:
35443:
35434:
35429:
35428:
35427:
35423:
35406:
35395:
35394:
35393:
35387:
35371:
35370:
35362:
35361:
35357:
35356:
35342:
35341:
35325:
35324:
35316:
35315:
35311:
35306:
35294:
35289:
35288:
35287:
35283:
35261:
35260:
35244:
35243:
35235:
35234:
35230:
35222:
35221:
35217:
35198:
35197:
35181:
35180:
35172:
35171:
35167:
35159:
35158:
35154:
35142:
35138:
35127:
35124:
35123:
35101:
35100:
35098:
35095:
35094:
35067:
35066:
35050:
35049:
35041:
35040:
35036:
35028:
35027:
35023:
35021:
35018:
35017:
34995:
34994:
34992:
34989:
34988:
34972:
34970:
34967:
34966:
34950:
34947:
34946:
34943:
34911:
34907:
34895:
34891:
34879:
34875:
34863:
34859:
34847:
34843:
34831:
34827:
34822:
34819:
34818:
34792:
34788:
34779:
34775:
34763:
34759:
34750:
34746:
34734:
34730:
34721:
34717:
34712:
34709:
34708:
34677:
34673:
34661:
34657:
34645:
34641:
34629:
34625:
34613:
34609:
34597:
34593:
34588:
34585:
34584:
34581:
34563:
34562:
34552:
34551:
34540:
34535:
34522:
34517:
34512:
34508:
34502:
34498:
34492:
34481:
34475:
34469:
34465:
34459:
34455:
34449:
34445:
34439:
34428:
34419:
34413:
34409:
34403:
34399:
34393:
34389:
34383:
34372:
34362:
34361:
34355:
34351:
34345:
34341:
34335:
34331:
34325:
34314:
34305:
34294:
34289:
34276:
34271:
34266:
34262:
34256:
34252:
34246:
34235:
34229:
34223:
34219:
34213:
34209:
34203:
34199:
34193:
34182:
34172:
34171:
34165:
34161:
34155:
34151:
34145:
34141:
34135:
34124:
34115:
34109:
34105:
34099:
34095:
34089:
34085:
34079:
34068:
34059:
34048:
34043:
34030:
34025:
34020:
34016:
34010:
34006:
34000:
33989:
33978:
33977:
33967:
33966:
33957:
33953:
33951:
33942:
33938:
33933:
33924:
33920:
33914:
33913:
33904:
33900:
33895:
33886:
33882:
33880:
33871:
33867:
33861:
33860:
33851:
33847:
33842:
33833:
33829:
33824:
33815:
33811:
33804:
33803:
33796:
33789:
33788:
33782:
33778:
33776:
33770:
33766:
33764:
33758:
33754:
33751:
33750:
33744:
33740:
33738:
33732:
33728:
33726:
33720:
33716:
33713:
33712:
33706:
33702:
33700:
33694:
33690:
33688:
33682:
33678:
33671:
33670:
33662:
33659:
33658:
33649:
33645:
33639:
33635:
33629:
33625:
33619:
33608:
33586:
33585:
33580:
33578:
33577:
33573:
33561:
33557:
33545:
33541:
33538:
33537:
33528:
33524:
33518:
33514:
33508:
33504:
33498:
33487:
33465:
33464:
33459:
33457:
33456:
33452:
33440:
33436:
33424:
33420:
33417:
33416:
33407:
33403:
33397:
33393:
33387:
33383:
33377:
33366:
33344:
33343:
33338:
33336:
33335:
33331:
33319:
33315:
33303:
33299:
33295:
33293:
33290:
33289:
33285:
33260:
33256:
33254:
33251:
33250:
33230:
33226:
33224:
33221:
33220:
33219:-axes, such as
33204:
33201:
33200:
33184:
33181:
33180:
33164:
33161:
33160:
33137:
33136:
33125:
33120:
33107:
33102:
33097:
33093:
33087:
33083:
33077:
33066:
33060:
33054:
33050:
33044:
33040:
33034:
33030:
33024:
33013:
33004:
32998:
32994:
32988:
32984:
32978:
32974:
32968:
32957:
32947:
32946:
32940:
32936:
32930:
32926:
32920:
32916:
32910:
32899:
32890:
32879:
32874:
32861:
32856:
32851:
32847:
32841:
32837:
32831:
32820:
32814:
32808:
32804:
32798:
32794:
32788:
32784:
32778:
32767:
32757:
32756:
32750:
32746:
32740:
32736:
32730:
32726:
32720:
32709:
32700:
32694:
32690:
32684:
32680:
32674:
32670:
32664:
32653:
32644:
32633:
32628:
32615:
32610:
32605:
32601:
32595:
32591:
32585:
32574:
32563:
32562:
32552:
32551:
32542:
32538:
32536:
32527:
32523:
32521:
32512:
32508:
32505:
32504:
32495:
32491:
32489:
32480:
32476:
32474:
32465:
32461:
32458:
32457:
32448:
32444:
32442:
32433:
32429:
32427:
32418:
32414:
32407:
32406:
32396:
32395:
32389:
32385:
32383:
32377:
32373:
32371:
32365:
32361:
32358:
32357:
32351:
32347:
32345:
32339:
32335:
32333:
32327:
32323:
32320:
32319:
32313:
32309:
32307:
32301:
32297:
32295:
32289:
32285:
32278:
32277:
32269:
32267:
32264:
32263:
32244:
32241:
32240:
32224:
32221:
32220:
32200:
32195:
32194:
32186:
32177:
32172:
32171:
32162:
32158:
32156:
32153:
32152:
32135:
32131:
32129:
32126:
32125:
32100:
32092:
32084:
32075:
32071:
32069:
32066:
32065:
32049:
32047:
32044:
32043:
32017:
32013:
32008:
31997:
31985:
31981:
31960:
31950:
31949:
31945:
31940:
31929:
31917:
31913:
31905:
31903:
31900:
31899:
31883:
31875:
31867:
31859:
31854:
31851:
31850:
31837:identity matrix
31834:
31809:
31801:
31799:
31796:
31795:
31748:
31740:
31731:
31726:
31725:
31719:
31715:
31710:
31706:
31702:
31672:
31668:
31660:
31658:
31655:
31654:
31635:
31632:
31631:
31615:
31612:
31611:
31591:
31587:
31585:
31582:
31581:
31565:
31562:
31561:
31541:
31537:
31535:
31532:
31531:
31515:
31514:
31505:
31501:
31495:
31491:
31485:
31481:
31475:
31464:
31439:
31438:
31433:
31431:
31430:
31426:
31414:
31410:
31398:
31394:
31391:
31390:
31381:
31377:
31371:
31367:
31361:
31357:
31351:
31340:
31315:
31314:
31309:
31307:
31306:
31302:
31290:
31286:
31274:
31270:
31267:
31266:
31257:
31253:
31247:
31243:
31237:
31233:
31227:
31216:
31191:
31190:
31185:
31183:
31182:
31178:
31166:
31162:
31150:
31146:
31142:
31140:
31137:
31136:
31116:
31115:
31101:
31096:
31083:
31078:
31073:
31069:
31063:
31059:
31053:
31042:
31020:
31019:
31014:
31012:
31011:
31007:
30995:
30991:
30988:
30987:
30973:
30968:
30955:
30950:
30945:
30941:
30935:
30931:
30925:
30914:
30892:
30891:
30886:
30884:
30883:
30879:
30867:
30863:
30860:
30859:
30845:
30840:
30827:
30822:
30817:
30813:
30807:
30803:
30797:
30786:
30764:
30763:
30758:
30756:
30755:
30751:
30739:
30735:
30731:
30729:
30726:
30725:
30702:
30700:
30697:
30696:
30689:Kronecker delta
30668:
30664:
30662:
30659:
30658:
30639:
30635:
30633:
30630:
30629:
30601:
30596:
30577:
30572:
30553:
30548:
30543:
30539:
30530:
30525:
30524:
30522:
30519:
30518:
30515:, respectively,
30500:
30497:
30496:
30480:
30477:
30476:
30460:
30457:
30456:
30440:
30437:
30436:
30420:
30417:
30416:
30383:
30378:
30362:
30357:
30341:
30337:
30331:
30321:
30316:
30315:
30311:
30310:
30309:
30305:
30299:
30295:
30289:
30278:
30256:
30255:
30250:
30248:
30247:
30235:
30231:
30229:
30226:
30225:
30202:
30201:
30195:
30191:
30189:
30183:
30179:
30177:
30171:
30167:
30164:
30163:
30157:
30153:
30151:
30145:
30141:
30139:
30133:
30129:
30126:
30125:
30119:
30115:
30113:
30107:
30103:
30101:
30095:
30091:
30084:
30083:
30075:
30073:
30070:
30069:
30048:
30044:
30042:
30039:
30038:
30022:
30019:
30018:
30015:
29998:
29974:
29972:
29969:
29968:
29950:
29946:
29945:
29943:
29940:
29939:
29925:
29924:
29910:
29909:
29902:
29901:
29896:
29895:
29888:
29887:
29876:
29875:
29874:
29860:
29859:
29852:
29851:
29846:
29845:
29831:
29830:
29821:
29820:
29809:
29808:
29797:
29786:
29781:
29780:
29776:
29772:
29771:
29765:
29761:
29755:
29744:
29736:
29732:
29718:
29717:
29703:
29702:
29696:
29685:
29680:
29679:
29675:
29671:
29670:
29656:
29655:
29649:
29645:
29639:
29628:
29612:
29611:
29605:
29594:
29589:
29584:
29579:
29575:
29574:
29568:
29564:
29558:
29547:
29536:
29530:
29526:
29522:
29520:
29517:
29516:
29494:
29493:
29491:
29488:
29487:
29471:
29469:
29466:
29465:
29449:
29446:
29445:
29429:
29428:
29414:
29413:
29407:
29396:
29391:
29390:
29386:
29382:
29381:
29367:
29366:
29359:
29357:
29351:
29350:
29334:
29333:
29324:
29319:
29318:
29306:
29301:
29300:
29296:
29292:
29278:
29277:
29270:
29268:
29262:
29261:
29245:
29244:
29235:
29230:
29229:
29222:
29218:
29204:
29199:
29198:
29181:
29180:
29179:
29175:
29173:
29171:
29165:
29164:
29153:
29148:
29147:
29130:
29129:
29128:
29124:
29105:
29104:
29085:
29080:
29079:
29062:
29061:
29060:
29056:
29042:
29041:
29040:
29036:
29035:
29031:
29029:
29027:
29021:
29020:
29004:
28999:
28998:
28981:
28980:
28979:
28975:
28961:
28960:
28959:
28955:
28936:
28931:
28930:
28913:
28912:
28911:
28907:
28893:
28892:
28891:
28887:
28882:
28880:
28877:
28876:
28854:
28853:
28851:
28848:
28847:
28830:
28825:
28824:
28819:
28816:
28815:
28790:
28785:
28784:
28767:
28766:
28765:
28761:
28742:
28741:
28722:
28717:
28716:
28699:
28698:
28697:
28693:
28679:
28678:
28677:
28673:
28672:
28668:
28649:
28644:
28643:
28626:
28625:
28624:
28620:
28606:
28605:
28604:
28600:
28581:
28576:
28575:
28558:
28557:
28556:
28552:
28538:
28537:
28536:
28532:
28530:
28527:
28526:
28510:
28509:
28493:
28488:
28487:
28470:
28469:
28468:
28464:
28450:
28449:
28448:
28444:
28425:
28420:
28419:
28402:
28401:
28400:
28396:
28382:
28381:
28380:
28376:
28367:
28366:
28355:
28350:
28349:
28340:
28325:
28324:
28320:
28319:
28315:
28311:
28297:
28292:
28291:
28282:
28267:
28266:
28262:
28261:
28257:
28253:
28246:
28240:
28229:
28224:
28219:
28214:
28210:
28209:
28205:
28203:
28200:
28199:
28174:
28173:
28171:
28168:
28167:
28146:
28145:
28134:
28133:
28132:
28121:
28120:
28112:
28097:
28096:
28085:
28084:
28083:
28072:
28071:
28063:
28062:
28058:
28052:
28037:
28036:
28032:
28031:
28022:
28007:
28006:
28002:
28001:
27996:
27993:
27992:
27976:
27962:
27961:
27953:
27938:
27937:
27933:
27931:
27928:
27927:
27901:
27900:
27896:
27894:
27891:
27890:
27871:
27868:
27867:
27866:along the line
27845:
27844:
27842:
27839:
27838:
27820:
27819:
27808:
27807:
27806:
27795:
27794:
27792:
27789:
27788:
27772:
27770:
27767:
27766:
27746:
27741:
27740:
27725:
27724:
27713:
27712:
27711:
27700:
27699:
27691:
27690:
27686:
27672:
27671:
27660:
27655:
27654:
27637:
27636:
27635:
27631:
27622:
27617:
27616:
27604:
27599:
27594:
27588:
27585:
27584:
27562:
27561:
27559:
27556:
27555:
27538:
27533:
27532:
27527:
27524:
27523:
27506:
27502:
27500:
27497:
27496:
27474:
27473:
27462:
27445:
27443:
27440:
27439:
27423:
27421:
27418:
27417:
27395:
27394:
27392:
27389:
27388:
27372:
27370:
27367:
27366:
27349:
27344:
27343:
27341:
27338:
27337:
27296:
27292:
27290:
27287:
27286:
27270:
27267:
27266:
27247:
27238:
27233:
27232:
27223:
27218:
27217:
27212:
27209:
27208:
27188:
27183:
27182:
27174:
27171:
27170:
27154:
27152:
27149:
27148:
27130:
27126:
27125:
27123:
27120:
27119:
27094:
27093:
27086:
27085:
27080:
27079:
27072:
27071:
27060:
27059:
27058:
27044:
27043:
27036:
27035:
27030:
27029:
27015:
27014:
27000:
26999:
26988:
26977:
26972:
26971:
26967:
26963:
26962:
26956:
26952:
26946:
26935:
26927:
26923:
26909:
26908:
26899:
26895:
26893:
26890:
26889:
26888:is as follows:
26873:
26871:
26868:
26867:
26845:
26844:
26842:
26839:
26838:
26821:
26817:
26815:
26812:
26811:
26808:
26784:
26779:
26776:
26775:
26755:
26751:
26744:
26743:
26734:
26730:
26729:
26724:
26721:
26720:
26703:
26699:
26694:
26691:
26690:
26673:
26664:
26660:
26659:
26651:
26648:
26647:
26625:
26623:
26620:
26619:
26603:
26601:
26598:
26597:
26581:
26579:
26576:
26575:
26555:
26551:
26546:
26530:
26529:
26524:
26523:
26513:
26512:
26507:
26506:
26504:
26501:
26500:
26481:
26479:
26476:
26475:
26456:
26451:
26448:
26447:
26431:
26429:
26426:
26425:
26407:
26403:
26402:
26400:
26397:
26396:
26374:
26370:
26365:
26351:
26347:
26341:
26330:
26325:
26321:
26305:
26296:
26291:
26290:
26281:
26277:
26271:
26260:
26255:
26251:
26243:
26229:
26213:
26204:
26199:
26198:
26189:
26185:
26179:
26168:
26163:
26159:
26145:
26141:
26136:
26127:
26122:
26121:
26112:
26108:
26102:
26091:
26086:
26082:
26069:
26068:
26063:
26062:
26060:
26057:
26056:
26034:
26030:
26025:
26012:
26003:
25998:
25997:
25996:
25992:
25983:
25979:
25973:
25962:
25946:
25942:
25932:
25924:
25923:
25919:
25910:
25905:
25904:
25895:
25891:
25885:
25874:
25857:
25856:
25851:
25850:
25848:
25845:
25844:
25828:
25826:
25823:
25822:
25806:
25804:
25801:
25800:
25784:
25782:
25779:
25778:
25759:
25751:
25743:
25732:
25724:
25713:
25711:
25708:
25707:
25691:
25689:
25686:
25685:
25663:
25653:
25644:
25639:
25638:
25637:
25633:
25632:
25626:
25622:
25616:
25605:
25588:
25587:
25582:
25581:
25579:
25576:
25575:
25559:
25557:
25554:
25553:
25535:
25531:
25530:
25528:
25525:
25524:
25505:
25503:
25500:
25499:
25483:
25481:
25478:
25477:
25474:
25468:
25445:
25441:
25440:
25438:
25435:
25434:
25415:
25408:
25407:
25402:
25401:
25393:
25385:
25378:
25377:
25372:
25371:
25363:
25361:
25358:
25357:
25354:
25340:
25339:
25334:
25323:
25319:
25313:
25309:
25297:
25293:
25287:
25276:
25268:
25264:
25256:
25248:
25237:
25233:
25227:
25223:
25211:
25207:
25201:
25190:
25182:
25178:
25169:
25168:
25157:
25148:
25143:
25142:
25127:
25122:
25121:
25112:
25108:
25102:
25091:
25076:
25062:
25053:
25048:
25047:
25032:
25027:
25026:
25014:
25010:
25004:
24993:
24979:
24974:
24970:
24968:
24965:
24964:
24948:
24947:
24941:
24937:
24932:
24927:
24917:
24906:
24902:
24896:
24892:
24880:
24876:
24870:
24859:
24851:
24847:
24840:
24829:
24820:
24815:
24814:
24799:
24794:
24793:
24781:
24777:
24771:
24760:
24750:
24749:
24744:
24738:
24734:
24728:
24724:
24712:
24708:
24702:
24691:
24675:
24674:
24667:
24666:
24660:
24656:
24653:
24652:
24646:
24642:
24639:
24638:
24632:
24628:
24621:
24620:
24613:
24612:
24603:
24592:
24576:
24565:
24550:
24538:
24534:
24519:
24515:
24510:
24498:
24494:
24479:
24475:
24469:
24468:
24456:
24452:
24437:
24433:
24428:
24419:
24408:
24392:
24381:
24366:
24354:
24350:
24335:
24331:
24325:
24324:
24312:
24308:
24293:
24289:
24284:
24272:
24268:
24253:
24249:
24244:
24235:
24224:
24208:
24197:
24177:
24176:
24170:
24166:
24160:
24149:
24133:
24132:
24125:
24124:
24118:
24114:
24104:
24093:
24077:
24066:
24047:
24043:
24030:
24026:
24011:
24007:
23995:
23991:
23978:
23974:
23959:
23955:
23946:
23945:
23939:
23935:
23922:
23918:
23903:
23899:
23887:
23883:
23873:
23862:
23846:
23835:
23816:
23812:
23799:
23795:
23780:
23776:
23767:
23766:
23760:
23756:
23743:
23739:
23724:
23720:
23708:
23704:
23691:
23687:
23672:
23668:
23656:
23652:
23642:
23631:
23615:
23604:
23584:
23583:
23577:
23573:
23567:
23556:
23540:
23539:
23532:
23531:
23525:
23521:
23514:
23503:
23487:
23483:
23470:
23466:
23451:
23447:
23435:
23431:
23424:
23413:
23397:
23393:
23380:
23376:
23361:
23357:
23348:
23347:
23341:
23337:
23330:
23319:
23303:
23299:
23286:
23282:
23267:
23263:
23251:
23247:
23234:
23230:
23215:
23211:
23199:
23195:
23188:
23177:
23164:
23163:
23157:
23153:
23140:
23136:
23121:
23117:
23105:
23101:
23094:
23083:
23067:
23063:
23050:
23046:
23031:
23027:
23015:
23011:
23004:
22993:
22976:
22975:
22969:
22965:
22959:
22948:
22932:
22931:
22924:
22923:
22914:
22910:
22897:
22893:
22881:
22877:
22864:
22860:
22839:
22835:
22820:
22816:
22803:
22799:
22787:
22783:
22770:
22766:
22745:
22741:
22732:
22731:
22722:
22718:
22705:
22701:
22689:
22685:
22672:
22668:
22647:
22643:
22628:
22624:
22611:
22607:
22595:
22591:
22578:
22574:
22553:
22549:
22540:
22539:
22530:
22526:
22513:
22509:
22497:
22493:
22480:
22476:
22455:
22451:
22436:
22432:
22419:
22415:
22403:
22399:
22386:
22382:
22361:
22357:
22344:
22343:
22337:
22333:
22327:
22316:
22300:
22299:
22287:
22286:
22280:
22276:
22263:
22259:
22247:
22243:
22230:
22226:
22217:
22216:
22210:
22206:
22193:
22189:
22177:
22173:
22160:
22156:
22147:
22146:
22140:
22136:
22123:
22119:
22107:
22103:
22090:
22086:
22073:
22072:
22065:
22064:
22059:
22047:
22043:
22038:
22026:
22022:
22013:
22012:
22000:
21996:
21988:
21983:
21971:
21967:
21961:
21960:
21948:
21944:
21939:
21927:
21923:
21915:
21905:
21904:
21903:
21899:
21893:
21889:
21883:
21872:
21856:
21855:
21850:
21829:
21828:
21822:
21818:
21815:
21814:
21808:
21804:
21801:
21800:
21794:
21790:
21783:
21782:
21775:
21774:
21769:
21757:
21753:
21748:
21736:
21732:
21723:
21722:
21710:
21706:
21698:
21693:
21681:
21677:
21671:
21670:
21658:
21654:
21649:
21637:
21633:
21625:
21615:
21614:
21613:
21609:
21602:
21601:
21596:
21584:
21580:
21575:
21563:
21559:
21550:
21549:
21537:
21533:
21525:
21520:
21508:
21504:
21498:
21497:
21485:
21481:
21476:
21464:
21460:
21452:
21442:
21441:
21440:
21436:
21430:
21426:
21420:
21409:
21395:
21384:
21375:
21370:
21369:
21354:
21349:
21348:
21336:
21332:
21326:
21315:
21304:
21302:
21299:
21298:
21282:
21280:
21277:
21276:
21260:
21259:
21253:
21249:
21244:
21239:
21223:
21214:
21209:
21208:
21193:
21188:
21187:
21175:
21171:
21165:
21154:
21139:
21125:
21116:
21111:
21110:
21095:
21090:
21089:
21077:
21073:
21067:
21056:
21042:
21037:
21034:
21033:
21028:
21010:
21001:
20996:
20995:
20980:
20975:
20974:
20960:
20951:
20947:
20941:
20930:
20909:
20900:
20895:
20894:
20879:
20874:
20873:
20861:
20857:
20851:
20840:
20824:
20823:
20818:
20800:
20791:
20786:
20785:
20770:
20765:
20764:
20750:
20739:
20730:
20725:
20724:
20709:
20704:
20703:
20691:
20687:
20681:
20670:
20654:
20653:
20648:
20630:
20621:
20616:
20615:
20597:
20592:
20591:
20577:
20566:
20557:
20552:
20551:
20533:
20528:
20527:
20515:
20511:
20505:
20494:
20481:
20480:
20475:
20456:
20451:
20450:
20439:
20427:
20422:
20421:
20407:
20395:
20390:
20389:
20378:
20366:
20361:
20360:
20348:
20344:
20338:
20327:
20314:
20313:
20299:
20290:
20285:
20284:
20272:
20267:
20266:
20251:
20246:
20245:
20233:
20228:
20227:
20216:
20205:
20193:
20188:
20187:
20176:
20164:
20159:
20158:
20146:
20142:
20136:
20125:
20112:
20111:
20106:
20081:
20076:
20075:
20063:
20058:
20057:
20037:
20028:
20023:
20022:
20010:
20005:
20004:
19989:
19984:
19983:
19971:
19966:
19965:
19954:
19943:
19931:
19926:
19925:
19914:
19902:
19897:
19896:
19884:
19880:
19874:
19863:
19850:
19849:
19844:
19829:
19821:
19809:
19804:
19803:
19791:
19786:
19785:
19765:
19756:
19751:
19750:
19738:
19733:
19732:
19717:
19712:
19711:
19699:
19694:
19693:
19682:
19671:
19659:
19654:
19653:
19642:
19630:
19625:
19624:
19612:
19608:
19602:
19591:
19578:
19577:
19572:
19556:
19551:
19550:
19538:
19533:
19532:
19521:
19513:
19499:
19490:
19485:
19484:
19472:
19467:
19466:
19451:
19446:
19445:
19433:
19428:
19427:
19416:
19405:
19393:
19388:
19387:
19376:
19364:
19359:
19358:
19346:
19342:
19336:
19325:
19314:
19309:
19305:
19303:
19300:
19299:
19286:
19285:
19280:
19261:
19256:
19255:
19243:
19238:
19237:
19226:
19212:
19203:
19198:
19197:
19185:
19180:
19179:
19164:
19159:
19158:
19146:
19141:
19140:
19129:
19115:
19103:
19098:
19097:
19086:
19074:
19069:
19068:
19056:
19052:
19046:
19035:
19022:
19021:
19006:
19001:
19000:
18988:
18983:
18982:
18971:
18959:
18954:
18953:
18941:
18936:
18935:
18924:
18905:
18896:
18891:
18890:
18878:
18873:
18872:
18854:
18849:
18848:
18836:
18831:
18830:
18819:
18807:
18802:
18801:
18789:
18784:
18783:
18772:
18758:
18746:
18741:
18740:
18729:
18717:
18712:
18711:
18699:
18695:
18689:
18678:
18665:
18664:
18650:
18641:
18636:
18635:
18623:
18618:
18617:
18597:
18585:
18580:
18579:
18568:
18556:
18551:
18550:
18538:
18534:
18528:
18517:
18504:
18503:
18489:
18480:
18475:
18474:
18462:
18457:
18456:
18439:
18427:
18422:
18421:
18410:
18398:
18393:
18392:
18380:
18376:
18370:
18359:
18346:
18345:
18330:
18325:
18324:
18313:
18302:
18290:
18285:
18284:
18269:
18264:
18263:
18252:
18240:
18235:
18234:
18222:
18218:
18212:
18201:
18190:
18185:
18181:
18179:
18176:
18175:
18159:
18158:
18153:
18138:
18129:
18124:
18123:
18111:
18106:
18105:
18088:
18081:
18069:
18064:
18063:
18052:
18041:
18029:
18024:
18023:
18017:
18016:
18011:
17995:
17990:
17989:
17978:
17969:
17964:
17963:
17946:
17937:
17936:
17931:
17912:
17907:
17906:
17895:
17886:
17881:
17880:
17866:
17851:
17850:
17838:
17833:
17832:
17821:
17809:
17804:
17803:
17792:
17774:
17773:
17768:
17740:
17735:
17734:
17722:
17717:
17716:
17701:
17696:
17695:
17684:
17672:
17667:
17666:
17655:
17637:
17636:
17631:
17613:
17605:
17593:
17588:
17587:
17575:
17570:
17569:
17554:
17549:
17548:
17537:
17525:
17520:
17519:
17508:
17490:
17489:
17484:
17465:
17460:
17459:
17448:
17437:
17428:
17423:
17422:
17407:
17402:
17401:
17390:
17378:
17373:
17372:
17361:
17343:
17342:
17330:
17325:
17324:
17309:
17304:
17303:
17292:
17278:
17265:
17253:
17248:
17247:
17236:
17225:
17213:
17208:
17207:
17200:
17198:
17195:
17194:
17191:Jacobi identity
17174:
17173:
17164:
17159:
17158:
17143:
17138:
17137:
17126:
17112:
17097:
17092:
17091:
17080:
17069:
17057:
17052:
17051:
17041:
17035:
17034:
17029:
17001:
16996:
16995:
16980:
16975:
16974:
16963:
16949:
16934:
16929:
16928:
16917:
16906:
16894:
16889:
16888:
16876:
16875:
16870:
16854:
16849:
16848:
16837:
16822:
16817:
16816:
16805:
16784:
16779:
16778:
16763:
16758:
16757:
16746:
16732:
16717:
16712:
16711:
16700:
16689:
16677:
16672:
16671:
16659:
16658:
16653:
16640:
16635:
16634:
16623:
16605:
16600:
16599:
16588:
16573:
16568:
16567:
16552:
16547:
16546:
16535:
16521:
16506:
16501:
16500:
16489:
16478:
16466:
16461:
16460:
16448:
16447:
16439:
16430:
16425:
16424:
16406:
16401:
16400:
16389:
16374:
16369:
16368:
16353:
16348:
16347:
16336:
16322:
16307:
16302:
16301:
16290:
16279:
16267:
16262:
16261:
16251:
16244:
16242:
16239:
16238:
16235:Jacobi identity
16218:
16217:
16202:
16197:
16196:
16185:
16174:
16162:
16157:
16156:
16141:
16136:
16135:
16124:
16112:
16107:
16106:
16094:
16090:
16084:
16073:
16062:
16057:
16054:
16053:
16034:
16033:
16028:
16027:
16019:
16007:
16002:
16001:
15986:
15981:
15980:
15969:
15957:
15952:
15951:
15939:
15935:
15929:
15918:
15907:
15902:
15899:
15898:
15882:
15881:
15876:
15875:
15866:
15861:
15860:
15841:
15840:
15835:
15834:
15825:
15820:
15819:
15807:
15803:
15797:
15786:
15775:
15770:
15766:
15764:
15761:
15760:
15744:
15742:
15739:
15738:
15722:
15720:
15717:
15716:
15692:
15691:
15686:
15685:
15683:
15680:
15679:
15665:
15664:
15639:
15638:
15633:
15632:
15616:
15615:
15610:
15609:
15597:
15592:
15591:
15579:
15575:
15569:
15558:
15545:
15544:
15531:
15530:
15525:
15524:
15508:
15507:
15502:
15501:
15485:
15484:
15479:
15478:
15466:
15461:
15460:
15448:
15444:
15438:
15427:
15414:
15413:
15408:
15395:
15390:
15389:
15380:
15375:
15374:
15362:
15358:
15352:
15341:
15328:
15327:
15318:
15313:
15312:
15306:
15302:
15290:
15285:
15284:
15275:
15264:
15251:
15250:
15241:
15236:
15235:
15229:
15225:
15211:
15201:
15197:
15196:
15187:
15176:
15165:
15160:
15156:
15154:
15151:
15150:
15139:for the triple
15137:Jacobi identity
15105:
15100:
15099:
15082:
15077:
15076:
15065:
15064:
15060:
15059:
15055:
15047:
15028:
15023:
15022:
15011:
15010:
15006:
14998:
14997:
14993:
14984:
14979:
14978:
14973:
14970:
14969:
14951:
14940:
14929:
14924:
14923:
14919:
14915:
14914:
14908:
14904:
14898:
14887:
14879:
14875:
14867:
14859:
14848:
14837:
14832:
14831:
14827:
14823:
14822:
14816:
14812:
14806:
14795:
14787:
14783:
14775:
14773:
14770:
14769:
14747:
14738:
14733:
14732:
14727:
14724:
14723:
14702:
14693:
14688:
14687:
14686:
14682:
14668:
14663:
14662:
14658:
14654:
14652:
14649:
14648:
14632:
14630:
14627:
14626:
14603:
14602:
14597:
14596:
14578:
14569:
14564:
14563:
14562:
14558:
14550:
14549:
14545:
14537:
14524:
14515:
14510:
14509:
14508:
14504:
14496:
14487:
14482:
14481:
14479:
14476:
14475:
14459:
14457:
14454:
14453:
14437:
14435:
14432:
14431:
14413:
14412:
14407:
14406:
14404:
14401:
14400:
14384:
14382:
14379:
14378:
14361:
14357:
14355:
14352:
14351:
14330:
14326:
14324:
14321:
14320:
14303:
14298:
14297:
14295:
14292:
14291:
14271:
14266:
14265:
14259:
14255:
14242:
14232:
14228:
14227:
14226:
14222:
14216:
14205:
14193:
14191:
14188:
14187:
14184:
14163:
14160:
14159:
14141:
14137:
14136:
14134:
14131:
14130:
14110:
14104:
14103:
14098:
14084:
14076:
14069:
14068:
14063:
14062:
14054:
14044:
14035:
14031:
14029:
14026:
14025:
14009:
14008:
13998:
13997:
13992:
13991:
13981:
13980:
13975:
13974:
13963:
13959:
13953:
13942:
13937:
13933:
13923:
13915:
13904:
13893:
13888:
13887:
13883:
13879:
13878:
13872:
13868:
13862:
13851:
13843:
13839:
13831:
13821:
13812:
13811:
13804:
13803:
13798:
13797:
13787:
13786:
13781:
13780:
13769:
13765:
13759:
13748:
13743:
13739:
13729:
13711:
13700:
13695:
13694:
13690:
13686:
13679:
13678:
13667:
13662:
13661:
13657:
13653:
13652:
13645:
13644:
13639:
13638:
13637:
13633:
13627:
13623:
13617:
13606:
13601:
13597:
13587:
13578:
13577:
13570:
13569:
13564:
13563:
13553:
13552:
13547:
13546:
13535:
13531:
13525:
13514:
13509:
13505:
13495:
13477:
13466:
13461:
13460:
13456:
13452:
13451:
13447:
13434:
13423:
13418:
13417:
13413:
13409:
13408:
13404:
13398:
13394:
13388:
13377:
13372:
13368:
13358:
13351:
13345:
13341:
13337:
13335:
13332:
13331:
13311:
13306:
13305:
13297:
13294:
13293:
13270:
13265:
13264:
13255:
13251:
13245:
13234:
13228:
13225:
13224:
13196:
13195:
13190:
13189:
13179:
13178:
13173:
13172:
13166:
13162:
13156:
13145:
13140:
13136:
13126:
13107:
13102:
13101:
13090:
13089:
13085:
13075:
13074:
13069:
13068:
13062:
13058:
13052:
13041:
13036:
13032:
13013:
13008:
13007:
12996:
12995:
12991:
12977:
12972:
12971:
12960:
12959:
12955:
12949:
12945:
12939:
12928:
12923:
12919:
12909:
12900:
12896:
12894:
12891:
12890:
12871:
12862:
12857:
12856:
12847:
12842:
12841:
12836:
12833:
12832:
12806:
12805:
12800:
12799:
12790:
12785:
12784:
12773:
12772:
12768:
12753:
12752:
12747:
12746:
12737:
12732:
12731:
12720:
12719:
12715:
12709:
12705:
12699:
12688:
12674:
12665:
12660:
12659:
12650:
12645:
12644:
12638:
12634:
12628:
12617:
12603:
12594:
12590:
12588:
12585:
12584:
12567:
12562:
12561:
12559:
12556:
12555:
12538:
12533:
12532:
12530:
12527:
12526:
12485:
12481:
12479:
12476:
12475:
12459:
12456:
12455:
12448:
12427:
12425:
12422:
12421:
12401:
12390:
12385:
12384:
12380:
12376:
12375:
12369:
12365:
12359:
12348:
12331:
12330:
12325:
12324:
12322:
12319:
12318:
12300:
12296:
12295:
12293:
12290:
12289:
12270:
12263:
12262:
12257:
12256:
12248:
12237:
12226:
12221:
12220:
12216:
12212:
12211:
12205:
12201:
12195:
12184:
12176:
12172:
12164:
12162:
12159:
12158:
12142:
12133:
12128:
12127:
12118:
12113:
12112:
12107:
12104:
12103:
12083:
12078:
12077:
12069:
12066:
12065:
12042:
12037:
12034:
12033:
12015:
12011:
12010:
12008:
12005:
12004:
11990:
11989:
11974:
11973:
11968:
11967:
11958:
11953:
11952:
11942:
11938:
11932:
11921:
11916:
11912:
11894:
11885:
11880:
11879:
11875:
11871:
11862:
11857:
11856:
11846:
11842:
11836:
11825:
11817:
11813:
11804:
11803:
11791:
11790:
11785:
11784:
11775:
11770:
11769:
11758:
11757:
11753:
11744:
11739:
11738:
11728:
11724:
11718:
11707:
11694:
11693:
11687:
11682:
11681:
11672:
11667:
11666:
11656:
11652:
11646:
11635:
11624:
11619:
11615:
11613:
11610:
11609:
11593:
11591:
11588:
11587:
11571:
11569:
11566:
11565:
11549:
11548:
11538:
11537:
11532:
11526:
11522:
11520:
11514:
11510:
11504:
11503:
11497:
11493:
11488:
11483:
11477:
11473:
11470:
11469:
11463:
11459:
11457:
11451:
11447:
11442:
11432:
11431:
11424:
11415:
11411:
11408:
11407:
11402:
11393:
11389:
11382:
11377:
11369:
11365:
11363:
11360:
11359:
11339:
11335:
11326:
11322:
11313:
11309:
11298:
11296:
11293:
11292:
11272:
11268:
11266:
11263:
11262:
11255:
11234:
11232:
11229:
11228:
11210:
11206:
11205:
11203:
11200:
11199:
11183:
11181:
11178:
11177:
11159:
11158:
11153:
11152:
11143:
11138:
11137:
11126:
11117:
11112:
11111:
11109:
11106:
11105:
11089:
11080:
11075:
11074:
11065:
11060:
11059:
11054:
11051:
11050:
11034:
11032:
11029:
11028:
11011:
11006:
11005:
11003:
11000:
10999:
10982:
10977:
10976:
10974:
10971:
10970:
10929:
10925:
10923:
10920:
10919:
10903:
10900:
10899:
10896:
10881:
10852:
10851:
10841:
10840:
10836:
10828:
10826:
10823:
10822:
10804:
10803:
10799:
10797:
10794:
10793:
10777:
10775:
10772:
10771:
10748:
10744:
10742:
10739:
10738:
10716:
10715:
10706:
10695:
10694:
10693:
10684:
10673:
10672:
10671:
10669:
10666:
10665:
10649:
10640:
10629:
10628:
10627:
10618:
10607:
10606:
10605:
10603:
10600:
10599:
10583:
10582:
10574:
10560:
10549:
10548:
10547:
10541:
10537:
10527:
10523:
10517:
10506:
10501:
10497:
10483:
10482:
10468:
10463:
10449:
10445:
10439:
10428:
10423:
10419:
10410:
10409:
10399:
10390:
10379:
10378:
10377:
10371:
10367:
10358:
10354:
10345:
10334:
10333:
10332:
10326:
10322:
10315:
10311:
10302:
10291:
10290:
10289:
10283:
10279:
10269:
10265:
10259:
10248:
10237:
10232:
10228:
10226:
10223:
10222:
10206:
10205:
10197:
10188:
10177:
10176:
10175:
10169:
10165:
10156:
10152:
10143:
10132:
10131:
10130:
10124:
10120:
10105:
10104:
10099:
10090:
10079:
10078:
10077:
10071:
10067:
10050:
10049:
10032:
10031:
10019:
10008:
10007:
10006:
10000:
9996:
9979:
9978:
9968:
9962:
9957:
9956:
9952:
9950:
9947:
9946:
9929:
9918:
9917:
9916:
9902:
9901:
9892:
9881:
9880:
9879:
9877:
9874:
9873:
9856:
9851:
9850:
9848:
9845:
9844:
9828:
9826:
9823:
9822:
9805:
9794:
9793:
9792:
9790:
9787:
9786:
9764:
9763:
9761:
9758:
9757:
9736:
9727:
9722:
9721:
9710:
9702:
9693:
9688:
9687:
9676:
9667:
9662:
9661:
9659:
9656:
9655:
9639:
9637:
9634:
9633:
9617:
9615:
9612:
9611:
9595:
9593:
9590:
9589:
9573:
9571:
9568:
9567:
9550:
9546:
9544:
9541:
9540:
9516:
9511:
9510:
9508:
9505:
9504:
9490:
9489:
9480:
9475:
9474:
9468:
9464:
9455:
9450:
9449:
9440:
9429:
9418:
9413:
9410:
9409:
9400:
9395:
9394:
9388:
9384:
9378:
9367:
9356:
9351:
9347:
9345:
9342:
9341:
9325:
9323:
9320:
9319:
9316:resultant force
9274:
9270:
9268:
9265:
9264:
9248:
9245:
9244:
9229:
9202:
9201:
9197:
9195:
9192:
9191:
9170:
9162:
9149:
9140:
9136:
9129:
9128:
9124:
9114:
9105:
9101:
9099:
9096:
9095:
9077:
9076:
9072:
9070:
9067:
9066:
9050:
9048:
9045:
9044:
9028:
9027:
9019:
9011:
9000:
8996:
8990:
8979:
8974:
8970:
8960:
8946:
8935:
8934:
8933:
8927:
8923:
8913:
8909:
8903:
8892:
8887:
8883:
8875:
8858:
8847:
8846:
8845:
8836:
8825:
8824:
8823:
8817:
8812:
8798:
8794:
8788:
8777:
8772:
8768:
8762:
8758:
8748:
8739:
8738:
8725:
8716:
8705:
8704:
8703:
8697:
8693:
8685:
8681:
8668:
8659:
8648:
8647:
8646:
8640:
8636:
8628:
8624:
8618:
8614:
8608:
8597:
8583:
8574:
8573:
8564:
8559:
8558:
8549:
8544:
8543:
8537:
8533:
8527:
8516:
8502:
8495:
8489:
8485:
8481:
8479:
8476:
8475:
8464:
8429:
8428:
8424:
8422:
8419:
8418:
8391:
8390:
8380:
8379:
8375:
8367:
8365:
8362:
8361:
8339:
8334:
8320:
8316:
8310:
8296:
8295:
8291:
8289:
8286:
8285:
8267:
8266:
8262:
8260:
8257:
8256:
8240:
8239:
8226:
8220:
8209:
8208:
8207:
8201:
8197:
8187:
8183:
8177:
8166:
8159:
8158:
8150:
8141:
8136:
8135:
8128:
8122:
8111:
8110:
8109:
8103:
8099:
8092:
8090:
8087:
8086:
8070:
8068:
8065:
8064:
8045:
8044:
8036:
8022:
8011:
8010:
8009:
8003:
7999:
7989:
7985:
7979:
7968:
7963:
7959:
7945:
7944:
7930:
7925:
7911:
7907:
7901:
7890:
7885:
7881:
7872:
7871:
7861:
7852:
7841:
7840:
7839:
7833:
7829:
7821:
7817:
7808:
7797:
7796:
7795:
7789:
7785:
7775:
7771:
7765:
7754:
7741:
7740:
7734:
7729:
7728:
7719:
7714:
7713:
7704:
7700:
7694:
7683:
7672:
7667:
7663:
7661:
7658:
7657:
7654:
7627:
7626:
7621:
7612:
7601:
7600:
7599:
7593:
7589:
7574:
7565:
7554:
7553:
7552:
7546:
7542:
7525:
7524:
7513:
7504:
7499:
7498:
7487:
7480:
7474:
7469:
7468:
7465:
7464:
7455:
7444:
7443:
7442:
7428:
7427:
7418:
7407:
7406:
7405:
7391:
7377:
7376:
7363:
7359:
7355:
7348:
7343:
7342:
7338:
7336:
7329:
7323:
7312:
7311:
7310:
7306:
7304:
7301:
7300:
7283:
7272:
7271:
7270:
7256:
7255:
7246:
7235:
7234:
7233:
7231:
7228:
7227:
7210:
7205:
7204:
7202:
7199:
7198:
7182:
7180:
7177:
7176:
7159:
7154:
7153:
7151:
7148:
7147:
7131:
7129:
7126:
7125:
7106:
7104:
7101:
7100:
7084:
7082:
7079:
7078:
7062:
7060:
7057:
7056:
7042:
7041:
7033:
7024:
7019:
7018:
7007:
6999:
6986:
6977:
6972:
6971:
6970:
6966:
6958:
6951:
6945:
6940:
6939:
6936:
6935:
6927:
6918:
6913:
6912:
6905:
6899:
6894:
6893:
6886:
6884:
6881:
6880:
6863:
6858:
6857:
6855:
6852:
6851:
6835:
6833:
6830:
6829:
6788:
6784:
6782:
6779:
6778:
6762:
6759:
6758:
6757:If a system of
6729:
6728:
6726:
6723:
6722:
6707:
6682:
6678:
6665:
6657:
6654:
6653:
6639:
6638:
6629:
6625:
6610:
6601:
6600:
6594:
6590:
6573:
6558:
6551:
6547:
6532:
6531:
6525:
6517:
6506:
6502:
6490:
6481:
6477:
6471:
6467:
6455:
6443:
6439:
6438:
6434:
6415:
6406:
6405:
6392:
6381:
6377:
6368:
6364:
6363:
6359:
6358:
6340:
6334:
6326:
6306:
6302:
6275:
6269:
6261:
6250:
6243:
6236:
6232:
6228:
6226:
6223:
6222:
6206:
6203:
6202:
6180:
6176:
6167:
6163:
6154:
6150:
6141:
6137:
6128:
6124:
6113:
6110:
6109:
6093:
6090:
6089:
6073:
6070:
6069:
6053:
6050:
6049:
6027:
6023:
6014:
6010:
6001:
5997:
5988:
5984:
5982:
5979:
5978:
5939:
5936:
5935:
5915:
5911:
5893:
5889:
5879:
5872:
5868:
5859:
5845:
5841:
5840:
5834:
5830:
5820:
5813:
5809:
5800:
5796:
5794:
5791:
5790:
5774:
5771:
5770:
5744:
5740:
5729:
5726:
5725:
5705:
5701:
5688:
5674:
5670:
5668:
5627:
5623:
5614:
5609:
5596:
5591:
5571:
5567:
5557:
5553:
5543:
5536:
5532:
5530:
5527:
5526:
5484:
5481:
5480:
5464:
5461:
5460:
5440:
5437:
5436:
5420:
5417:
5416:
5400:
5397:
5396:
5361:
5358:
5357:
5331:
5327:
5323:
5321:
5302:
5298:
5296:
5282:
5278:
5276:
5275:
5271:
5258:
5256:
5242:
5231:
5227:
5211:
5207:
5205:
5198:
5195:
5175:
5171:
5156:
5145:
5141:
5121:
5117:
5107:
5103:
5093:
5086:
5082:
5080:
5077:
5076:
5060:
5057:
5056:
5040:
5037:
5036:
5020:
5017:
5016:
5000:
4997:
4996:
4973:
4967:
4946:
4943:
4942:
4926:
4923:
4922:
4902:
4899:
4898:
4882:
4879:
4878:
4858:
4855:
4854:
4828:
4825:
4824:
4808:
4805:
4804:
4770:
4767:
4766:
4750:
4748:
4745:
4744:
4710:
4707:
4706:
4690:
4687:
4686:
4658:
4649:
4645:
4644:
4614:
4610:
4601:
4597:
4595:
4592:
4591:
4568:
4563:
4553:
4549:
4543:
4530:
4526:
4524:
4521:
4520:
4493:
4488:
4478:
4474:
4468:
4457:
4444:
4440:
4438:
4435:
4434:
4433:terms, that is
4417:
4413:
4408:
4405:
4404:
4382:
4377:
4367:
4363:
4357:
4346:
4336:
4332:
4321:
4312:
4301:
4297:
4293:
4289:
4288:
4282:
4278:
4267:
4261:
4250:
4237:
4232:
4231:
4222:
4217:
4216:
4210:
4206:
4195:
4189:
4178:
4165:
4161:
4159:
4156:
4155:
4137:
4134:
4133:
4116:
4112:
4110:
4107:
4106:
4089:
4085:
4083:
4080:
4079:
4063:
4060:
4059:
4039:
4036:
4035:
4018:
4014:
4009:
4006:
4005:
3995:
3992:solid cylinder.
3989:
3983:
3977:
3971:
3961:
3956:
3934:
3930:
3925:
3915:
3906:
3902:
3897:
3887:
3875:
3871:
3859:
3855:
3853:
3850:
3849:
3824:
3820:
3816:
3814:
3802:
3798:
3796:
3793:
3792:
3767:
3763:
3759:
3757:
3745:
3741:
3739:
3736:
3735:
3712:
3708:
3704:
3702:
3690:
3686:
3684:
3681:
3680:
3679:Rectangular:
3656:
3652:
3650:
3638:
3634:
3622:
3618:
3616:
3613:
3612:
3608:
3604:
3600:
3598:
3597:
3577:
3574:
3573:
3568:) and breadth (
3563:
3543:
3539:
3537:
3534:
3533:
3513:
3509:
3507:
3504:
3503:
3496:
3484:
3450:
3432:
3428:
3419:
3409:
3399:
3391:
3387:
3382:
3378:
3371:
3369:
3358:
3353:
3344:
3336:
3333:
3332:
3316:
3313:
3312:
3292:
3282:
3274:
3271:
3270:
3240:
3234:
3230:
3228:
3217:
3212:
3203:
3195:
3192:
3191:
3168:
3164:
3153:
3150:
3135:
3126:
3122:
3120:
3117:
3116:
3100:
3097:
3096:
3072:
3069:
3068:
3065:
3044:
3041:
3040:
3017:
3013:
3009:
3002:
2998:
2988:
2986:
2975:
2970:
2955:
2953:
2944:
2940:
2938:
2935:
2934:
2918:
2915:
2914:
2898:
2895:
2894:
2874:
2871:
2870:
2854:
2851:
2850:
2834:
2831:
2830:
2807:
2803:
2792:
2789:
2780:
2776:
2774:
2771:
2770:
2753:
2749:
2747:
2744:
2743:
2726:
2722:
2720:
2717:
2716:
2689:
2667:
2663:
2658:
2655:
2654:
2637:
2633:
2622:
2619:
2618:
2596:
2592:
2579:
2570:
2566:
2555:
2551:
2547:
2543:
2533:
2525:
2517:
2504:
2495:
2491:
2489:
2486:
2485:
2468:
2467:
2453:
2452:
2438:
2432:
2428:
2416:
2415:
2405:
2395:
2387:
2386:
2382:
2374:
2364:
2356:
2355:
2351:
2350:
2346:
2334:
2333:
2323:
2315:
2311:
2307:
2299:
2291:
2283:
2276:
2271:
2267:
2265:
2262:
2261:
2241:
2239:
2236:
2235:
2219:
2211:
2203:
2201:
2198:
2197:
2176:
2172:
2161:
2158:
2157:
2133:
2129:
2118:
2115:
2114:
2098:
2096:
2093:
2092:
2076:
2074:
2071:
2070:
2044:
2043:
2041:
2038:
2037:
2021:
2020:
2006:
2005:
1991:
1985:
1981:
1969:
1968:
1958:
1948:
1940:
1939:
1935:
1927:
1917:
1909:
1908:
1904:
1903:
1899:
1887:
1886:
1878:
1870:
1856:
1848:
1840:
1833:
1828:
1824:
1822:
1819:
1818:
1802:
1791:
1789:
1786:
1785:
1769:
1761:
1753:
1751:
1748:
1747:
1731:
1729:
1726:
1725:
1705:
1703:
1700:
1699:
1683:
1681:
1678:
1677:
1661:
1653:
1645:
1643:
1640:
1639:
1617:
1613:
1602:
1599:
1598:
1582:
1579:
1578:
1562:
1559:
1558:
1554:
1552:Simple pendulum
1549:
1543:
1530:
1510:
1506:
1495:
1492:
1491:
1487:
1483:
1476:
1472:
1466:
1459:
1455:
1433:
1429:
1418:
1415:
1414:
1410:
1406:
1402:
1399:simple pendulum
1367:
1364:
1358:
1351:
1302:
1294:
1291:
1285:
1278:
1271:
1253:
1228:
1147:
1144:
1143:
1127:
1124:
1123:
1106:
1102:
1097:
1094:
1093:
1055:
957:
956:
955:
954:
950:
949:
948:
940:
939:
909:
868:
855:
854:
847:
846:
845:
720:
712:
711:
691:
645:Circular motion
639:
629:
628:
627:
584:
554:
551:
530:
509:
501:
500:
497:
496:
454:
444:
436:
435:
434:
393:
389:Mechanical work
382:
366:
304:
296:
295:
294:
249:
241:
218:
194:
188:
184:
182:
173:
172:
170:
167:
166:
111:
103:
100:
99:
96:
94:
84:
62:
55:
35:
28:
23:
22:
15:
12:
11:
5:
40641:
40631:
40630:
40625:
40620:
40615:
40601:
40600:
40591:
40590:
40588:
40587:
40582:
40580:Woldemar Voigt
40577:
40572:
40567:
40562:
40557:
40552:
40547:
40545:Leonhard Euler
40542:
40537:
40532:
40527:
40521:
40519:
40517:Mathematicians
40513:
40512:
40509:
40508:
40506:
40505:
40500:
40495:
40490:
40485:
40480:
40475:
40470:
40465:
40459:
40457:
40453:
40452:
40450:
40449:
40444:
40442:Torsion tensor
40439:
40434:
40429:
40424:
40419:
40414:
40408:
40406:
40399:
40395:
40394:
40392:
40391:
40386:
40381:
40376:
40371:
40366:
40361:
40356:
40351:
40346:
40341:
40336:
40331:
40326:
40321:
40316:
40311:
40306:
40301:
40295:
40293:
40287:
40286:
40284:
40283:
40277:
40275:Tensor product
40272:
40267:
40265:Symmetrization
40262:
40257:
40255:Lie derivative
40252:
40247:
40242:
40237:
40232:
40226:
40224:
40218:
40217:
40215:
40214:
40209:
40204:
40199:
40194:
40189:
40184:
40179:
40177:Tensor density
40174:
40169:
40163:
40161:
40155:
40154:
40152:
40151:
40149:Voigt notation
40146:
40141:
40136:
40134:Ricci calculus
40131:
40126:
40121:
40119:Index notation
40116:
40111:
40105:
40103:
40099:
40098:
40095:
40094:
40092:
40091:
40086:
40081:
40076:
40071:
40065:
40063:
40061:
40060:
40055:
40049:
40046:
40045:
40043:
40042:
40037:
40035:Tensor algebra
40032:
40027:
40022:
40017:
40015:Dyadic algebra
40012:
40007:
40001:
39999:
39990:
39986:
39985:
39978:
39975:
39974:
39967:
39966:
39959:
39952:
39944:
39935:
39934:
39930:
39929:
39910:
39894:
39892:
39889:
39870:
39854:
39852:
39848:
39847:
39808:
39777:
39775:
39772:
39733:
39721:
39701:
39699:
39695:
39694:
39665:
39635:
39633:
39630:
39601:
39568:
39545:
39542:
39527:Mass flow rate
39524:
39520:
39519:
39517:
39515:
39496:
39493:
39491:
39472:
39457:
39453:
39452:
39450:
39431:
39429:
39426:
39424:
39411:
39409:
39405:
39404:
39402:
39381:
39370:
39367:
39365:
39350:
39348:
39344:
39343:
39341:
39310:
39263:
39260:
39236:
39211:
39190:
39186:
39185:
39170:
39143:
39141:
39138:
39123:
39080:
39078:
39074:
39073:
39071:
39069:
39054:
39051:
39049:
39034:
39019:
39015:
39014:
39007:
39000:
38997:
38994:
38991:
38988:
38985:
38981:
38980:
38977:
38975:
38969:
38966:
38965:
38955:
38954:
38947:
38940:
38932:
38926:
38925:
38920:
38915:
38910:
38905:
38900:
38893:
38892:External links
38890:
38888:
38887:
38873:
38853:
38846:
38824:
38802:. 4th Series.
38783:
38761:
38752:
38739:
38732:
38711:
38698:
38692:978-0077295493
38691:
38656:
38646:
38640:978-0135034521
38639:
38621:
38608:
38595:
38589:978-0195371239
38588:
38559:
38552:
38534:
38527:
38509:
38503:978-0135160626
38502:
38475:
38468:
38448:
38438:
38412:
38405:
38385:
38359:
38332:
38326:978-0983563334
38325:
38302:
38296:978-1449063337
38295:
38272:
38242:
38235:
38214:
38207:
38186:
38179:
38146:
38134:
38109:
38077:
38070:
38041:
38039:
38036:
38035:
38034:
38029:
38024:
38019:
38014:
38012:Central moment
38007:
38004:
37992:
37983:
37978:
37973:
37968:
37965:
37961:
37957:
37936:
37914:
37891:
37888:
37882:
37877:
37871:
37867:
37862:
37858:
37855:
37851:
37846:
37839:
37833:
37826:
37822:
37817:
37813:
37810:
37806:
37801:
37794:
37788:
37765:
37741:
37736:
37714:
37692:
37688:
37684:
37680:
37677:
37673:
37651:
37628:
37620:
37616:
37611:
37606:
37603:
37599:
37591:
37587:
37582:
37577:
37574:
37570:
37562:
37558:
37553:
37548:
37545:
37525:
37522:
37519:
37514:
37509:
37499:
37495:
37488:
37484:
37480:
37475:
37470:
37465:
37460:
37450:
37446:
37439:
37435:
37431:
37426:
37421:
37416:
37411:
37401:
37397:
37390:
37386:
37382:
37377:
37351:
37348:
37345:
37340:
37336:
37330:
37326:
37322:
37317:
37313:
37307:
37303:
37299:
37294:
37290:
37284:
37280:
37259:
37256:
37253:
37249:
37244:
37237:
37231:
37208:
37179:
37159:
37139:
37123:
37120:
37083:Platonic solid
37070:
37067:
37064:
37034:
37012:asymmetric top
37008:center of mass
36993:axis of figure
36962:
36958:
36935:
36931:
36908:
36904:
36882:
36859:
36854:
36846:
36842:
36838:
36836:
36833:
36831:
36828:
36827:
36824:
36821:
36817:
36813:
36809:
36807:
36804:
36803:
36800:
36797:
36795:
36792:
36788:
36784:
36780:
36779:
36777:
36772:
36768:
36747:
36741:
36735:
36729:
36724:
36720:
36715:
36709:
36703:
36680:
36658:
36641:
36640:Principal axes
36638:
36623:
36617:
36611:
36588:
36565:
36559:
36553:
36546:
36540:
36534:
36528:
36524:
36518:
36512:
36489:
36467:
36457:where vectors
36446:
36442:
36437:
36433:
36429:
36407:
36384:
36378:
36372:
36358:
36355:
36349:
36346:
36325:
36317:
36313:
36307:
36303:
36299:
36273:
36260:
36257:
36240:
36228:
36213:
36193:
36189:
36185:
36181:
36177:
36172:
36167:
36162:
36158:
36154:
36150:
36146:
36143:
36140:
36137:
36132:
36127:
36122:
36118:
36096:
36084:center of mass
36069:
36064:
36047:Main article:
36044:
36041:
36026:
36021:
36013:
36009:
36005:
36004:
35999:
35995:
35991:
35990:
35985:
35981:
35977:
35976:
35974:
35967:
35959:
35955:
35951:
35946:
35942:
35938:
35936:
35933:
35930:
35927:
35925:
35922:
35919:
35916:
35915:
35912:
35909:
35906:
35903:
35899:
35895:
35891:
35886:
35882:
35878:
35876:
35873:
35870:
35867:
35866:
35863:
35860:
35857:
35854:
35852:
35849:
35846:
35843:
35839:
35835:
35831:
35826:
35822:
35818:
35817:
35815:
35808:
35800:
35796:
35792:
35788:
35784:
35780:
35776:
35772:
35768:
35767:
35765:
35760:
35757:
35754:
35733:
35716:
35710:
35707:
35700:
35687:
35681:
35677:
35671:
35667:
35662:
35658:
35645:
35639:
35624:
35621:
35614:
35611:
35608:
35604:
35597:
35594:
35579:
35573:
35558:
35555:
35548:
35544:
35531:
35525:
35521:
35518:
35515:
35487:
35483:
35477:
35472:
35465:
35462:
35456:
35452:
35447:
35442:
35437:
35432:
35426:
35422:
35419:
35415:
35409:
35402:
35399:
35390:
35385:
35378:
35375:
35369:
35365:
35360:
35355:
35349:
35346:
35339:
35332:
35329:
35323:
35319:
35314:
35309:
35305:
35302:
35297:
35292:
35286:
35282:
35279:
35275:
35268:
35265:
35258:
35251:
35248:
35242:
35238:
35233:
35229:
35225:
35220:
35216:
35212:
35205:
35202:
35195:
35188:
35185:
35179:
35175:
35170:
35166:
35162:
35157:
35153:
35150:
35145:
35141:
35137:
35134:
35131:
35108:
35105:
35081:
35074:
35071:
35064:
35057:
35054:
35048:
35044:
35039:
35035:
35031:
35026:
35002:
34999:
34975:
34954:
34942:
34939:
34935:
34934:
34922:
34917:
34914:
34910:
34906:
34903:
34898:
34894:
34890:
34885:
34882:
34878:
34874:
34871:
34866:
34862:
34858:
34853:
34850:
34846:
34842:
34839:
34834:
34830:
34826:
34815:
34803:
34798:
34795:
34791:
34787:
34782:
34778:
34774:
34769:
34766:
34762:
34758:
34753:
34749:
34745:
34740:
34737:
34733:
34729:
34724:
34720:
34716:
34701:principal axes
34688:
34683:
34680:
34676:
34672:
34667:
34664:
34660:
34656:
34651:
34648:
34644:
34640:
34635:
34632:
34628:
34624:
34619:
34616:
34612:
34608:
34603:
34600:
34596:
34592:
34580:
34577:
34561:
34556:
34549:
34543:
34538:
34534:
34530:
34525:
34520:
34516:
34511:
34505:
34501:
34495:
34490:
34487:
34484:
34480:
34476:
34472:
34468:
34462:
34458:
34452:
34448:
34442:
34437:
34434:
34431:
34427:
34423:
34420:
34416:
34412:
34406:
34402:
34396:
34392:
34386:
34381:
34378:
34375:
34371:
34367:
34364:
34363:
34358:
34354:
34348:
34344:
34338:
34334:
34328:
34323:
34320:
34317:
34313:
34309:
34306:
34303:
34297:
34292:
34288:
34284:
34279:
34274:
34270:
34265:
34259:
34255:
34249:
34244:
34241:
34238:
34234:
34230:
34226:
34222:
34216:
34212:
34206:
34202:
34196:
34191:
34188:
34185:
34181:
34177:
34174:
34173:
34168:
34164:
34158:
34154:
34148:
34144:
34138:
34133:
34130:
34127:
34123:
34119:
34116:
34112:
34108:
34102:
34098:
34092:
34088:
34082:
34077:
34074:
34071:
34067:
34063:
34060:
34057:
34051:
34046:
34042:
34038:
34033:
34028:
34024:
34019:
34013:
34009:
34003:
33998:
33995:
33992:
33988:
33984:
33983:
33981:
33976:
33971:
33963:
33960:
33956:
33952:
33948:
33945:
33941:
33937:
33934:
33930:
33927:
33923:
33919:
33916:
33915:
33910:
33907:
33903:
33899:
33896:
33892:
33889:
33885:
33881:
33877:
33874:
33870:
33866:
33863:
33862:
33857:
33854:
33850:
33846:
33843:
33839:
33836:
33832:
33828:
33825:
33821:
33818:
33814:
33810:
33809:
33807:
33802:
33799:
33797:
33793:
33785:
33781:
33777:
33773:
33769:
33765:
33761:
33757:
33753:
33752:
33747:
33743:
33739:
33735:
33731:
33727:
33723:
33719:
33715:
33714:
33709:
33705:
33701:
33697:
33693:
33689:
33685:
33681:
33677:
33676:
33674:
33669:
33665:
33661:
33660:
33657:
33652:
33648:
33642:
33638:
33632:
33628:
33622:
33617:
33614:
33611:
33607:
33595:
33592:
33589:
33583:
33576:
33574:
33567:
33564:
33560:
33556:
33551:
33548:
33544:
33540:
33539:
33536:
33531:
33527:
33521:
33517:
33511:
33507:
33501:
33496:
33493:
33490:
33486:
33474:
33471:
33468:
33462:
33455:
33453:
33446:
33443:
33439:
33435:
33430:
33427:
33423:
33419:
33418:
33415:
33410:
33406:
33400:
33396:
33390:
33386:
33380:
33375:
33372:
33369:
33365:
33353:
33350:
33347:
33341:
33334:
33332:
33325:
33322:
33318:
33314:
33309:
33306:
33302:
33298:
33297:
33284:
33281:
33266:
33263:
33259:
33236:
33233:
33229:
33208:
33188:
33168:
33146:
33141:
33134:
33128:
33123:
33119:
33115:
33110:
33105:
33101:
33096:
33090:
33086:
33080:
33075:
33072:
33069:
33065:
33061:
33057:
33053:
33047:
33043:
33037:
33033:
33027:
33022:
33019:
33016:
33012:
33008:
33005:
33001:
32997:
32991:
32987:
32981:
32977:
32971:
32966:
32963:
32960:
32956:
32952:
32949:
32948:
32943:
32939:
32933:
32929:
32923:
32919:
32913:
32908:
32905:
32902:
32898:
32894:
32891:
32888:
32882:
32877:
32873:
32869:
32864:
32859:
32855:
32850:
32844:
32840:
32834:
32829:
32826:
32823:
32819:
32815:
32811:
32807:
32801:
32797:
32791:
32787:
32781:
32776:
32773:
32770:
32766:
32762:
32759:
32758:
32753:
32749:
32743:
32739:
32733:
32729:
32723:
32718:
32715:
32712:
32708:
32704:
32701:
32697:
32693:
32687:
32683:
32677:
32673:
32667:
32662:
32659:
32656:
32652:
32648:
32645:
32642:
32636:
32631:
32627:
32623:
32618:
32613:
32609:
32604:
32598:
32594:
32588:
32583:
32580:
32577:
32573:
32569:
32568:
32566:
32561:
32556:
32548:
32545:
32541:
32537:
32533:
32530:
32526:
32522:
32518:
32515:
32511:
32507:
32506:
32501:
32498:
32494:
32490:
32486:
32483:
32479:
32475:
32471:
32468:
32464:
32460:
32459:
32454:
32451:
32447:
32443:
32439:
32436:
32432:
32428:
32424:
32421:
32417:
32413:
32412:
32410:
32405:
32400:
32392:
32388:
32384:
32380:
32376:
32372:
32368:
32364:
32360:
32359:
32354:
32350:
32346:
32342:
32338:
32334:
32330:
32326:
32322:
32321:
32316:
32312:
32308:
32304:
32300:
32296:
32292:
32288:
32284:
32283:
32281:
32276:
32272:
32248:
32228:
32208:
32203:
32198:
32193:
32189:
32185:
32180:
32175:
32170:
32165:
32161:
32138:
32134:
32107:
32103:
32099:
32095:
32091:
32087:
32083:
32078:
32074:
32052:
32029:
32026:
32020:
32016:
32011:
32007:
32004:
32000:
31996:
31993:
31988:
31984:
31980:
31977:
31974:
31971:
31967:
31963:
31959:
31948:
31943:
31939:
31936:
31932:
31928:
31925:
31920:
31916:
31912:
31908:
31886:
31882:
31878:
31874:
31870:
31866:
31862:
31858:
31832:
31812:
31808:
31804:
31781:
31778:
31775:
31771:
31768:
31764:
31761:
31756:
31751:
31747:
31743:
31739:
31734:
31729:
31722:
31718:
31713:
31709:
31705:
31701:
31698:
31695:
31692:
31689:
31686:
31683:
31680:
31675:
31671:
31667:
31663:
31639:
31619:
31597:
31594:
31590:
31569:
31547:
31544:
31540:
31513:
31508:
31504:
31498:
31494:
31488:
31484:
31478:
31473:
31470:
31467:
31463:
31459:
31448:
31445:
31442:
31436:
31429:
31427:
31420:
31417:
31413:
31409:
31404:
31401:
31397:
31393:
31392:
31389:
31384:
31380:
31374:
31370:
31364:
31360:
31354:
31349:
31346:
31343:
31339:
31335:
31324:
31321:
31318:
31312:
31305:
31303:
31296:
31293:
31289:
31285:
31280:
31277:
31273:
31269:
31268:
31265:
31260:
31256:
31250:
31246:
31240:
31236:
31230:
31225:
31222:
31219:
31215:
31211:
31200:
31197:
31194:
31188:
31181:
31179:
31172:
31169:
31165:
31161:
31156:
31153:
31149:
31145:
31144:
31114:
31110:
31104:
31099:
31095:
31091:
31086:
31081:
31077:
31072:
31066:
31062:
31056:
31051:
31048:
31045:
31041:
31029:
31026:
31023:
31017:
31010:
31008:
31001:
30998:
30994:
30990:
30989:
30986:
30982:
30976:
30971:
30967:
30963:
30958:
30953:
30949:
30944:
30938:
30934:
30928:
30923:
30920:
30917:
30913:
30901:
30898:
30895:
30889:
30882:
30880:
30873:
30870:
30866:
30862:
30861:
30858:
30854:
30848:
30843:
30839:
30835:
30830:
30825:
30821:
30816:
30810:
30806:
30800:
30795:
30792:
30789:
30785:
30773:
30770:
30767:
30761:
30754:
30752:
30745:
30742:
30738:
30734:
30733:
30705:
30693:
30692:
30674:
30671:
30667:
30656:
30642:
30638:
30616:
30610:
30607:
30604:
30599:
30595:
30591:
30586:
30583:
30580:
30575:
30571:
30567:
30562:
30559:
30556:
30551:
30547:
30542:
30538:
30533:
30528:
30516:
30504:
30484:
30464:
30444:
30424:
30398:
30392:
30389:
30386:
30381:
30377:
30371:
30368:
30365:
30360:
30356:
30352:
30347:
30344:
30340:
30334:
30329:
30324:
30319:
30314:
30308:
30302:
30298:
30292:
30287:
30284:
30281:
30277:
30265:
30262:
30259:
30253:
30241:
30238:
30234:
30211:
30206:
30198:
30194:
30190:
30186:
30182:
30178:
30174:
30170:
30166:
30165:
30160:
30156:
30152:
30148:
30144:
30140:
30136:
30132:
30128:
30127:
30122:
30118:
30114:
30110:
30106:
30102:
30098:
30094:
30090:
30089:
30087:
30082:
30078:
30051:
30047:
30026:
30014:
30011:
29997:
29996:Inertia tensor
29994:
29977:
29953:
29949:
29923:
29917:
29914:
29905:
29899:
29891:
29883:
29880:
29873:
29867:
29864:
29855:
29849:
29844:
29838:
29835:
29829:
29826:
29824:
29822:
29816:
29813:
29806:
29800:
29795:
29789:
29784:
29779:
29775:
29768:
29764:
29758:
29753:
29750:
29747:
29743:
29739:
29735:
29731:
29725:
29722:
29716:
29710:
29707:
29699:
29694:
29688:
29683:
29678:
29674:
29669:
29663:
29660:
29652:
29648:
29642:
29637:
29634:
29631:
29627:
29623:
29620:
29617:
29615:
29613:
29608:
29603:
29597:
29592:
29587:
29582:
29578:
29571:
29567:
29561:
29556:
29553:
29550:
29546:
29542:
29539:
29537:
29533:
29529:
29525:
29524:
29501:
29498:
29474:
29453:
29427:
29421:
29418:
29410:
29405:
29399:
29394:
29389:
29385:
29380:
29374:
29371:
29365:
29362:
29360:
29356:
29353:
29352:
29348:
29341:
29338:
29332:
29327:
29322:
29317:
29314:
29309:
29304:
29299:
29295:
29291:
29285:
29282:
29276:
29273:
29271:
29267:
29264:
29263:
29259:
29252:
29249:
29243:
29238:
29233:
29228:
29225:
29221:
29217:
29213:
29207:
29202:
29197:
29194:
29188:
29185:
29178:
29174:
29170:
29167:
29166:
29162:
29156:
29151:
29146:
29143:
29137:
29134:
29127:
29123:
29119:
29112:
29109:
29103:
29099:
29094:
29088:
29083:
29078:
29075:
29069:
29066:
29059:
29055:
29049:
29046:
29039:
29034:
29030:
29026:
29023:
29022:
29018:
29013:
29007:
29002:
28997:
28994:
28988:
28985:
28978:
28974:
28968:
28965:
28958:
28954:
28950:
28945:
28939:
28934:
28929:
28926:
28920:
28917:
28910:
28906:
28900:
28897:
28890:
28886:
28884:
28861:
28858:
28833:
28828:
28823:
28803:
28799:
28793:
28788:
28783:
28780:
28774:
28771:
28764:
28760:
28756:
28749:
28746:
28740:
28736:
28731:
28725:
28720:
28715:
28712:
28706:
28703:
28696:
28692:
28686:
28683:
28676:
28671:
28667:
28663:
28658:
28652:
28647:
28642:
28639:
28633:
28630:
28623:
28619:
28613:
28610:
28603:
28599:
28595:
28590:
28584:
28579:
28574:
28571:
28565:
28562:
28555:
28551:
28545:
28542:
28535:
28507:
28502:
28496:
28491:
28486:
28483:
28477:
28474:
28467:
28463:
28457:
28454:
28447:
28443:
28439:
28434:
28428:
28423:
28418:
28415:
28409:
28406:
28399:
28395:
28389:
28386:
28379:
28375:
28372:
28370:
28368:
28364:
28358:
28353:
28348:
28343:
28338:
28332:
28329:
28323:
28318:
28314:
28310:
28306:
28300:
28295:
28290:
28285:
28280:
28274:
28271:
28265:
28260:
28256:
28252:
28249:
28247:
28243:
28238:
28232:
28227:
28222:
28217:
28213:
28208:
28207:
28181:
28178:
28155:
28149:
28141:
28138:
28128:
28125:
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13097:
13093:
13088:
13084:
13078:
13072:
13065:
13061:
13055:
13050:
13047:
13044:
13040:
13035:
13031:
13027:
13022:
13016:
13011:
13006:
13003:
12999:
12994:
12990:
12986:
12980:
12975:
12970:
12967:
12963:
12958:
12952:
12948:
12942:
12937:
12934:
12931:
12927:
12922:
12916:
12913:
12908:
12899:
12874:
12870:
12865:
12860:
12855:
12850:
12845:
12840:
12820:
12816:
12809:
12803:
12798:
12793:
12788:
12783:
12780:
12776:
12771:
12767:
12763:
12756:
12750:
12745:
12740:
12735:
12730:
12727:
12723:
12718:
12712:
12708:
12702:
12697:
12694:
12691:
12687:
12681:
12678:
12673:
12668:
12663:
12658:
12653:
12648:
12641:
12637:
12631:
12626:
12623:
12620:
12616:
12610:
12607:
12602:
12593:
12570:
12565:
12541:
12536:
12514:
12511:
12508:
12505:
12502:
12499:
12496:
12493:
12488:
12484:
12463:
12452:center of mass
12447:
12446:Kinetic energy
12444:
12430:
12409:
12404:
12399:
12393:
12388:
12383:
12379:
12372:
12368:
12362:
12357:
12354:
12351:
12347:
12343:
12340:
12334:
12328:
12303:
12299:
12277:
12273:
12266:
12260:
12255:
12251:
12246:
12240:
12235:
12229:
12224:
12219:
12215:
12208:
12204:
12198:
12193:
12190:
12187:
12183:
12179:
12175:
12171:
12167:
12145:
12141:
12136:
12131:
12126:
12121:
12116:
12111:
12091:
12086:
12081:
12076:
12073:
12059:center of mass
12045:
12041:
12018:
12014:
11988:
11984:
11977:
11971:
11966:
11961:
11956:
11951:
11945:
11941:
11935:
11930:
11927:
11924:
11920:
11915:
11911:
11907:
11902:
11897:
11893:
11888:
11883:
11878:
11874:
11870:
11865:
11860:
11855:
11849:
11845:
11839:
11834:
11831:
11828:
11824:
11820:
11816:
11812:
11809:
11807:
11805:
11801:
11794:
11788:
11783:
11778:
11773:
11768:
11765:
11761:
11756:
11752:
11747:
11742:
11737:
11731:
11727:
11721:
11716:
11713:
11710:
11706:
11702:
11699:
11697:
11695:
11690:
11685:
11680:
11675:
11670:
11665:
11659:
11655:
11649:
11644:
11641:
11638:
11634:
11630:
11627:
11625:
11622:
11618:
11617:
11596:
11574:
11547:
11542:
11536:
11533:
11529:
11525:
11521:
11517:
11513:
11509:
11506:
11505:
11500:
11496:
11492:
11489:
11487:
11484:
11480:
11476:
11472:
11471:
11466:
11462:
11458:
11454:
11450:
11446:
11443:
11441:
11438:
11437:
11435:
11430:
11427:
11425:
11422:
11418:
11414:
11410:
11409:
11405:
11400:
11396:
11392:
11388:
11385:
11383:
11380:
11376:
11372:
11368:
11367:
11347:
11342:
11338:
11334:
11329:
11325:
11321:
11316:
11312:
11308:
11305:
11301:
11279:
11275:
11271:
11257:Note that the
11254:
11251:
11237:
11213:
11209:
11186:
11162:
11156:
11151:
11146:
11141:
11136:
11133:
11129:
11125:
11120:
11115:
11092:
11088:
11083:
11078:
11073:
11068:
11063:
11058:
11037:
11014:
11009:
10985:
10980:
10958:
10955:
10952:
10949:
10946:
10943:
10940:
10937:
10932:
10928:
10907:
10880:
10877:
10865:
10859:
10856:
10850:
10844:
10839:
10835:
10831:
10807:
10802:
10780:
10769:center of mass
10751:
10747:
10723:
10720:
10714:
10709:
10702:
10699:
10692:
10687:
10680:
10677:
10652:
10648:
10643:
10636:
10633:
10626:
10621:
10614:
10611:
10581:
10577:
10573:
10569:
10563:
10556:
10553:
10544:
10540:
10536:
10530:
10526:
10520:
10515:
10512:
10509:
10505:
10500:
10496:
10490:
10487:
10481:
10477:
10471:
10466:
10462:
10458:
10452:
10448:
10442:
10437:
10434:
10431:
10427:
10422:
10418:
10415:
10413:
10411:
10407:
10402:
10398:
10393:
10386:
10383:
10374:
10370:
10366:
10361:
10357:
10353:
10348:
10341:
10338:
10329:
10325:
10321:
10318:
10314:
10310:
10305:
10298:
10295:
10286:
10282:
10278:
10272:
10268:
10262:
10257:
10254:
10251:
10247:
10243:
10240:
10238:
10235:
10231:
10230:
10204:
10200:
10196:
10191:
10184:
10181:
10172:
10168:
10164:
10159:
10155:
10151:
10146:
10139:
10136:
10127:
10123:
10119:
10116:
10113:
10110:
10108:
10106:
10102:
10098:
10093:
10086:
10083:
10074:
10070:
10066:
10063:
10057:
10054:
10048:
10045:
10039:
10036:
10030:
10027:
10022:
10015:
10012:
10003:
9999:
9995:
9992:
9986:
9983:
9977:
9974:
9971:
9969:
9965:
9960:
9955:
9954:
9932:
9925:
9922:
9915:
9909:
9906:
9900:
9895:
9888:
9885:
9859:
9854:
9831:
9808:
9801:
9798:
9771:
9768:
9743:
9739:
9735:
9730:
9725:
9720:
9717:
9713:
9709:
9705:
9701:
9696:
9691:
9686:
9683:
9679:
9675:
9670:
9665:
9642:
9620:
9598:
9576:
9553:
9549:
9519:
9514:
9488:
9483:
9478:
9471:
9467:
9463:
9458:
9453:
9448:
9443:
9438:
9435:
9432:
9428:
9424:
9421:
9419:
9416:
9412:
9411:
9408:
9403:
9398:
9391:
9387:
9381:
9376:
9373:
9370:
9366:
9362:
9359:
9357:
9354:
9350:
9349:
9328:
9303:
9300:
9297:
9294:
9291:
9288:
9285:
9282:
9277:
9273:
9252:
9228:
9225:
9205:
9200:
9177:
9173:
9169:
9165:
9161:
9156:
9153:
9148:
9143:
9139:
9132:
9127:
9121:
9118:
9113:
9104:
9080:
9075:
9053:
9026:
9022:
9018:
9014:
9009:
9003:
8999:
8993:
8988:
8985:
8982:
8978:
8973:
8967:
8964:
8959:
8955:
8949:
8942:
8939:
8930:
8926:
8922:
8916:
8912:
8906:
8901:
8898:
8895:
8891:
8886:
8882:
8878:
8874:
8871:
8867:
8861:
8854:
8851:
8844:
8839:
8832:
8829:
8820:
8815:
8811:
8807:
8801:
8797:
8791:
8786:
8783:
8780:
8776:
8771:
8765:
8761:
8755:
8752:
8747:
8744:
8742:
8740:
8737:
8733:
8728:
8724:
8719:
8712:
8709:
8700:
8696:
8692:
8688:
8684:
8680:
8676:
8671:
8667:
8662:
8655:
8652:
8643:
8639:
8635:
8631:
8627:
8621:
8617:
8611:
8606:
8603:
8600:
8596:
8590:
8587:
8582:
8579:
8577:
8575:
8572:
8567:
8562:
8557:
8552:
8547:
8540:
8536:
8530:
8525:
8522:
8519:
8515:
8509:
8506:
8501:
8498:
8496:
8488:
8484:
8483:
8463:
8462:Kinetic energy
8460:
8432:
8427:
8404:
8398:
8395:
8389:
8383:
8378:
8374:
8370:
8347:
8342:
8337:
8333:
8329:
8323:
8319:
8313:
8309:
8305:
8299:
8294:
8270:
8265:
8238:
8235:
8232:
8229:
8227:
8223:
8216:
8213:
8204:
8200:
8196:
8190:
8186:
8180:
8175:
8172:
8169:
8165:
8161:
8160:
8157:
8153:
8149:
8144:
8139:
8134:
8131:
8129:
8125:
8118:
8115:
8106:
8102:
8098:
8095:
8094:
8073:
8062:center of mass
8043:
8039:
8035:
8031:
8025:
8018:
8015:
8006:
8002:
7998:
7992:
7988:
7982:
7977:
7974:
7971:
7967:
7962:
7958:
7952:
7949:
7943:
7939:
7933:
7928:
7924:
7920:
7914:
7910:
7904:
7899:
7896:
7893:
7889:
7884:
7880:
7877:
7875:
7873:
7869:
7864:
7860:
7855:
7848:
7845:
7836:
7832:
7828:
7824:
7820:
7816:
7811:
7804:
7801:
7792:
7788:
7784:
7778:
7774:
7768:
7763:
7760:
7757:
7753:
7749:
7746:
7744:
7742:
7737:
7732:
7727:
7722:
7717:
7712:
7707:
7703:
7697:
7692:
7689:
7686:
7682:
7678:
7675:
7673:
7670:
7666:
7665:
7653:
7650:
7624:
7620:
7615:
7608:
7605:
7596:
7592:
7588:
7584:
7581:
7577:
7573:
7568:
7561:
7558:
7549:
7545:
7541:
7538:
7532:
7529:
7523:
7520:
7516:
7512:
7507:
7502:
7497:
7494:
7490:
7486:
7483:
7481:
7477:
7472:
7467:
7466:
7463:
7458:
7451:
7448:
7441:
7435:
7432:
7426:
7421:
7414:
7411:
7403:
7398:
7395:
7390:
7384:
7381:
7374:
7366:
7362:
7358:
7351:
7346:
7341:
7335:
7332:
7330:
7326:
7319:
7316:
7309:
7308:
7286:
7279:
7276:
7269:
7263:
7260:
7254:
7249:
7242:
7239:
7213:
7208:
7185:
7162:
7157:
7134:
7109:
7087:
7065:
7040:
7036:
7032:
7027:
7022:
7017:
7014:
7010:
7006:
7002:
6998:
6994:
6989:
6985:
6980:
6975:
6969:
6965:
6961:
6957:
6954:
6952:
6948:
6943:
6938:
6937:
6934:
6930:
6926:
6921:
6916:
6911:
6908:
6906:
6902:
6897:
6892:
6889:
6888:
6866:
6861:
6838:
6817:
6814:
6811:
6808:
6805:
6802:
6799:
6796:
6791:
6787:
6766:
6736:
6733:
6706:
6703:
6690:
6685:
6681:
6677:
6672:
6669:
6664:
6661:
6637:
6632:
6628:
6624:
6618:
6615:
6609:
6606:
6604:
6602:
6597:
6593:
6588:
6581:
6578:
6572:
6566:
6563:
6557:
6554:
6550:
6546:
6543:
6540:
6537:
6535:
6533:
6528:
6523:
6520:
6515:
6509:
6505:
6498:
6495:
6489:
6484:
6480:
6474:
6470:
6463:
6460:
6454:
6451:
6446:
6442:
6437:
6432:
6429:
6423:
6420:
6414:
6411:
6409:
6407:
6404:
6401:
6395:
6390:
6384:
6380:
6376:
6371:
6367:
6362:
6357:
6354:
6348:
6345:
6337:
6332:
6329:
6325:
6321:
6318:
6315:
6309:
6305:
6301:
6298:
6295:
6292:
6289:
6283:
6280:
6272:
6267:
6264:
6260:
6256:
6253:
6251:
6242:
6239:
6235:
6231:
6230:
6210:
6188:
6183:
6179:
6175:
6170:
6166:
6162:
6157:
6153:
6149:
6144:
6140:
6136:
6131:
6127:
6123:
6120:
6117:
6097:
6077:
6057:
6035:
6030:
6026:
6022:
6017:
6013:
6009:
6004:
6000:
5996:
5991:
5987:
5956:
5955:
5943:
5923:
5918:
5914:
5910:
5907:
5904:
5901:
5892:
5888:
5878:
5875:
5871:
5867:
5862:
5857:
5852:
5849:
5844:
5833:
5829:
5819:
5816:
5812:
5808:
5803:
5799:
5778:
5767:
5755:
5752:
5747:
5743:
5739:
5736:
5733:
5713:
5708:
5704:
5700:
5695:
5692:
5687:
5682:
5677:
5673:
5667:
5664:
5661:
5658:
5655:
5652:
5649:
5645:
5642:
5638:
5635:
5630:
5626:
5622:
5617:
5612:
5608:
5602:
5599:
5594:
5590:
5586:
5583:
5580:
5574:
5570:
5565:
5560:
5556:
5552:
5542:
5539:
5535:
5514:
5511:
5507:
5504:
5500:
5497:
5494:
5491:
5488:
5468:
5444:
5435:, and density
5424:
5404:
5389:
5377:
5374:
5371:
5368:
5365:
5345:
5340:
5334:
5330:
5326:
5320:
5316:
5310:
5305:
5301:
5295:
5290:
5285:
5281:
5274:
5268:
5264:
5261:
5255:
5249:
5246:
5238:
5235:
5230:
5225:
5219:
5214:
5210:
5204:
5201:
5197:
5193:
5190:
5187:
5183:
5178:
5174:
5169:
5163:
5160:
5152:
5149:
5144:
5140:
5136:
5133:
5130:
5124:
5120:
5115:
5110:
5106:
5102:
5092:
5089:
5085:
5064:
5044:
5024:
5004:
4969:Main article:
4966:
4963:
4950:
4930:
4906:
4886:
4862:
4832:
4812:
4792:
4789:
4786:
4783:
4780:
4777:
4774:
4753:
4732:
4729:
4726:
4723:
4720:
4717:
4714:
4694:
4669:
4666:
4661:
4656:
4652:
4648:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4622:
4617:
4613:
4609:
4604:
4600:
4571:
4566:
4562:
4556:
4552:
4546:
4542:
4538:
4533:
4529:
4501:
4496:
4491:
4487:
4481:
4477:
4471:
4466:
4463:
4460:
4456:
4452:
4447:
4443:
4420:
4416:
4412:
4390:
4385:
4380:
4376:
4370:
4366:
4360:
4355:
4352:
4349:
4345:
4339:
4335:
4328:
4325:
4320:
4315:
4310:
4304:
4300:
4296:
4292:
4285:
4281:
4274:
4271:
4264:
4259:
4256:
4253:
4249:
4245:
4240:
4235:
4230:
4225:
4220:
4213:
4209:
4202:
4199:
4192:
4187:
4184:
4181:
4177:
4173:
4164:
4141:
4119:
4115:
4092:
4088:
4067:
4043:
4021:
4017:
4013:
3994:
3993:
3987:
3981:
3975:
3968:
3960:
3957:
3955:
3952:
3951:
3950:
3937:
3933:
3928:
3922:
3919:
3914:
3909:
3905:
3900:
3894:
3891:
3886:
3881:
3878:
3874:
3870:
3865:
3862:
3858:
3846:
3833:
3827:
3823:
3819:
3813:
3808:
3805:
3801:
3789:
3776:
3770:
3766:
3762:
3756:
3751:
3748:
3744:
3721:
3715:
3711:
3707:
3701:
3696:
3693:
3689:
3677:
3664:
3659:
3655:
3649:
3644:
3641:
3637:
3633:
3628:
3625:
3621:
3596:
3593:
3581:
3549:
3546:
3542:
3519:
3516:
3512:
3495:
3492:
3483:
3480:
3457:
3453:
3446:
3443:
3435:
3431:
3426:
3422:
3418:
3415:
3412:
3405:
3402:
3394:
3390:
3385:
3381:
3374:
3368:
3361:
3352:
3348:
3343:
3340:
3320:
3299:
3295:
3291:
3288:
3285:
3278:
3252:
3246:
3243:
3237:
3233:
3227:
3220:
3211:
3207:
3202:
3199:
3179:
3171:
3167:
3162:
3159:
3156:
3149:
3143:
3140:
3134:
3125:
3104:
3076:
3064:
3061:
3048:
3028:
3020:
3016:
3012:
3005:
3001:
2997:
2994:
2991:
2985:
2978:
2969:
2964:
2961:
2958:
2952:
2947:
2943:
2922:
2902:
2878:
2858:
2838:
2818:
2810:
2806:
2801:
2798:
2795:
2788:
2779:
2756:
2752:
2725:
2688:
2685:
2670:
2666:
2662:
2640:
2636:
2632:
2629:
2626:
2604:
2599:
2595:
2591:
2586:
2583:
2578:
2573:
2569:
2564:
2558:
2554:
2550:
2546:
2540:
2537:
2532:
2528:
2524:
2520:
2516:
2511:
2508:
2503:
2494:
2466:
2460:
2457:
2451:
2448:
2445:
2441:
2435:
2431:
2427:
2424:
2421:
2419:
2417:
2413:
2408:
2403:
2398:
2394:
2390:
2385:
2381:
2377:
2372:
2367:
2363:
2359:
2354:
2349:
2345:
2342:
2339:
2337:
2335:
2331:
2326:
2322:
2318:
2314:
2310:
2306:
2302:
2298:
2294:
2290:
2286:
2282:
2279:
2277:
2274:
2270:
2269:
2244:
2222:
2218:
2214:
2210:
2206:
2179:
2175:
2171:
2168:
2165:
2136:
2132:
2128:
2125:
2122:
2101:
2079:
2051:
2048:
2019:
2013:
2010:
2004:
2001:
1998:
1994:
1988:
1984:
1980:
1977:
1974:
1972:
1970:
1966:
1961:
1956:
1951:
1947:
1943:
1938:
1934:
1930:
1925:
1920:
1916:
1912:
1907:
1902:
1898:
1895:
1892:
1890:
1888:
1885:
1881:
1877:
1873:
1869:
1866:
1863:
1859:
1855:
1851:
1847:
1843:
1839:
1836:
1834:
1831:
1827:
1826:
1805:
1801:
1798:
1794:
1772:
1768:
1764:
1760:
1756:
1734:
1708:
1686:
1664:
1660:
1656:
1652:
1648:
1625:
1620:
1616:
1612:
1609:
1606:
1586:
1566:
1553:
1550:
1542:
1539:
1518:
1513:
1509:
1505:
1502:
1499:
1441:
1436:
1432:
1428:
1425:
1422:
1383:
1380:
1377:
1374:
1371:
1349:applied torque
1330:figure skaters
1314:
1309:
1306:
1301:
1298:
1227:
1224:
1207:kinetic energy
1184:Leonhard Euler
1151:
1131:
1109:
1105:
1101:
1054:
1051:
1032:principal axes
952:
951:
942:
941:
933:
932:
931:
930:
929:
911:
910:
908:
907:
900:
893:
885:
882:
881:
880:
879:
866:
849:
848:
844:
843:
838:
833:
828:
823:
818:
813:
808:
803:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
738:
733:
728:
722:
721:
718:
717:
714:
713:
710:
709:
690:
689:
684:
679:
674:
672:Coriolis force
669:
668:
667:
657:
652:
647:
641:
640:
635:
634:
631:
630:
626:
625:
620:
615:
614:
613:
608:
598:
593:
588:
581:
570:
569:
568:
563:
550:
549:
544:
539:
534:
527:
522:
517:
511:
510:
507:
506:
503:
502:
499:
498:
495:
494:
489:
484:
479:
474:
469:
463:
457:
455:
448:
445:
442:
441:
438:
437:
433:
432:
427:
422:
417:
412:
407:
402:
397:
391:
386:
380:
375:
364:
359:
354:
349:
344:
343:
342:
337:
327:
322:
317:
312:
306:
305:
302:
301:
298:
297:
293:
292:
287:
282:
277:
272:
267:
262:
257:
251:
250:
247:
246:
243:
242:
240:
239:
234:
229:
223:
220:
219:
214:
200:
197:
191:
187:
181:
163:
162:
156:
155:
147:
146:
138:
132:
131:
118:
115:
110:
107:
97:
92:
89:
88:
85:
82:
79:
78:
75:
69:
68:
63:
61:Common symbols
60:
57:
56:
50:
42:
41:
26:
9:
6:
4:
3:
2:
40640:
40629:
40626:
40624:
40621:
40619:
40616:
40614:
40611:
40610:
40608:
40599:
40596:
40595:
40586:
40583:
40581:
40578:
40576:
40573:
40571:
40568:
40566:
40563:
40561:
40558:
40556:
40553:
40551:
40548:
40546:
40543:
40541:
40538:
40536:
40533:
40531:
40528:
40526:
40523:
40522:
40520:
40518:
40514:
40504:
40501:
40499:
40496:
40494:
40491:
40489:
40486:
40484:
40481:
40479:
40476:
40474:
40471:
40469:
40466:
40464:
40461:
40460:
40458:
40454:
40448:
40445:
40443:
40440:
40438:
40435:
40433:
40430:
40428:
40425:
40423:
40422:Metric tensor
40420:
40418:
40415:
40413:
40410:
40409:
40407:
40403:
40400:
40396:
40390:
40387:
40385:
40382:
40380:
40377:
40375:
40372:
40370:
40367:
40365:
40362:
40360:
40357:
40355:
40352:
40350:
40347:
40345:
40342:
40340:
40337:
40335:
40334:Exterior form
40332:
40330:
40327:
40325:
40322:
40320:
40317:
40315:
40312:
40310:
40307:
40305:
40302:
40300:
40297:
40296:
40294:
40288:
40281:
40278:
40276:
40273:
40271:
40268:
40266:
40263:
40261:
40258:
40256:
40253:
40251:
40248:
40246:
40243:
40241:
40238:
40236:
40233:
40231:
40228:
40227:
40225:
40223:
40219:
40213:
40210:
40208:
40207:Tensor bundle
40205:
40203:
40200:
40198:
40195:
40193:
40190:
40188:
40185:
40183:
40180:
40178:
40175:
40173:
40170:
40168:
40165:
40164:
40162:
40156:
40150:
40147:
40145:
40142:
40140:
40137:
40135:
40132:
40130:
40127:
40125:
40122:
40120:
40117:
40115:
40112:
40110:
40107:
40106:
40104:
40100:
40090:
40087:
40085:
40082:
40080:
40077:
40075:
40072:
40070:
40067:
40066:
40064:
40059:
40056:
40054:
40051:
40050:
40047:
40041:
40038:
40036:
40033:
40031:
40028:
40026:
40023:
40021:
40018:
40016:
40013:
40011:
40008:
40006:
40003:
40002:
40000:
39998:
39994:
39991:
39987:
39983:
39982:
39976:
39972:
39965:
39960:
39958:
39953:
39951:
39946:
39945:
39942:
39928:
39919:
39914:
39911:
39903:
39898:
39895:
39893:
39890:
39888:
39879:
39874:
39871:
39863:
39858:
39855:
39853:
39850:
39849:
39846:
39837:
39832:
39827:
39822:
39817:
39812:
39809:
39807:
39798:
39793:
39788:
39787:
39781:
39778:
39776:
39773:
39771:
39762:
39757:
39752:
39747:
39742:
39737:
39734:
39732:
39720:
39715:
39710:
39705:
39702:
39700:
39697:
39696:
39693:
39689:
39677:
39669:
39666:
39664:
39655:
39649:
39644:
39639:
39636:
39634:
39631:
39629:
39625:
39613:
39605:
39602:
39600:
39596:
39587:
39582:
39577:
39572:
39569:
39567:
39543:
39540:
39528:
39525:
39522:
39521:
39518:
39516:
39514:
39505:
39500:
39497:
39494:
39492:
39489:
39486:
39482:
39478:
39473:
39471:
39466:
39461:
39458:
39455:
39454:
39451:
39442:
39441:
39435:
39432:
39430:
39427:
39425:
39420:
39415:
39412:
39410:
39407:
39406:
39403:
39401:
39392:
39391:
39385:
39382:
39380:
39375:
39371:
39368:
39366:
39364:
39359:
39354:
39351:
39349:
39346:
39345:
39342:
39340:
39331:
39330:
39324:
39319:
39314:
39313:angular speed
39311:
39309:
39305:
39299:
39298:
39292:
39285:
39280:
39273:
39268:
39264:
39261:
39256:
39251:
39245:
39240:
39237:
39235:
39230:
39225:
39220:
39215:
39212:
39210:
39206:
39200:
39195:
39191:
39188:
39187:
39183:
39175:
39171:
39169:
39164:
39163:
39157:
39152:
39147:
39144:
39142:
39139:
39137:
39132:
39127:
39124:
39122:
39118:
39112:
39106:
39100:
39095:
39089:
39084:
39081:
39079:
39076:
39075:
39072:
39070:
39068:
39063:
39058:
39055:
39052:
39050:
39048:
39043:
39038:
39035:
39033:
39028:
39023:
39020:
39017:
39016:
39012:
39008:
39005:
39001:
38998:
38995:
38992:
38989:
38986:
38983:
38982:
38972:
38967:
38963:
38960:
38953:
38948:
38946:
38941:
38939:
38934:
38933:
38930:
38924:
38921:
38919:
38916:
38914:
38911:
38909:
38906:
38904:
38901:
38899:
38896:
38895:
38876:
38870:
38867:. MIT Press.
38866:
38865:
38857:
38849:
38847:0-19-853248-2
38843:
38839:
38835:
38828:
38813:
38809:
38805:
38801:
38794:
38787:
38779:
38772:
38765:
38756:
38749:
38743:
38735:
38733:0-201-02918-9
38729:
38725:
38718:
38716:
38708:
38702:
38694:
38688:
38684:
38677:
38675:
38673:
38671:
38669:
38667:
38665:
38663:
38661:
38650:
38642:
38636:
38632:
38625:
38618:
38612:
38605:
38599:
38591:
38585:
38581:
38574:
38572:
38570:
38568:
38566:
38564:
38555:
38553:9780748744473
38549:
38545:
38538:
38530:
38528:9780471216438
38524:
38520:
38513:
38505:
38499:
38495:
38488:
38486:
38484:
38482:
38480:
38471:
38465:
38461:
38460:
38452:
38445:
38441:
38435:
38431:
38426:
38425:
38416:
38408:
38402:
38398:
38397:
38389:
38378:September 30,
38374:
38370:
38363:
38352:September 30,
38347:
38343:
38336:
38328:
38322:
38318:
38317:
38309:
38307:
38298:
38292:
38288:
38287:
38279:
38277:
38268:
38261:
38259:
38257:
38255:
38253:
38251:
38249:
38247:
38238:
38236:0-387-00887-X
38232:
38228:
38221:
38219:
38210:
38208:0-201-07392-7
38204:
38200:
38193:
38191:
38182:
38180:0-03-097302-3
38176:
38171:
38170:
38161:
38159:
38157:
38155:
38153:
38151:
38143:
38137:
38131:
38127:
38123:
38122:
38113:
38097:
38093:
38092:
38084:
38082:
38073:
38067:
38063:
38059:
38055:
38054:
38046:
38042:
38033:
38030:
38028:
38025:
38023:
38022:Planar lamina
38020:
38018:
38015:
38013:
38010:
38009:
38003:
37990:
37976:
37971:
37966:
37902:
37889:
37886:
37875:
37869:
37853:
37824:
37808:
37734:
37675:
37639:
37626:
37618:
37614:
37609:
37604:
37601:
37597:
37589:
37585:
37580:
37575:
37572:
37568:
37560:
37556:
37551:
37546:
37543:
37523:
37520:
37517:
37512:
37507:
37497:
37493:
37486:
37482:
37478:
37473:
37468:
37463:
37458:
37448:
37444:
37437:
37433:
37429:
37424:
37419:
37414:
37409:
37399:
37395:
37388:
37384:
37380:
37375:
37365:
37349:
37346:
37343:
37338:
37334:
37328:
37324:
37320:
37315:
37311:
37305:
37301:
37297:
37292:
37288:
37282:
37278:
37257:
37254:
37251:
37197:
37177:
37157:
37137:
37128:
37119:
37117:
37113:
37109:
37107:
37102:
37099:
37095:
37091:
37086:
37084:
37068:
37065:
37062:
37053:
37049:
37032:
37023:
37021:
37020:spherical top
37017:
37016:symmetric top
37013:
37009:
37005:
37001:
36996:
36994:
36990:
36986:
36982:
36978:
36960:
36956:
36933:
36929:
36906:
36902:
36870:
36857:
36852:
36844:
36840:
36834:
36829:
36822:
36815:
36811:
36805:
36798:
36793:
36786:
36782:
36775:
36770:
36745:
36718:
36713:
36647:
36637:
36621:
36576:
36563:
36544:
36522:
36444:
36431:
36419:, such that,
36382:
36354:
36345:
36301:
36288:
36256:
36254:
36253:outer product
36238:
36227:
36211:
36183:
36175:
36170:
36152:
36138:
36135:
36130:
36120:
36085:
36067:
36050:
36040:
36037:
36024:
36019:
36011:
36007:
35997:
35993:
35983:
35979:
35972:
35965:
35957:
35953:
35949:
35944:
35940:
35934:
35931:
35928:
35923:
35920:
35917:
35910:
35907:
35904:
35897:
35893:
35889:
35884:
35880:
35874:
35871:
35868:
35861:
35858:
35855:
35850:
35847:
35844:
35837:
35833:
35829:
35824:
35820:
35813:
35806:
35798:
35794:
35786:
35782:
35774:
35770:
35763:
35758:
35755:
35752:
35744:
35741:
35739:
35732:
35727:
35714:
35698:
35675:
35660:
35637:
35612:
35609:
35606:
35602:
35546:
35523:
35519:
35516:
35513:
35505:
35503:
35498:
35485:
35481:
35475:
35470:
35454:
35445:
35440:
35435:
35424:
35420:
35417:
35413:
35407:
35388:
35383:
35367:
35358:
35353:
35337:
35321:
35312:
35303:
35300:
35295:
35284:
35280:
35277:
35273:
35256:
35240:
35231:
35227:
35218:
35214:
35210:
35193:
35177:
35168:
35164:
35155:
35151:
35148:
35143:
35139:
35135:
35132:
35129:
35079:
35062:
35046:
35037:
35033:
35024:
35016:direction is
34952:
34945:The distance
34938:
34915:
34912:
34908:
34904:
34901:
34896:
34892:
34888:
34883:
34880:
34876:
34872:
34869:
34864:
34860:
34856:
34851:
34848:
34844:
34840:
34837:
34832:
34828:
34816:
34796:
34793:
34789:
34785:
34780:
34776:
34772:
34767:
34764:
34760:
34756:
34751:
34747:
34743:
34738:
34735:
34731:
34727:
34722:
34718:
34706:
34705:
34704:
34702:
34681:
34678:
34674:
34670:
34665:
34662:
34658:
34654:
34649:
34646:
34642:
34638:
34633:
34630:
34626:
34622:
34617:
34614:
34610:
34606:
34601:
34598:
34594:
34576:
34559:
34554:
34547:
34541:
34536:
34532:
34528:
34523:
34518:
34514:
34509:
34503:
34499:
34493:
34488:
34485:
34482:
34478:
34470:
34466:
34460:
34456:
34450:
34446:
34440:
34435:
34432:
34429:
34425:
34421:
34414:
34410:
34404:
34400:
34394:
34390:
34384:
34379:
34376:
34373:
34369:
34365:
34356:
34352:
34346:
34342:
34336:
34332:
34326:
34321:
34318:
34315:
34311:
34307:
34301:
34295:
34290:
34286:
34282:
34277:
34272:
34268:
34263:
34257:
34253:
34247:
34242:
34239:
34236:
34232:
34224:
34220:
34214:
34210:
34204:
34200:
34194:
34189:
34186:
34183:
34179:
34175:
34166:
34162:
34156:
34152:
34146:
34142:
34136:
34131:
34128:
34125:
34121:
34117:
34110:
34106:
34100:
34096:
34090:
34086:
34080:
34075:
34072:
34069:
34065:
34061:
34055:
34049:
34044:
34040:
34036:
34031:
34026:
34022:
34017:
34011:
34007:
34001:
33996:
33993:
33990:
33986:
33979:
33974:
33969:
33961:
33958:
33954:
33946:
33943:
33939:
33935:
33928:
33925:
33921:
33917:
33908:
33905:
33901:
33897:
33890:
33887:
33883:
33875:
33872:
33868:
33864:
33855:
33852:
33848:
33844:
33837:
33834:
33830:
33826:
33819:
33816:
33812:
33805:
33800:
33798:
33791:
33783:
33779:
33771:
33767:
33759:
33755:
33745:
33741:
33733:
33729:
33721:
33717:
33707:
33703:
33695:
33691:
33683:
33679:
33672:
33667:
33655:
33650:
33646:
33640:
33636:
33630:
33626:
33620:
33615:
33612:
33609:
33605:
33581:
33575:
33565:
33562:
33558:
33554:
33549:
33546:
33542:
33534:
33529:
33525:
33519:
33515:
33509:
33505:
33499:
33494:
33491:
33488:
33484:
33460:
33454:
33444:
33441:
33437:
33433:
33428:
33425:
33421:
33413:
33408:
33404:
33398:
33394:
33388:
33384:
33378:
33373:
33370:
33367:
33363:
33339:
33333:
33323:
33320:
33316:
33312:
33307:
33304:
33300:
33280:
33264:
33261:
33257:
33234:
33231:
33227:
33206:
33186:
33166:
33157:
33144:
33139:
33132:
33126:
33121:
33117:
33113:
33108:
33103:
33099:
33094:
33088:
33084:
33078:
33073:
33070:
33067:
33063:
33055:
33051:
33045:
33041:
33035:
33031:
33025:
33020:
33017:
33014:
33010:
33006:
32999:
32995:
32989:
32985:
32979:
32975:
32969:
32964:
32961:
32958:
32954:
32950:
32941:
32937:
32931:
32927:
32921:
32917:
32911:
32906:
32903:
32900:
32896:
32892:
32886:
32880:
32875:
32871:
32867:
32862:
32857:
32853:
32848:
32842:
32838:
32832:
32827:
32824:
32821:
32817:
32809:
32805:
32799:
32795:
32789:
32785:
32779:
32774:
32771:
32768:
32764:
32760:
32751:
32747:
32741:
32737:
32731:
32727:
32721:
32716:
32713:
32710:
32706:
32702:
32695:
32691:
32685:
32681:
32675:
32671:
32665:
32660:
32657:
32654:
32650:
32646:
32640:
32634:
32629:
32625:
32621:
32616:
32611:
32607:
32602:
32596:
32592:
32586:
32581:
32578:
32575:
32571:
32564:
32559:
32554:
32546:
32543:
32539:
32531:
32528:
32524:
32516:
32513:
32509:
32499:
32496:
32492:
32484:
32481:
32477:
32469:
32466:
32462:
32452:
32449:
32445:
32437:
32434:
32430:
32422:
32419:
32415:
32408:
32403:
32398:
32390:
32386:
32378:
32374:
32366:
32362:
32352:
32348:
32340:
32336:
32328:
32324:
32314:
32310:
32302:
32298:
32290:
32286:
32279:
32274:
32260:
32246:
32226:
32206:
32201:
32191:
32183:
32178:
32168:
32163:
32159:
32136:
32132:
32123:
32118:
32105:
32097:
32089:
32081:
32076:
32072:
32040:
32027:
32024:
32018:
31991:
31986:
31982:
31978:
31975:
31972:
31969:
31923:
31918:
31914:
31910:
31880:
31872:
31849:
31844:
31842:
31838:
31831:
31827:
31826:outer product
31806:
31792:
31779:
31776:
31773:
31769:
31766:
31762:
31759:
31754:
31745:
31737:
31732:
31720:
31703:
31696:
31693:
31690:
31687:
31684:
31678:
31673:
31669:
31665:
31651:
31637:
31617:
31595:
31592:
31588:
31567:
31545:
31542:
31538:
31528:
31511:
31506:
31502:
31496:
31492:
31486:
31482:
31476:
31471:
31468:
31465:
31461:
31457:
31434:
31428:
31418:
31415:
31411:
31407:
31402:
31399:
31395:
31387:
31382:
31378:
31372:
31368:
31362:
31358:
31352:
31347:
31344:
31341:
31337:
31333:
31310:
31304:
31294:
31291:
31287:
31283:
31278:
31275:
31271:
31263:
31258:
31254:
31248:
31244:
31238:
31234:
31228:
31223:
31220:
31217:
31213:
31209:
31186:
31180:
31170:
31167:
31163:
31159:
31154:
31151:
31147:
31134:
31129:
31112:
31108:
31102:
31097:
31093:
31089:
31084:
31079:
31075:
31070:
31064:
31060:
31054:
31049:
31046:
31043:
31039:
31015:
31009:
30999:
30996:
30992:
30984:
30980:
30974:
30969:
30965:
30961:
30956:
30951:
30947:
30942:
30936:
30932:
30926:
30921:
30918:
30915:
30911:
30887:
30881:
30871:
30868:
30864:
30856:
30852:
30846:
30841:
30837:
30833:
30828:
30823:
30819:
30814:
30808:
30804:
30798:
30793:
30790:
30787:
30783:
30759:
30753:
30743:
30740:
30736:
30722:
30720:
30690:
30672:
30669:
30665:
30657:
30640:
30636:
30614:
30605:
30597:
30593:
30589:
30581:
30573:
30569:
30565:
30557:
30549:
30545:
30540:
30536:
30531:
30517:
30502:
30482:
30462:
30442:
30422:
30415:
30414:
30413:
30410:
30396:
30387:
30379:
30375:
30366:
30358:
30354:
30350:
30345:
30342:
30338:
30332:
30322:
30306:
30300:
30296:
30290:
30285:
30282:
30279:
30275:
30251:
30239:
30236:
30232:
30222:
30209:
30204:
30196:
30192:
30184:
30180:
30172:
30168:
30158:
30154:
30146:
30142:
30134:
30130:
30120:
30116:
30108:
30104:
30096:
30092:
30085:
30080:
30067:
30049:
30045:
30037:point masses
30024:
30010:
30007:
30006:
29993:
29990:
29921:
29871:
29842:
29827:
29825:
29804:
29798:
29793:
29787:
29773:
29766:
29762:
29756:
29751:
29748:
29745:
29741:
29737:
29733:
29729:
29714:
29697:
29692:
29686:
29672:
29667:
29650:
29646:
29640:
29635:
29632:
29629:
29625:
29621:
29618:
29616:
29606:
29601:
29595:
29590:
29576:
29569:
29565:
29559:
29554:
29551:
29548:
29544:
29540:
29538:
29531:
29527:
29451:
29442:
29425:
29408:
29403:
29397:
29383:
29378:
29363:
29361:
29354:
29346:
29330:
29325:
29312:
29307:
29293:
29289:
29274:
29272:
29265:
29257:
29241:
29236:
29223:
29219:
29215:
29211:
29205:
29192:
29176:
29168:
29160:
29154:
29141:
29125:
29121:
29117:
29101:
29097:
29092:
29086:
29073:
29057:
29053:
29037:
29032:
29024:
29016:
29011:
29005:
28992:
28976:
28972:
28956:
28952:
28948:
28943:
28937:
28924:
28908:
28904:
28888:
28831:
28801:
28797:
28791:
28778:
28762:
28758:
28754:
28738:
28734:
28729:
28723:
28710:
28694:
28690:
28674:
28669:
28665:
28661:
28656:
28650:
28637:
28621:
28617:
28601:
28597:
28593:
28588:
28582:
28569:
28553:
28549:
28533:
28523:
28505:
28500:
28494:
28481:
28465:
28461:
28445:
28441:
28437:
28432:
28426:
28413:
28397:
28393:
28377:
28373:
28371:
28362:
28356:
28341:
28336:
28321:
28316:
28312:
28308:
28304:
28298:
28283:
28278:
28263:
28258:
28254:
28250:
28248:
28241:
28236:
28230:
28225:
28211:
28196:
28153:
28117:
28109:
28105:
28068:
28059:
28053:
28048:
28033:
28028:
28023:
28018:
28003:
27998:
27973:
27958:
27949:
27934:
27912:
27897:
27887:
27873:
27752:
27747:
27733:
27696:
27687:
27683:
27667:
27661:
27648:
27632:
27628:
27623:
27610:
27605:
27600:
27539:
27507:
27503:
27470:
27467:
27459:
27453:
27350:
27323:
27320:
27317:
27314:
27311:
27308:
27305:
27302:
27297:
27293:
27272:
27263:
27244:
27239:
27229:
27224:
27189:
27105:
27055:
27026:
27011:
26995:
26989:
26984:
26978:
26964:
26957:
26953:
26947:
26942:
26939:
26936:
26932:
26928:
26924:
26920:
26905:
26900:
26896:
26822:
26818:
26803:
26760:
26752:
26739:
26731:
26726:
26704:
26700:
26696:
26674:
26669:
26661:
26656:
26653:
26645:
26641:
26561:
26556:
26540:
26537:
26520:
26497:
26393:
26380:
26375:
26358:
26352:
26348:
26342:
26337:
26334:
26331:
26327:
26322:
26318:
26314:
26302:
26297:
26282:
26278:
26272:
26267:
26264:
26261:
26257:
26252:
26237:
26222:
26210:
26205:
26190:
26186:
26180:
26175:
26172:
26169:
26165:
26160:
26156:
26152:
26146:
26133:
26128:
26113:
26109:
26103:
26098:
26095:
26092:
26088:
26083:
26079:
26076:
26053:
26040:
26035:
26022:
26018:
26009:
26004:
25993:
25984:
25980:
25974:
25969:
25966:
25963:
25959:
25955:
25952:
25947:
25938:
25929:
25920:
25916:
25911:
25896:
25892:
25886:
25881:
25878:
25875:
25871:
25867:
25864:
25764:
25756:
25748:
25740:
25729:
25718:
25682:
25669:
25664:
25659:
25650:
25645:
25634:
25627:
25623:
25617:
25612:
25609:
25606:
25602:
25598:
25595:
25521:
25473:
25463:
25420:
25398:
25390:
25368:
25353:
25330:
25324:
25314:
25310:
25298:
25294:
25288:
25283:
25280:
25277:
25273:
25269:
25265:
25261:
25253:
25244:
25238:
25228:
25224:
25212:
25208:
25202:
25197:
25194:
25191:
25187:
25183:
25179:
25175:
25173:
25154:
25149:
25133:
25128:
25113:
25109:
25103:
25098:
25095:
25092:
25088:
25084:
25081:
25073:
25059:
25054:
25038:
25033:
25015:
25011:
25005:
25000:
24997:
24994:
24990:
24986:
24983:
24981:
24961:
24942:
24938:
24923:
24913:
24907:
24897:
24893:
24881:
24877:
24871:
24866:
24863:
24860:
24856:
24852:
24848:
24844:
24842:
24826:
24821:
24805:
24800:
24782:
24778:
24772:
24767:
24764:
24761:
24757:
24753:
24739:
24729:
24725:
24713:
24709:
24703:
24698:
24695:
24692:
24688:
24684:
24681:
24679:
24669:
24661:
24657:
24647:
24643:
24633:
24629:
24622:
24615:
24604:
24599:
24596:
24593:
24589:
24582:
24577:
24572:
24569:
24566:
24562:
24552:
24545:
24542:
24539:
24535:
24526:
24523:
24520:
24516:
24505:
24502:
24499:
24495:
24486:
24483:
24480:
24476:
24463:
24460:
24457:
24453:
24444:
24441:
24438:
24434:
24420:
24415:
24412:
24409:
24405:
24398:
24393:
24388:
24385:
24382:
24378:
24368:
24361:
24358:
24355:
24351:
24342:
24339:
24336:
24332:
24319:
24316:
24313:
24309:
24300:
24297:
24294:
24290:
24279:
24276:
24273:
24269:
24260:
24257:
24254:
24250:
24236:
24231:
24228:
24225:
24221:
24214:
24209:
24204:
24201:
24198:
24194:
24184:
24178:
24171:
24167:
24161:
24156:
24153:
24150:
24146:
24142:
24139:
24137:
24127:
24119:
24115:
24105:
24100:
24097:
24094:
24090:
24083:
24078:
24073:
24070:
24067:
24063:
24053:
24048:
24044:
24037:
24034:
24031:
24027:
24018:
24015:
24012:
24008:
24001:
23996:
23992:
23985:
23982:
23979:
23975:
23966:
23963:
23960:
23956:
23949:
23940:
23936:
23929:
23926:
23923:
23919:
23910:
23907:
23904:
23900:
23893:
23888:
23884:
23874:
23869:
23866:
23863:
23859:
23852:
23847:
23842:
23839:
23836:
23832:
23822:
23817:
23813:
23806:
23803:
23800:
23796:
23787:
23784:
23781:
23777:
23770:
23761:
23757:
23750:
23747:
23744:
23740:
23731:
23728:
23725:
23721:
23714:
23709:
23705:
23698:
23695:
23692:
23688:
23679:
23676:
23673:
23669:
23662:
23657:
23653:
23643:
23638:
23635:
23632:
23628:
23621:
23616:
23611:
23608:
23605:
23601:
23591:
23585:
23578:
23574:
23568:
23563:
23560:
23557:
23553:
23549:
23546:
23544:
23534:
23526:
23522:
23515:
23510:
23507:
23504:
23500:
23493:
23488:
23484:
23477:
23474:
23471:
23467:
23458:
23455:
23452:
23448:
23441:
23436:
23432:
23425:
23420:
23417:
23414:
23410:
23403:
23398:
23394:
23387:
23384:
23381:
23377:
23368:
23365:
23362:
23358:
23351:
23342:
23338:
23331:
23326:
23323:
23320:
23316:
23309:
23304:
23300:
23293:
23290:
23287:
23283:
23274:
23271:
23268:
23264:
23257:
23252:
23248:
23241:
23238:
23235:
23231:
23222:
23219:
23216:
23212:
23205:
23200:
23196:
23189:
23184:
23181:
23178:
23174:
23167:
23158:
23154:
23147:
23144:
23141:
23137:
23128:
23125:
23122:
23118:
23111:
23106:
23102:
23095:
23090:
23087:
23084:
23080:
23073:
23068:
23064:
23057:
23054:
23051:
23047:
23038:
23035:
23032:
23028:
23021:
23016:
23012:
23005:
23000:
22997:
22994:
22990:
22983:
22977:
22970:
22966:
22960:
22955:
22952:
22949:
22945:
22941:
22938:
22936:
22926:
22915:
22911:
22904:
22901:
22898:
22894:
22887:
22882:
22878:
22871:
22868:
22865:
22861:
22854:
22846:
22843:
22840:
22836:
22829:
22821:
22817:
22810:
22807:
22804:
22800:
22793:
22788:
22784:
22777:
22774:
22771:
22767:
22760:
22752:
22749:
22746:
22742:
22735:
22723:
22719:
22712:
22709:
22706:
22702:
22695:
22690:
22686:
22679:
22676:
22673:
22669:
22662:
22654:
22651:
22648:
22644:
22637:
22629:
22625:
22618:
22615:
22612:
22608:
22601:
22596:
22592:
22585:
22582:
22579:
22575:
22568:
22560:
22557:
22554:
22550:
22543:
22531:
22527:
22520:
22517:
22514:
22510:
22503:
22498:
22494:
22487:
22484:
22481:
22477:
22470:
22462:
22459:
22456:
22452:
22445:
22437:
22433:
22426:
22423:
22420:
22416:
22409:
22404:
22400:
22393:
22390:
22387:
22383:
22376:
22368:
22365:
22362:
22358:
22351:
22345:
22338:
22334:
22328:
22323:
22320:
22317:
22313:
22309:
22306:
22304:
22295:
22289:
22281:
22277:
22270:
22267:
22264:
22260:
22253:
22248:
22244:
22237:
22234:
22231:
22227:
22220:
22211:
22207:
22200:
22197:
22194:
22190:
22183:
22178:
22174:
22167:
22164:
22161:
22157:
22150:
22141:
22137:
22130:
22127:
22124:
22120:
22113:
22108:
22104:
22097:
22094:
22091:
22087:
22080:
22074:
22067:
22061:
22054:
22051:
22048:
22044:
22033:
22030:
22027:
22023:
22016:
22007:
22004:
22001:
21997:
21990:
21985:
21978:
21975:
21972:
21968:
21955:
21952:
21949:
21945:
21934:
21931:
21928:
21924:
21917:
21912:
21906:
21900:
21894:
21890:
21884:
21879:
21876:
21873:
21869:
21865:
21862:
21860:
21847:
21842:
21837:
21831:
21823:
21819:
21809:
21805:
21795:
21791:
21784:
21777:
21771:
21764:
21761:
21758:
21754:
21743:
21740:
21737:
21733:
21726:
21717:
21714:
21711:
21707:
21700:
21695:
21688:
21685:
21682:
21678:
21665:
21662:
21659:
21655:
21644:
21641:
21638:
21634:
21627:
21622:
21616:
21610:
21604:
21598:
21591:
21588:
21585:
21581:
21570:
21567:
21564:
21560:
21553:
21544:
21541:
21538:
21534:
21527:
21522:
21515:
21512:
21509:
21505:
21492:
21489:
21486:
21482:
21471:
21468:
21465:
21461:
21454:
21449:
21443:
21437:
21431:
21427:
21421:
21416:
21413:
21410:
21406:
21402:
21399:
21397:
21381:
21376:
21360:
21355:
21337:
21333:
21327:
21322:
21319:
21316:
21312:
21308:
21273:
21254:
21250:
21235:
21220:
21215:
21199:
21194:
21176:
21172:
21166:
21161:
21158:
21155:
21151:
21147:
21144:
21136:
21122:
21117:
21101:
21096:
21078:
21074:
21068:
21063:
21060:
21057:
21053:
21049:
21046:
21044:
21025:
21007:
21002:
20986:
20981:
20965:
20952:
20948:
20942:
20937:
20934:
20931:
20927:
20923:
20920:
20906:
20901:
20885:
20880:
20862:
20858:
20852:
20847:
20844:
20841:
20837:
20833:
20830:
20828:
20815:
20797:
20792:
20776:
20771:
20755:
20747:
20736:
20731:
20715:
20710:
20692:
20688:
20682:
20677:
20674:
20671:
20667:
20663:
20660:
20658:
20645:
20627:
20622:
20606:
20603:
20598:
20582:
20574:
20563:
20558:
20542:
20539:
20534:
20516:
20512:
20506:
20501:
20498:
20495:
20491:
20487:
20485:
20472:
20457:
20444:
20433:
20428:
20412:
20404:
20396:
20383:
20372:
20367:
20349:
20345:
20339:
20334:
20331:
20328:
20324:
20320:
20318:
20296:
20291:
20273:
20260:
20252:
20239:
20234:
20210:
20202:
20194:
20181:
20170:
20165:
20147:
20143:
20137:
20132:
20129:
20126:
20122:
20118:
20116:
20103:
20093:
20082:
20069:
20064:
20048:
20034:
20029:
20011:
19998:
19990:
19977:
19972:
19948:
19940:
19932:
19919:
19908:
19903:
19885:
19881:
19875:
19870:
19867:
19864:
19860:
19856:
19854:
19841:
19826:
19810:
19797:
19792:
19776:
19762:
19757:
19739:
19726:
19718:
19705:
19700:
19676:
19668:
19660:
19647:
19636:
19631:
19613:
19609:
19603:
19598:
19595:
19592:
19588:
19584:
19582:
19569:
19557:
19544:
19539:
19518:
19510:
19496:
19491:
19473:
19460:
19452:
19439:
19434:
19410:
19402:
19394:
19381:
19370:
19365:
19347:
19343:
19337:
19332:
19329:
19326:
19322:
19318:
19316:
19277:
19262:
19249:
19244:
19223:
19209:
19204:
19186:
19173:
19165:
19152:
19147:
19120:
19112:
19104:
19091:
19080:
19075:
19057:
19053:
19047:
19042:
19039:
19036:
19032:
19028:
19026:
19018:
19015:
19007:
18994:
18989:
18968:
18960:
18947:
18942:
18920:
18902:
18897:
18879:
18866:
18855:
18842:
18837:
18816:
18808:
18795:
18790:
18763:
18755:
18747:
18734:
18723:
18718:
18700:
18696:
18690:
18685:
18682:
18679:
18675:
18671:
18669:
18647:
18642:
18624:
18611:
18608:
18602:
18594:
18586:
18573:
18562:
18557:
18539:
18535:
18529:
18524:
18521:
18518:
18514:
18510:
18508:
18486:
18481:
18463:
18447:
18444:
18436:
18428:
18415:
18404:
18399:
18381:
18377:
18371:
18366:
18363:
18360:
18356:
18352:
18350:
18331:
18318:
18307:
18296:
18291:
18278:
18270:
18257:
18246:
18241:
18223:
18219:
18213:
18208:
18205:
18202:
18198:
18194:
18192:
18172:
18150:
18135:
18130:
18112:
18096:
18093:
18085:
18083:
18070:
18057:
18046:
18035:
18030:
18008:
17996:
17983:
17970:
17954:
17951:
17943:
17941:
17928:
17913:
17900:
17887:
17871:
17860:
17857:
17855:
17839:
17826:
17810:
17797:
17783:
17780:
17778:
17765:
17752:
17746:
17741:
17723:
17710:
17702:
17689:
17673:
17660:
17646:
17643:
17641:
17628:
17610:
17599:
17594:
17576:
17563:
17555:
17542:
17526:
17513:
17499:
17496:
17494:
17481:
17466:
17453:
17442:
17429:
17416:
17408:
17395:
17379:
17366:
17352:
17349:
17347:
17331:
17318:
17310:
17297:
17283:
17272:
17269:
17267:
17254:
17241:
17230:
17219:
17214:
17192:
17187:
17165:
17152:
17144:
17131:
17117:
17109:
17098:
17085:
17074:
17063:
17058:
17045:
17043:
17038:
17026:
17019:
17013:
17010:
17002:
16989:
16981:
16968:
16954:
16946:
16935:
16922:
16911:
16900:
16895:
16882:
16880:
16867:
16855:
16842:
16831:
16823:
16810:
16796:
16793:
16785:
16772:
16764:
16751:
16737:
16729:
16718:
16705:
16694:
16683:
16678:
16665:
16663:
16650:
16641:
16628:
16617:
16614:
16606:
16593:
16582:
16574:
16561:
16553:
16540:
16526:
16518:
16507:
16494:
16483:
16472:
16467:
16454:
16452:
16436:
16431:
16415:
16407:
16394:
16383:
16375:
16362:
16354:
16341:
16327:
16319:
16308:
16295:
16284:
16273:
16268:
16255:
16253:
16248:
16236:
16231:
16203:
16190:
16179:
16168:
16163:
16150:
16142:
16129:
16118:
16113:
16095:
16091:
16085:
16080:
16077:
16074:
16070:
16066:
16064:
16042:
16039:
16024:
16013:
16008:
15995:
15987:
15974:
15963:
15958:
15940:
15936:
15930:
15925:
15922:
15919:
15915:
15911:
15909:
15890:
15887:
15872:
15867:
15854:
15849:
15846:
15831:
15826:
15808:
15804:
15798:
15793:
15790:
15787:
15783:
15779:
15777:
15702:
15699:
15655:
15652:
15647:
15644:
15629:
15624:
15621:
15603:
15598:
15580:
15576:
15570:
15565:
15562:
15559:
15555:
15551:
15549:
15521:
15516:
15513:
15498:
15493:
15490:
15472:
15467:
15449:
15445:
15439:
15434:
15431:
15428:
15424:
15420:
15418:
15405:
15396:
15386:
15381:
15363:
15359:
15353:
15348:
15345:
15342:
15338:
15334:
15332:
15319:
15307:
15303:
15296:
15291:
15276:
15271:
15268:
15265:
15261:
15257:
15255:
15242:
15230:
15226:
15219:
15208:
15188:
15183:
15180:
15177:
15173:
15169:
15167:
15144:
15142:
15141:cross product
15138:
15122:
15119:
15116:
15112:
15106:
15093:
15089:
15083:
15070:
15061:
15056:
15052:
15044:
15040:
15035:
15029:
15016:
15007:
15003:
14994:
14990:
14985:
14966:
14947:
14941:
14936:
14930:
14916:
14909:
14905:
14899:
14894:
14891:
14888:
14884:
14880:
14876:
14872:
14864:
14855:
14849:
14844:
14838:
14824:
14817:
14813:
14807:
14802:
14799:
14796:
14792:
14788:
14784:
14780:
14755:
14744:
14739:
14708:
14699:
14694:
14683:
14679:
14675:
14669:
14655:
14623:
14610:
14593:
14589:
14584:
14575:
14570:
14559:
14555:
14546:
14542:
14534:
14530:
14521:
14516:
14505:
14501:
14493:
14488:
14362:
14358:
14349:
14331:
14327:
14304:
14277:
14272:
14260:
14256:
14252:
14248:
14239:
14223:
14217:
14212:
14209:
14206:
14202:
14198:
14179:
14165:
14116:
14111:
14094:
14089:
14086:
14081:
14059:
14049:
14046:
14041:
14032:
14022:
14005:
13988:
13970:
13964:
13960:
13954:
13949:
13946:
13943:
13939:
13934:
13928:
13925:
13920:
13911:
13905:
13900:
13894:
13880:
13873:
13869:
13863:
13858:
13855:
13852:
13848:
13844:
13840:
13836:
13826:
13823:
13818:
13816:
13794:
13776:
13770:
13766:
13760:
13755:
13752:
13749:
13745:
13740:
13734:
13731:
13726:
13722:
13717:
13707:
13701:
13687:
13674:
13668:
13654:
13634:
13628:
13624:
13618:
13613:
13610:
13607:
13603:
13598:
13592:
13589:
13584:
13582:
13560:
13542:
13536:
13532:
13526:
13521:
13518:
13515:
13511:
13506:
13500:
13497:
13492:
13488:
13483:
13473:
13467:
13453:
13448:
13444:
13440:
13430:
13424:
13410:
13405:
13399:
13395:
13389:
13384:
13381:
13378:
13374:
13369:
13363:
13360:
13355:
13353:
13342:
13312:
13279:
13276:
13271:
13256:
13252:
13246:
13241:
13238:
13235:
13231:
13221:
13208:
13204:
13186:
13167:
13163:
13157:
13152:
13149:
13146:
13142:
13137:
13131:
13128:
13123:
13119:
13114:
13108:
13095:
13086:
13082:
13063:
13059:
13053:
13048:
13045:
13042:
13038:
13033:
13029:
13025:
13020:
13014:
13001:
12992:
12988:
12984:
12978:
12965:
12956:
12950:
12946:
12940:
12935:
12932:
12929:
12925:
12920:
12914:
12911:
12906:
12897:
12887:
12868:
12863:
12853:
12848:
12818:
12814:
12796:
12791:
12778:
12769:
12765:
12761:
12743:
12738:
12725:
12716:
12710:
12706:
12700:
12695:
12692:
12689:
12685:
12679:
12676:
12671:
12666:
12656:
12651:
12639:
12635:
12629:
12624:
12621:
12618:
12614:
12608:
12605:
12600:
12591:
12568:
12539:
12512:
12509:
12506:
12503:
12500:
12497:
12494:
12491:
12486:
12482:
12461:
12453:
12443:
12407:
12402:
12397:
12391:
12377:
12370:
12366:
12360:
12355:
12352:
12349:
12345:
12341:
12338:
12275:
12253:
12244:
12238:
12233:
12227:
12213:
12206:
12202:
12196:
12191:
12188:
12185:
12181:
12177:
12173:
12169:
12139:
12134:
12124:
12119:
12084:
12062:
12060:
12039:
11986:
11982:
11964:
11959:
11943:
11939:
11933:
11928:
11925:
11922:
11918:
11913:
11909:
11905:
11900:
11891:
11886:
11872:
11868:
11863:
11847:
11843:
11837:
11832:
11829:
11826:
11822:
11818:
11814:
11810:
11808:
11799:
11781:
11776:
11763:
11754:
11750:
11745:
11729:
11725:
11719:
11714:
11711:
11708:
11704:
11700:
11698:
11688:
11678:
11673:
11657:
11653:
11647:
11642:
11639:
11636:
11632:
11628:
11626:
11562:
11545:
11540:
11534:
11527:
11523:
11515:
11511:
11507:
11498:
11494:
11490:
11485:
11478:
11474:
11464:
11460:
11452:
11448:
11444:
11439:
11433:
11428:
11426:
11420:
11412:
11398:
11390:
11386:
11384:
11374:
11340:
11336:
11332:
11327:
11323:
11319:
11314:
11310:
11303:
11277:
11269:
11260:
11250:
11149:
11144:
11131:
11123:
11118:
11086:
11081:
11071:
11066:
11012:
10983:
10956:
10953:
10950:
10947:
10944:
10941:
10938:
10935:
10930:
10926:
10905:
10894:
10890:
10885:
10876:
10863:
10848:
10837:
10833:
10800:
10770:
10765:
10749:
10745:
10712:
10707:
10690:
10685:
10646:
10641:
10624:
10619:
10596:
10579:
10571:
10567:
10561:
10542:
10538:
10528:
10524:
10518:
10513:
10510:
10507:
10503:
10498:
10494:
10479:
10475:
10469:
10464:
10460:
10450:
10446:
10440:
10435:
10432:
10429:
10425:
10420:
10416:
10414:
10405:
10396:
10391:
10372:
10368:
10359:
10355:
10351:
10346:
10327:
10323:
10316:
10312:
10308:
10303:
10284:
10280:
10270:
10266:
10260:
10255:
10252:
10249:
10245:
10241:
10239:
10219:
10202:
10194:
10189:
10170:
10166:
10157:
10153:
10149:
10144:
10125:
10121:
10114:
10111:
10109:
10096:
10091:
10072:
10068:
10061:
10046:
10043:
10028:
10025:
10020:
10001:
9997:
9990:
9975:
9972:
9970:
9963:
9930:
9913:
9898:
9893:
9857:
9806:
9754:
9741:
9733:
9728:
9715:
9707:
9699:
9694:
9681:
9673:
9668:
9551:
9547:
9538:
9533:
9517:
9486:
9481:
9469:
9465:
9461:
9456:
9441:
9436:
9433:
9430:
9426:
9422:
9420:
9406:
9401:
9389:
9385:
9379:
9374:
9371:
9368:
9364:
9360:
9358:
9317:
9301:
9298:
9295:
9292:
9289:
9286:
9283:
9280:
9275:
9271:
9250:
9238:
9233:
9227:Newton's laws
9224:
9223:of the body.
9222:
9198:
9188:
9175:
9167:
9159:
9154:
9151:
9146:
9141:
9137:
9125:
9119:
9116:
9111:
9102:
9073:
9041:
9024:
9016:
9007:
9001:
8997:
8991:
8986:
8983:
8980:
8976:
8971:
8965:
8962:
8957:
8953:
8947:
8928:
8924:
8914:
8910:
8904:
8899:
8896:
8893:
8889:
8884:
8880:
8872:
8869:
8865:
8859:
8842:
8837:
8818:
8813:
8809:
8799:
8795:
8789:
8784:
8781:
8778:
8774:
8769:
8763:
8759:
8753:
8750:
8745:
8743:
8735:
8731:
8722:
8717:
8698:
8694:
8686:
8682:
8678:
8674:
8665:
8660:
8641:
8637:
8629:
8625:
8619:
8615:
8609:
8604:
8601:
8598:
8594:
8588:
8585:
8580:
8578:
8570:
8565:
8555:
8550:
8538:
8534:
8528:
8523:
8520:
8517:
8513:
8507:
8504:
8499:
8497:
8486:
8468:
8459:
8455:
8453:
8449:
8425:
8415:
8402:
8387:
8376:
8372:
8358:
8345:
8340:
8335:
8331:
8321:
8317:
8311:
8307:
8303:
8292:
8263:
8253:
8236:
8233:
8230:
8228:
8221:
8202:
8198:
8188:
8184:
8178:
8173:
8170:
8167:
8163:
8155:
8147:
8142:
8132:
8130:
8123:
8104:
8100:
8063:
8058:
8041:
8033:
8029:
8023:
8004:
8000:
7990:
7986:
7980:
7975:
7972:
7969:
7965:
7960:
7956:
7941:
7937:
7931:
7926:
7922:
7912:
7908:
7902:
7897:
7894:
7891:
7887:
7882:
7878:
7876:
7867:
7858:
7853:
7834:
7830:
7822:
7818:
7814:
7809:
7790:
7786:
7776:
7772:
7766:
7761:
7758:
7755:
7751:
7747:
7745:
7735:
7725:
7720:
7705:
7701:
7695:
7690:
7687:
7684:
7680:
7676:
7674:
7649:
7647:
7643:
7640:
7618:
7613:
7594:
7590:
7582:
7579:
7571:
7566:
7547:
7543:
7536:
7521:
7518:
7510:
7505:
7492:
7484:
7482:
7475:
7461:
7456:
7439:
7424:
7419:
7401:
7396:
7388:
7372:
7364:
7360:
7349:
7333:
7331:
7324:
7284:
7267:
7252:
7247:
7211:
7160:
7122:
7038:
7030:
7025:
7012:
7004:
6996:
6992:
6983:
6978:
6967:
6963:
6955:
6953:
6946:
6932:
6924:
6919:
6909:
6907:
6900:
6864:
6815:
6812:
6809:
6806:
6803:
6800:
6797:
6794:
6789:
6785:
6764:
6755:
6753:
6720:
6711:
6702:
6688:
6683:
6679:
6675:
6670:
6667:
6662:
6659:
6635:
6630:
6626:
6622:
6616:
6613:
6607:
6605:
6595:
6591:
6586:
6579:
6576:
6570:
6564:
6561:
6555:
6552:
6548:
6544:
6541:
6538:
6536:
6526:
6521:
6518:
6513:
6507:
6503:
6496:
6493:
6487:
6482:
6478:
6472:
6468:
6461:
6458:
6452:
6449:
6444:
6440:
6435:
6430:
6427:
6421:
6418:
6412:
6410:
6402:
6399:
6393:
6388:
6382:
6378:
6374:
6369:
6365:
6360:
6355:
6352:
6346:
6343:
6335:
6330:
6327:
6323:
6319:
6316:
6313:
6307:
6299:
6293:
6290:
6287:
6281:
6278:
6270:
6265:
6262:
6258:
6254:
6252:
6240:
6237:
6233:
6208:
6199:
6186:
6181:
6177:
6173:
6168:
6164:
6160:
6155:
6151:
6147:
6142:
6138:
6134:
6129:
6121:
6115:
6095:
6075:
6055:
6046:
6033:
6028:
6024:
6020:
6015:
6011:
6007:
6002:
5998:
5994:
5989:
5985:
5971:
5967:
5965:
5961:
5941:
5921:
5916:
5908:
5905:
5902:
5890:
5886:
5876:
5873:
5869:
5865:
5860:
5855:
5850:
5847:
5842:
5831:
5827:
5817:
5814:
5810:
5806:
5801:
5797:
5776:
5768:
5753:
5750:
5745:
5741:
5737:
5734:
5731:
5711:
5706:
5702:
5698:
5693:
5690:
5685:
5680:
5675:
5671:
5665:
5662:
5659:
5656:
5653:
5650:
5647:
5643:
5640:
5636:
5633:
5628:
5624:
5620:
5615:
5610:
5606:
5600:
5597:
5592:
5588:
5584:
5581:
5578:
5572:
5568:
5563:
5558:
5554:
5550:
5540:
5537:
5533:
5512:
5509:
5505:
5502:
5498:
5495:
5492:
5489:
5486:
5466:
5458:
5442:
5422:
5402:
5394:
5390:
5375:
5372:
5369:
5366:
5363:
5343:
5338:
5332:
5328:
5324:
5318:
5314:
5308:
5303:
5299:
5293:
5288:
5283:
5279:
5272:
5266:
5262:
5259:
5253:
5247:
5244:
5236:
5233:
5228:
5223:
5217:
5212:
5208:
5202:
5199:
5191:
5188:
5185:
5181:
5176:
5172:
5167:
5161:
5158:
5150:
5147:
5142:
5138:
5134:
5131:
5128:
5122:
5118:
5113:
5108:
5104:
5100:
5090:
5087:
5083:
5062:
5042:
5022:
5002:
4994:
4990:
4989:
4988:
4985:
4977:
4972:
4962:
4948:
4928:
4920:
4904:
4884:
4876:
4860:
4852:
4848:
4844:
4830:
4810:
4787:
4784:
4781:
4778:
4775:
4727:
4724:
4721:
4718:
4715:
4692:
4685:
4680:
4667:
4664:
4659:
4638:
4635:
4632:
4629:
4626:
4620:
4615:
4611:
4607:
4602:
4598:
4589:
4584:
4569:
4564:
4560:
4554:
4550:
4544:
4540:
4536:
4531:
4527:
4516:
4512:
4499:
4494:
4489:
4485:
4479:
4475:
4469:
4464:
4461:
4458:
4454:
4450:
4445:
4441:
4418:
4414:
4410:
4401:
4388:
4383:
4378:
4374:
4368:
4364:
4358:
4353:
4350:
4347:
4343:
4337:
4333:
4326:
4323:
4318:
4313:
4308:
4302:
4298:
4294:
4290:
4283:
4279:
4272:
4269:
4262:
4257:
4254:
4251:
4247:
4243:
4238:
4228:
4223:
4211:
4207:
4200:
4197:
4190:
4185:
4182:
4179:
4175:
4171:
4162:
4153:
4139:
4117:
4113:
4090:
4086:
4065:
4056:
4041:
4019:
4015:
4011:
3999:
3988:
3982:
3980:solid sphere,
3976:
3970:
3969:
3965:
3935:
3931:
3926:
3920:
3917:
3912:
3907:
3903:
3898:
3892:
3889:
3884:
3879:
3876:
3872:
3868:
3863:
3860:
3856:
3847:
3831:
3825:
3821:
3817:
3811:
3806:
3803:
3799:
3791:Triangular:
3790:
3774:
3768:
3764:
3760:
3754:
3749:
3746:
3742:
3719:
3713:
3709:
3705:
3699:
3694:
3691:
3687:
3678:
3662:
3657:
3653:
3647:
3642:
3639:
3635:
3631:
3626:
3623:
3619:
3610:
3609:
3603:
3592:
3579:
3571:
3567:
3547:
3544:
3540:
3517:
3514:
3510:
3501:
3491:
3489:
3479:
3477:
3473:
3468:
3455:
3444:
3441:
3433:
3420:
3403:
3392:
3383:
3372:
3366:
3359:
3350:
3346:
3341:
3338:
3318:
3293:
3276:
3268:
3263:
3250:
3244:
3241:
3235:
3231:
3225:
3218:
3209:
3205:
3200:
3197:
3177:
3169:
3165:
3160:
3157:
3154:
3147:
3141:
3138:
3132:
3123:
3102:
3095:. The length
3094:
3090:
3074:
3060:
3046:
3026:
3018:
3014:
3010:
3003:
2999:
2995:
2992:
2989:
2983:
2976:
2967:
2962:
2959:
2956:
2950:
2945:
2941:
2933:), to obtain
2920:
2900:
2891:
2876:
2856:
2836:
2816:
2808:
2804:
2799:
2796:
2793:
2786:
2777:
2754:
2750:
2723:
2714:
2711:
2706:
2698:
2693:
2684:
2668:
2664:
2660:
2638:
2634:
2630:
2627:
2624:
2615:
2602:
2597:
2593:
2589:
2584:
2581:
2576:
2571:
2567:
2562:
2556:
2552:
2548:
2544:
2538:
2535:
2530:
2522:
2514:
2509:
2506:
2501:
2492:
2482:
2464:
2449:
2446:
2443:
2433:
2429:
2425:
2422:
2420:
2411:
2401:
2392:
2383:
2379:
2370:
2361:
2352:
2347:
2343:
2340:
2338:
2329:
2320:
2312:
2308:
2304:
2296:
2288:
2280:
2278:
2259:
2216:
2208:
2195:
2177:
2173:
2169:
2166:
2163:
2156:The quantity
2154:
2152:
2134:
2130:
2126:
2123:
2120:
2068:
2034:
2017:
2002:
1999:
1996:
1986:
1982:
1978:
1975:
1973:
1964:
1954:
1945:
1936:
1932:
1923:
1914:
1905:
1900:
1896:
1893:
1891:
1875:
1867:
1861:
1853:
1845:
1837:
1835:
1799:
1796:
1766:
1758:
1723:
1658:
1650:
1636:
1623:
1618:
1614:
1610:
1607:
1604:
1584:
1564:
1548:
1538:
1536:
1516:
1511:
1507:
1503:
1500:
1497:
1480:
1469:
1463:
1452:
1439:
1434:
1430:
1426:
1423:
1420:
1400:
1395:
1393:
1381:
1378:
1375:
1372:
1369:
1357:
1350:
1346:
1341:
1339:
1338:tuck position
1335:
1331:
1326:
1324:
1312:
1307:
1304:
1299:
1296:
1284:
1277:
1248:
1243:
1239:
1237:
1233:
1223:
1220:
1215:
1212:
1208:
1204:
1199:
1195:
1193:
1189:
1185:
1181:
1177:
1173:
1169:
1164:
1149:
1129:
1107:
1103:
1099:
1091:
1086:
1084:
1080:
1076:
1072:
1068:
1064:
1060:
1050:
1048:
1047:inertial mass
1044:
1039:
1037:
1033:
1029:
1025:
1020:
1018:
1014:
1009:
1005:
1001:
996:
994:
990:
986:
982:
978:
974:
970:
966:
962:
946:
937:
925:
924:Niagara River
921:
917:
906:
901:
899:
894:
892:
887:
886:
884:
883:
877:
867:
864:
859:
853:
852:
851:
850:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
723:
716:
715:
708:
704:
700:
696:
693:
692:
688:
685:
683:
680:
678:
675:
673:
670:
666:
663:
662:
661:
658:
656:
653:
651:
648:
646:
643:
642:
638:
633:
632:
624:
621:
619:
616:
612:
609:
607:
604:
603:
602:
599:
597:
594:
592:
589:
587:
582:
579:
575:
572:
571:
567:
564:
561:
557:
553:
552:
548:
545:
543:
540:
538:
535:
533:
528:
526:
523:
521:
518:
516:
513:
512:
505:
504:
493:
490:
488:
485:
483:
480:
478:
475:
473:
470:
468:
465:
464:
462:
461:
456:
453:
452:
447:
446:
440:
439:
431:
428:
426:
423:
421:
418:
416:
413:
411:
408:
406:
403:
401:
398:
396:
392:
390:
387:
385:
381:
379:
376:
373:
369:
365:
363:
360:
358:
355:
353:
350:
348:
345:
341:
338:
336:
333:
332:
331:
328:
326:
323:
321:
318:
316:
313:
311:
308:
307:
300:
299:
291:
288:
286:
283:
281:
278:
276:
273:
271:
268:
266:
263:
261:
258:
256:
253:
252:
245:
244:
238:
235:
233:
230:
228:
225:
224:
222:
221:
217:
198:
195:
185:
179:
165:
164:
161:
158:
157:
153:
152:
145:
142:
139:
137:
133:
116:
113:
108:
105:
98:
90:
86:
80:
76:
74:
70:
67:
64:
58:
53:
48:
43:
38:
33:
19:
40618:Rigid bodies
40585:Hermann Weyl
40389:Vector space
40374:Pseudotensor
40339:Fiber bundle
40292:abstractions
40187:Mixed tensor
40172:Tensor field
39979:
39917:
39901:
39877:
39861:
39835:
39825:
39815:
39796:
39785:
39784:
39760:
39750:
39740:
39718:
39708:
39653:
39642:
39585:
39575:
39503:
39498:
39487:
39484:
39480:
39476:
39464:
39439:
39438:
39434:angular jerk
39418:
39389:
39388:
39357:
39353:acceleration
39328:
39327:
39317:
39296:
39295:
39283:
39271:
39254:
39243:
39228:
39218:
39198:
39161:
39160:
39150:
39130:
39117:displacement
39110:
39104:
39098:
39087:
39061:
39041:
39026:
39010:
39003:
38880:November 21,
38878:. Retrieved
38863:
38856:
38833:
38827:
38815:. Retrieved
38803:
38799:
38786:
38777:
38764:
38755:
38742:
38723:
38701:
38682:
38649:
38630:
38624:
38611:
38598:
38579:
38543:
38537:
38518:
38512:
38493:
38458:
38451:
38443:
38423:
38415:
38395:
38388:
38376:. Retrieved
38372:
38362:
38350:. Retrieved
38345:
38335:
38315:
38285:
38266:
38229:. Springer.
38226:
38198:
38168:
38141:
38119:
38112:
38102:November 21,
38100:. Retrieved
38090:
38052:
38045:
37903:
37641:Let a point
37640:
37193:
37110:
37103:
37087:
37051:
37024:
37019:
37015:
37011:
37003:
36997:
36992:
36988:
36976:
36871:
36643:
36578:Notice that
36577:
36360:
36351:
36262:
36225:
36052:
36038:
35745:
35742:
35730:
35728:
35506:
35499:
34944:
34936:
34582:
33286:
33158:
32261:
32119:
32041:
31845:
31840:
31829:
31793:
31652:
31529:
31132:
31130:
30723:
30694:
30411:
30223:
30068:is given by
30016:
30002:
29999:
29991:
29443:
28524:
28197:
28166:noting that
27888:
27264:
26809:
26643:
26642:
26498:
26394:
26054:
25683:
25522:
25475:
25355:
24962:
21274:
18173:
17188:
16232:
15149:
14967:
14768:, to obtain
14624:
14185:
14023:
13222:
12888:
12449:
12063:
11563:
11256:
10897:
10882:
10766:
10597:
10220:
9755:
9534:
9242:
9220:
9189:
9042:
8473:
8456:
8447:
8416:
8359:
8254:
8059:
7655:
7645:
7644:
7641:
7123:
6756:
6751:
6716:
6200:
6047:
5975:
5957:
5766:is its mass.
5392:
5015:and density
4992:
4983:
4981:
4874:
4846:
4845:
4823:of the body
4681:
4585:
4517:
4513:
4402:
4154:
4057:
4003:
3569:
3565:
3497:
3485:
3469:
3264:
3066:
2892:
2702:
2616:
2483:
2155:
2150:
2035:
1637:
1555:
1481:
1467:
1464:
1453:
1396:
1342:
1327:
1269:
1231:
1229:
1216:
1200:
1196:
1187:
1186:in his book
1175:
1165:
1087:
1056:
1053:Introduction
1040:
1021:
997:
976:
972:
968:
964:
960:
958:
705: /
701: /
699:displacement
697: /
558: /
520:Displacement
458:
449:
443:Formulations
430:Virtual work
371:
370: /
310:Acceleration
303:Fundamentals
143:
140:
73:SI unit
65:
40525:Élie Cartan
40473:Spin tensor
40447:Weyl tensor
40405:Mathematics
40369:Multivector
40160:definitions
40058:Engineering
39997:Mathematics
39174:solid angle
38094:. pp.
37362:defines an
36989:figure axis
36692:, given by
35736:is the 3×3
32122:dot product
31835:is the 3×3
27285:particles,
25574:, given by
15884:centripetal
15641:centripetal
15510:centripetal
12317:defined by
10918:particles,
9843:to a point
9340:, to yield
9263:particles,
7197:to a point
6777:particles,
3998:OGV version
3848:Circular:
1475:to an axis
1174:. The term
841:von Neumann
508:Core topics
83:Other units
40607:Categories
40354:Linear map
40222:Operations
39909:m s, N m s
39885:m s,
39831:Lagrangian
39756:Lagrangian
38996:Dimensions
38984:Dimensions
38469:1572594918
38439:0970467028
38406:0748743146
38038:References
36357:Body frame
32120:where the
30013:Definition
27926:such that
16036:tangential
15843:tangential
15618:tangential
15487:tangential
14348:kinematics
12474:particles
9537:kinematics
6705:Rigid body
6088:along the
4682:Here, the
3959:Point mass
3476:gravimeter
2697:gravimeter
1545:See also:
1226:Definition
1004:point mass
981:rigid body
776:d'Alembert
756:Maupertuis
719:Scientists
601:Rigid body
275:Kinematics
40493:EM tensor
40329:Dimension
40280:Transpose
39925:ms,
39544:˙
39267:frequency
39194:frequency
38748:Mechanics
38199:Mechanics
37964:‖
37956:‖
37866:‖
37857:‖
37845:Λ
37821:‖
37812:‖
37800:Λ
37777:, yields
37687:‖
37679:‖
37364:ellipsoid
37243:Λ
37207:Λ
37122:Ellipsoid
36767:Λ
36728:Λ
36679:Λ
36239:⊗
36184:⊗
36176:−
36153:⋅
35929:−
35918:−
35905:−
35869:−
35856:−
35845:−
35709:^
35676:−
35661:⋅
35623:^
35613:⋅
35596:^
35557:^
35547:−
35464:^
35455:⋅
35441:−
35401:^
35377:^
35368:⋅
35348:^
35331:^
35322:⋅
35301:−
35267:^
35250:^
35241:⋅
35228:−
35215:⋅
35204:^
35187:^
35178:⋅
35165:−
35107:^
35073:^
35056:^
35047:⋅
35034:−
35001:^
34905:−
34873:−
34841:−
34479:∑
34426:∑
34422:−
34370:∑
34366:−
34312:∑
34308:−
34233:∑
34180:∑
34176:−
34122:∑
34118:−
34066:∑
34062:−
33987:∑
33936:−
33918:−
33898:−
33865:−
33845:−
33827:−
33606:∑
33485:∑
33364:∑
33064:∑
33011:∑
33007:−
32955:∑
32951:−
32897:∑
32893:−
32818:∑
32765:∑
32761:−
32707:∑
32703:−
32651:∑
32647:−
32572:∑
32192:⋅
32184:⋅
32098:⋅
32090:⋅
31992:ρ
31983:∭
31979:−
31924:ρ
31915:∭
31881:×
31824:is their
31807:⊗
31746:⊗
31738:−
31717:‖
31708:‖
31679:ρ
31670:∭
31462:∑
31458:−
31338:∑
31334:−
31214:∑
31210:−
31040:∑
30912:∑
30784:∑
30666:δ
30351:−
30339:δ
30276:∑
29916:^
29882:^
29866:^
29843:⋅
29837:^
29815:^
29778:Δ
29742:∑
29738:−
29730:⋅
29724:^
29709:^
29677:Δ
29668:⋅
29662:^
29626:∑
29622:−
29596:⊥
29581:Δ
29545:∑
29500:^
29420:^
29388:Δ
29379:⋅
29373:^
29364:−
29340:^
29331:×
29316:Δ
29313:×
29298:Δ
29290:⋅
29284:^
29275:−
29251:^
29242:×
29227:Δ
29224:−
29216:⋅
29196:Δ
29193:×
29187:^
29145:Δ
29142:×
29136:^
29122:⋅
29111:^
29102:×
29077:Δ
29074:×
29068:^
29054:×
29048:^
28996:Δ
28993:×
28987:^
28973:×
28967:^
28953:⋅
28928:Δ
28925:×
28919:^
28905:×
28899:^
28860:^
28822:Δ
28782:Δ
28779:×
28773:^
28759:⋅
28748:^
28739:×
28714:Δ
28711:×
28705:^
28691:×
28685:^
28666:≡
28641:Δ
28638:×
28632:^
28618:×
28612:^
28598:⋅
28573:Δ
28570:×
28564:^
28550:×
28544:^
28485:Δ
28482:×
28476:^
28462:×
28456:^
28442:⋅
28417:Δ
28414:×
28408:^
28394:×
28388:^
28347:Δ
28331:^
28317:−
28309:⋅
28289:Δ
28273:^
28259:−
28231:⊥
28216:Δ
28180:^
28140:^
28127:^
28118:−
28091:^
28078:^
28069:−
28043:^
28029:≡
28013:^
27999:−
27974:×
27968:^
27944:^
27907:^
27851:^
27814:^
27801:^
27738:Δ
27719:^
27706:^
27697:−
27678:^
27652:Δ
27649:⋅
27643:^
27629:−
27614:Δ
27606:⊥
27591:Δ
27568:^
27530:Δ
27480:^
27401:^
27365:. Choose
27318:…
27245:−
27215:Δ
27180:Δ
27100:^
27066:^
27050:^
27027:⋅
27021:^
27006:^
26969:Δ
26933:∑
26929:−
26921:⋅
26915:^
26851:^
26654:−
26538:−
26328:∑
26319:−
26303:−
26258:∑
26211:−
26166:∑
26134:−
26089:∑
26080:−
26023:−
26010:−
25960:∑
25956:−
25917:−
25872:∑
25868:−
25730:−
25651:−
25603:∑
25599:−
25417:ω
25399:×
25395:ω
25387:α
25365:τ
25336:ω
25307:Δ
25274:∑
25270:−
25262:×
25258:ω
25250:α
25221:Δ
25188:∑
25184:−
25159:ω
25155:×
25140:Δ
25134:×
25119:Δ
25089:∑
25085:−
25082:×
25078:ω
25064:α
25060:×
25045:Δ
25039:×
25024:Δ
24991:∑
24987:−
24976:τ
24924:…
24890:Δ
24857:∑
24853:−
24827:×
24812:Δ
24806:×
24791:Δ
24758:∑
24754:−
24722:Δ
24689:∑
24685:−
24586:Δ
24559:Δ
24553:−
24532:Δ
24513:Δ
24492:Δ
24473:Δ
24450:Δ
24431:Δ
24402:Δ
24375:Δ
24369:−
24348:Δ
24329:Δ
24306:Δ
24287:Δ
24266:Δ
24247:Δ
24218:Δ
24191:Δ
24185:−
24147:∑
24143:−
24087:Δ
24060:Δ
24054:−
24024:Δ
24005:Δ
23972:Δ
23953:Δ
23916:Δ
23897:Δ
23856:Δ
23829:Δ
23823:−
23793:Δ
23774:Δ
23737:Δ
23718:Δ
23685:Δ
23666:Δ
23625:Δ
23598:Δ
23592:−
23554:∑
23550:−
23497:Δ
23494:−
23464:Δ
23445:Δ
23407:Δ
23404:−
23374:Δ
23355:Δ
23313:Δ
23310:−
23280:Δ
23261:Δ
23228:Δ
23209:Δ
23171:Δ
23168:−
23134:Δ
23115:Δ
23077:Δ
23074:−
23044:Δ
23025:Δ
22987:Δ
22984:−
22946:∑
22942:−
22891:Δ
22888:−
22858:Δ
22833:Δ
22797:Δ
22764:Δ
22761:−
22739:Δ
22736:−
22699:Δ
22666:Δ
22663:−
22641:Δ
22638:−
22605:Δ
22572:Δ
22569:−
22547:Δ
22507:Δ
22474:Δ
22471:−
22449:Δ
22413:Δ
22410:−
22380:Δ
22355:Δ
22352:−
22314:∑
22310:−
22257:Δ
22224:Δ
22221:−
22187:Δ
22184:−
22154:Δ
22117:Δ
22084:Δ
22081:−
22041:Δ
22020:Δ
22017:−
21994:Δ
21991:−
21965:Δ
21942:Δ
21921:Δ
21918:−
21870:∑
21866:−
21848:…
21751:Δ
21730:Δ
21727:−
21704:Δ
21701:−
21675:Δ
21652:Δ
21631:Δ
21628:−
21578:Δ
21557:Δ
21554:−
21531:Δ
21528:−
21502:Δ
21479:Δ
21458:Δ
21455:−
21407:∑
21403:−
21382:×
21367:Δ
21361:×
21346:Δ
21313:∑
21309:−
21241:ω
21236:…
21225:ω
21221:×
21206:Δ
21200:×
21185:Δ
21152:∑
21148:−
21145:×
21141:ω
21127:α
21123:×
21108:Δ
21102:×
21087:Δ
21054:∑
21050:−
21039:τ
21026:…
21012:ω
21008:×
20993:Δ
20987:×
20972:Δ
20966:×
20962:ω
20928:∑
20924:−
20911:α
20907:×
20892:Δ
20886:×
20871:Δ
20838:∑
20834:−
20816:…
20802:ω
20798:×
20783:Δ
20777:×
20762:Δ
20756:×
20752:ω
20741:α
20737:×
20722:Δ
20716:×
20701:Δ
20668:∑
20664:−
20646:…
20632:ω
20628:×
20613:Δ
20607:−
20604:×
20589:Δ
20583:×
20579:ω
20568:α
20564:×
20549:Δ
20543:−
20540:×
20525:Δ
20492:∑
20473:…
20448:Δ
20445:×
20441:ω
20434:×
20419:Δ
20413:×
20409:ω
20387:Δ
20384:×
20380:α
20373:×
20358:Δ
20325:∑
20301:ω
20297:⋅
20282:Δ
20264:Δ
20261:−
20243:Δ
20240:⋅
20225:Δ
20218:ω
20211:×
20207:ω
20185:Δ
20182:×
20178:α
20171:×
20156:Δ
20123:∑
20104:…
20073:Δ
20070:⋅
20055:Δ
20049:−
20039:ω
20035:⋅
20020:Δ
20002:Δ
19999:−
19981:Δ
19978:⋅
19963:Δ
19956:ω
19949:×
19945:ω
19923:Δ
19920:×
19916:α
19909:×
19894:Δ
19861:∑
19842:…
19831:ω
19827:×
19823:ω
19801:Δ
19798:⋅
19783:Δ
19777:−
19767:ω
19763:⋅
19748:Δ
19730:Δ
19727:−
19709:Δ
19706:⋅
19691:Δ
19684:ω
19677:×
19673:ω
19651:Δ
19648:×
19644:α
19637:×
19622:Δ
19589:∑
19570:…
19548:Δ
19545:⋅
19530:Δ
19523:ω
19519:×
19515:ω
19511:−
19501:ω
19497:⋅
19482:Δ
19464:Δ
19461:−
19443:Δ
19440:⋅
19425:Δ
19418:ω
19411:×
19407:ω
19385:Δ
19382:×
19378:α
19371:×
19356:Δ
19323:∑
19311:τ
19278:…
19253:Δ
19250:⋅
19235:Δ
19228:ω
19224:−
19214:ω
19210:⋅
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19177:Δ
19174:−
19156:Δ
19153:⋅
19138:Δ
19131:ω
19121:×
19117:ω
19095:Δ
19092:×
19088:α
19081:×
19066:Δ
19033:∑
18998:Δ
18995:⋅
18980:Δ
18973:ω
18969:−
18951:Δ
18948:⋅
18933:Δ
18926:ω
18921:…
18907:ω
18903:⋅
18888:Δ
18870:Δ
18867:−
18846:Δ
18843:⋅
18828:Δ
18821:ω
18817:−
18799:Δ
18796:⋅
18781:Δ
18774:ω
18764:×
18760:ω
18738:Δ
18735:×
18731:α
18724:×
18709:Δ
18676:∑
18652:ω
18648:⋅
18633:Δ
18615:Δ
18612:−
18603:×
18599:ω
18577:Δ
18574:×
18570:α
18563:×
18548:Δ
18515:∑
18491:ω
18487:⋅
18472:Δ
18454:Δ
18448:−
18445:×
18441:ω
18419:Δ
18416:×
18412:α
18405:×
18390:Δ
18357:∑
18322:Δ
18319:×
18315:ω
18308:×
18304:ω
18297:×
18282:Δ
18261:Δ
18258:×
18254:α
18247:×
18232:Δ
18199:∑
18187:τ
18151:…
18140:ω
18136:⋅
18121:Δ
18103:Δ
18097:−
18094:×
18090:ω
18061:Δ
18058:×
18054:ω
18047:×
18043:ω
18036:×
18021:Δ
18009:…
17987:Δ
17984:⋅
17980:ω
17961:Δ
17955:−
17952:×
17948:ω
17929:…
17904:Δ
17901:⋅
17897:ω
17878:Δ
17872:×
17868:ω
17861:−
17830:Δ
17827:⋅
17823:ω
17801:Δ
17798:×
17794:ω
17784:−
17766:…
17747:⋅
17732:Δ
17714:Δ
17711:−
17693:Δ
17690:⋅
17686:ω
17664:Δ
17661:×
17657:ω
17647:−
17629:…
17615:ω
17611:×
17607:ω
17600:⋅
17585:Δ
17567:Δ
17564:−
17546:Δ
17543:⋅
17539:ω
17517:Δ
17514:×
17510:ω
17500:−
17482:…
17457:Δ
17454:×
17450:ω
17443:⋅
17439:ω
17420:Δ
17417:−
17399:Δ
17396:⋅
17392:ω
17370:Δ
17367:×
17363:ω
17353:−
17322:Δ
17319:×
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17298:×
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17284:×
17280:ω
17273:−
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17231:×
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17220:×
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17156:Δ
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17132:×
17128:ω
17118:×
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17089:Δ
17086:×
17082:ω
17075:×
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17064:×
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17027:…
17014:−
16993:Δ
16990:×
16972:Δ
16969:×
16965:ω
16955:×
16951:ω
16926:Δ
16923:×
16919:ω
16912:×
16908:ω
16901:×
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16868:…
16846:Δ
16843:×
16839:ω
16832:×
16814:Δ
16811:×
16807:ω
16797:−
16776:Δ
16773:×
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16752:×
16748:ω
16738:×
16734:ω
16709:Δ
16706:×
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16695:×
16691:ω
16684:×
16669:Δ
16651:…
16632:Δ
16629:×
16625:ω
16618:−
16615:×
16597:Δ
16594:×
16590:ω
16565:Δ
16562:×
16544:Δ
16541:×
16537:ω
16527:×
16523:ω
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16495:×
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16484:×
16480:ω
16473:×
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16437:×
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16416:×
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16328:×
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16299:Δ
16296:×
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16285:×
16281:ω
16274:×
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16126:α
16119:×
16104:Δ
16071:∑
16059:τ
16025:×
16021:ω
16014:×
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15978:Δ
15975:×
15971:α
15964:×
15949:Δ
15916:∑
15904:τ
15873:×
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15832:×
15817:Δ
15784:∑
15772:τ
15604:×
15589:Δ
15556:∑
15473:×
15458:Δ
15425:∑
15406:…
15387:×
15372:Δ
15339:∑
15297:×
15282:Δ
15262:∑
15220:×
15209:−
15174:∑
15162:τ
15097:Δ
15094:×
15074:Δ
15071:×
15067:ω
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15020:Δ
15017:×
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15004:×
15000:ω
14991:×
14976:Δ
14953:ω
14921:Δ
14885:∑
14881:−
14873:×
14869:ω
14861:α
14829:Δ
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14789:−
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14756:×
14745:−
14700:−
14660:Δ
14576:−
14556:×
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14543:×
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14522:−
14502:×
14498:α
14461:α
14439:ω
14253:×
14240:−
14203:∑
14195:τ
14078:ω
14060:⋅
14056:ω
13989:⋅
13940:∑
13917:ω
13885:Δ
13849:∑
13845:−
13837:⋅
13833:ω
13795:⋅
13746:∑
13713:ω
13692:Δ
13659:Δ
13641:ω
13604:∑
13561:⋅
13512:∑
13479:ω
13458:Δ
13445:⋅
13436:ω
13415:Δ
13375:∑
13303:Δ
13262:Δ
13232:∑
13187:⋅
13143:∑
13099:Δ
13096:×
13092:ω
13083:⋅
13039:∑
13005:Δ
13002:×
12998:ω
12989:⋅
12969:Δ
12966:×
12962:ω
12926:∑
12869:−
12839:Δ
12782:Δ
12779:×
12775:ω
12766:⋅
12729:Δ
12726:×
12722:ω
12686:∑
12657:⋅
12615:∑
12507:…
12382:Δ
12346:∑
12342:−
12272:ω
12250:ω
12218:Δ
12182:∑
12178:−
12140:−
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12075:Δ
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11950:Δ
11919:∑
11896:ω
11892:×
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11869:×
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11823:∑
11819:−
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11764:×
11760:ω
11751:×
11736:Δ
11705:∑
11679:×
11664:Δ
11633:∑
11508:−
11491:−
11445:−
11429:≡
11387:≡
11375:×
11185:ω
11135:Δ
11132:×
11128:ω
11087:−
11057:Δ
10951:…
10858:^
10849:α
10830:τ
10722:^
10701:^
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10679:^
10635:^
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10613:^
10572:×
10555:^
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10504:∑
10489:^
10480:α
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10426:∑
10385:^
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10352:−
10340:^
10320:Δ
10317:α
10309:×
10297:^
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10234:τ
10183:^
10163:Δ
10154:ω
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10138:^
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10115:α
10085:^
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10056:^
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10038:^
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10014:^
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9991:×
9985:^
9976:α
9924:^
9914:×
9908:^
9887:^
9800:^
9770:^
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9716:×
9712:ω
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9641:α
9619:ω
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9427:∑
9415:τ
9365:∑
9296:…
9168:⋅
9138:ω
9017:⋅
8977:∑
8941:^
8921:Δ
8890:∑
8881:⋅
8873:ω
8853:^
8843:⋅
8831:^
8806:Δ
8775:∑
8760:ω
8711:^
8691:Δ
8687:ω
8679:⋅
8654:^
8634:Δ
8630:ω
8595:∑
8556:⋅
8514:∑
8397:^
8388:ω
8328:Δ
8308:∑
8215:^
8195:Δ
8164:∑
8148:−
8117:^
8097:Δ
8034:×
8017:^
7997:Δ
7966:∑
7951:^
7942:ω
7919:Δ
7888:∑
7847:^
7827:Δ
7823:ω
7815:×
7803:^
7783:Δ
7752:∑
7726:×
7711:Δ
7681:∑
7607:^
7587:Δ
7583:ω
7560:^
7540:Δ
7537:×
7531:^
7522:ω
7496:Δ
7493:×
7489:ω
7450:^
7440:×
7434:^
7413:^
7397:ω
7394:ω
7383:^
7357:Δ
7340:Δ
7318:^
7278:^
7268:×
7262:^
7241:^
7064:ω
7016:Δ
7013:×
7009:ω
6984:−
6964:×
6960:ω
6925:−
6891:Δ
6810:…
6735:^
6689:ρ
6676:π
6556:−
6545:ρ
6542:π
6519:−
6453:−
6431:ρ
6428:π
6375:−
6356:ρ
6353:π
6328:−
6324:∫
6291:ρ
6288:π
6263:−
6259:∫
6174:−
6108:-axis is
5751:ρ
5738:π
5663:ρ
5660:π
5651:θ
5621:ρ
5607:∫
5601:π
5589:∫
5564:ρ
5555:∭
5513:θ
5443:ρ
5415:, radius
5393:thin disc
5376:ℓ
5370:ρ
5329:ℓ
5300:ℓ
5280:ℓ
5260:ρ
5245:ℓ
5234:ℓ
5229:−
5200:ρ
5168:ρ
5159:ℓ
5148:ℓ
5143:−
5139:∫
5114:ρ
5105:∭
5043:ℓ
5023:ρ
4941:-axis or
4693:ρ
4621:ρ
4612:∭
4541:∑
4455:∑
4344:∑
4334:ω
4295:ω
4248:∑
4229:⋅
4176:∑
3927:π
3899:π
3442:≈
3367:≈
3351:ω
3277:π
3210:ω
3124:ω
3015:π
2968:ω
2778:ω
2724:ω
2713:frequency
2594:ω
2568:ω
2523:⋅
2459:^
2450:ω
2440:ω
2397:ω
2393:⋅
2380:−
2376:ω
2362:⋅
2321:×
2317:ω
2305:×
2289:×
2243:ω
2217:×
2213:ω
2078:α
2050:^
2012:^
2003:α
1993:α
1950:α
1946:⋅
1933:−
1929:α
1915:⋅
1876:×
1872:α
1862:×
1846:×
1830:τ
1767:×
1763:α
1733:α
1659:×
1647:τ
1379:α
1370:τ
1308:ω
1209:, and in
1166:In 1673,
1028:symmetric
1000:extensive
998:It is an
821:Liouville
703:frequency
623:Vibration
340:potential
265:Continuum
260:Celestial
237:Textbooks
136:Dimension
117:ω
52:Flywheels
40623:Rotation
40359:Manifold
40344:Geodesic
40102:Notation
39869:m s, N s
39692:J s
39688:m s
39663:m s
39624:m s
39595:m s
39571:momentum
39224:velocity
39094:position
39083:distance
39037:absement
38962:SI units
38817:June 27,
38006:See also
37704:, where
37094:vehicles
35093:, where
30328:‖
30313:‖
29464:through
15715:because
10767:Use the
9237:flywheel
8060:Use the
4993:thin rod
4965:Examples
4684:function
4655:‖
4647:‖
4588:integral
3611:Square:
3564:Height (
3488:pendulum
1784:. Since
1541:Examples
1236:centroid
1219:flywheel
1203:momentum
1122:, where
1079:imperial
1036:diagonal
926:in 1890.
876:Category
801:Hamilton
786:Lagrange
781:Clairaut
746:Horrocks
707:velocity
677:Pendulum
665:reactive
637:Rotation
606:dynamics
556:Inertial
542:Friction
425:Velocity
400:Momentum
280:Kinetics
270:Dynamics
248:Branches
232:Timeline
87:lbf·ft·s
40456:Physics
40290:Related
40053:Physics
39971:Tensors
39897:rotatum
39676:actergy
39612:actergy
39581:impulse
39501::
39252::
39182:rad, sr
36285:be the
36251:is the
32259:-axis.
30687:is the
9219:is the
6221:-axis,
5525:, then
4078:masses
2710:natural
2256:is the
2149:is the
1085:units.
979:, of a
836:Koopman
796:Poisson
791:Laplace
736:Huygens
731:Galileo
576: (
515:Damping
368:Inertia
362:Impulse
335:kinetic
285:Statics
255:Applied
227:History
40384:Vector
40379:Spinor
40364:Matrix
40158:Tensor
39811:energy
39792:moment
39780:torque
39736:energy
39714:weight
39686:
39668:action
39661:
39622:
39604:action
39593:
39564:
39511:
39483:⟩ = ∑
39447:
39398:
39337:
38871:
38844:
38730:
38689:
38637:
38619:, 1948
38586:
38550:
38525:
38500:
38466:
38436:
38403:
38323:
38293:
38233:
38205:
38177:
38132:
38068:
37198:. Let
37170:, and
37090:motion
36998:A toy
36948:, and
36758:where
36287:matrix
36204:where
36086:, and
35729:where
33602:
33571:
33481:
33450:
33360:
33329:
33199:, and
31839:, and
31794:where
31455:
31424:
31331:
31300:
31207:
31176:
31135:, are
31036:
31005:
30908:
30877:
30780:
30749:
30495:, and
30412:where
30272:
30245:
30066:tensor
30005:tensor
29938:where
27765:where
27169:, and
27118:where
26574:where
25777:where
25433:where
14346:. The
14290:where
14129:where
12831:where
12288:where
11176:where
10891:, and
10664:, and
10598:where
9503:where
7055:where
6652:where
6245:sphere
5934:where
5724:where
5356:where
4897:- and
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3599:": -->
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3407:
3376:
3280:
3039:where
2829:where
2036:where
1529:where
1397:For a
1334:divers
1059:torque
1024:scalar
1013:moment
985:torque
874:
826:Appell
811:Cauchy
806:Jacobi
751:Halley
741:Newton
726:Kepler
578:linear
574:Motion
420:Torque
395:Moment
330:Energy
320:Couple
40304:Basis
39989:Scope
39913:power
39873:power
39843:m s,
39804:m s,
39768:m s,
39729:m s,
39704:force
39214:speed
39146:angle
38796:(PDF)
38774:(PDF)
38654:axes.
31530:Here
30717:is a
15147:Proof
9945:, so
7299:, so
6717:If a
3734:and;
1180:Latin
831:Gibbs
816:Routh
771:Euler
410:Speed
405:Space
347:Force
39857:yank
39821:work
39746:work
39460:mass
39414:jerk
39126:area
39057:time
39022:time
38882:2014
38869:ISBN
38842:ISBN
38819:2008
38728:ISBN
38687:ISBN
38635:ISBN
38584:ISBN
38548:ISBN
38523:ISBN
38498:ISBN
38464:ISBN
38434:ISBN
38401:ISBN
38380:2014
38354:2014
38321:ISBN
38291:ISBN
38231:ISBN
38203:ISBN
38175:ISBN
38130:ISBN
38104:2014
38098:–187
38066:ISBN
37088:The
37066:>
37048:360°
36263:Let
36053:Let
33249:and
28846:and
25684:Let
9535:The
5895:disc
5881:disc
5789:as,
5545:disc
4919:beam
3601:edit
3445:0.99
3404:3.14
3373:9.81
3265:The
2091:and
1230:The
1090:mass
1017:axis
993:mass
959:The
415:Time
378:Mass
77:kg⋅m
39891:MLT
39806:N m
39774:MLT
39727:kg
39632:MLT
39628:J s
39599:N s
39445:rad
39423:m s
39396:rad
39363:m s
39335:rad
39259:m s
39234:m s
39168:rad
39047:m s
38808:doi
38430:107
38126:166
38096:173
38058:doi
37270:or
37092:of
37004:top
37000:top
36991:or
8284:as
5836:rod
5822:rod
5095:rod
3190:or
1081:or
1041:In
40609::
39923:kg
39915::
39907:kg
39899::
39883:kg
39875::
39867:kg
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39829:,
39823::
39819:,
39813::
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39670::
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39606::
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39583::
39579:,
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39470:kg
39462::
39436::
39416::
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39321:,
39315::
39308:Hz
39306:,
39293::
39288:,
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38836:.
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38776:.
38714:^
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38080:^
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1083:US
1075:SI
1019:.
971:,
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39428:T
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38074:.
38060::
37991:.
37982:n
37977:I
37972:1
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37960:x
37935:n
37913:x
37887:=
37881:n
37876:I
37870:2
37861:x
37854:=
37850:n
37838:T
37832:n
37825:2
37816:x
37809:=
37805:x
37793:T
37787:x
37764:n
37740:n
37735:I
37713:n
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37683:x
37676:=
37672:x
37650:x
37627:.
37619:3
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37610:1
37605:=
37602:c
37598:,
37590:2
37586:I
37581:1
37576:=
37573:b
37569:,
37561:1
37557:I
37552:1
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37544:a
37524:,
37521:1
37518:=
37513:2
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37498:3
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37487:/
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37474:(
37469:+
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37425:(
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37302:I
37298:+
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37258:,
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37252:=
37248:x
37236:T
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37190:.
37178:c
37158:b
37138:a
37069:2
37063:m
37052:m
37050:/
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36934:2
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36830:0
36823:0
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36799:0
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36776:[
36771:=
36746:,
36740:T
36734:Q
36723:Q
36719:=
36714:B
36708:C
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36622:B
36616:C
36610:I
36587:A
36564:.
36558:T
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36545:B
36539:C
36533:I
36527:A
36523:=
36517:C
36511:I
36488:x
36466:y
36445:,
36441:y
36436:A
36432:=
36428:x
36406:A
36383:B
36377:C
36371:I
36330:T
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36316:0
36312:I
36306:R
36302:=
36298:I
36272:R
36229:3
36226:E
36212:m
36192:]
36188:R
36180:R
36171:3
36166:E
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36157:R
36149:R
36145:(
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36117:I
36095:R
36068:0
36063:I
36025:.
36020:]
36012:3
36008:n
35998:2
35994:n
35984:1
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35973:[
35966:]
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35807:]
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35631:T
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35610:m
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35517:=
35514:I
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35414:)
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34893:I
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34877:I
34870:=
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34829:I
34825:(
34814:.
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34786:=
34777:I
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34765:x
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34748:I
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34728:=
34719:I
34715:(
34687:)
34682:z
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34510:(
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34486:=
34483:k
34471:k
34467:z
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34436:1
34433:=
34430:k
34415:k
34411:z
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34401:x
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34374:k
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34322:1
34319:=
34316:k
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34283:+
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34258:k
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34248:N
34243:1
34240:=
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34225:k
34221:y
34215:k
34211:x
34205:k
34201:m
34195:N
34190:1
34187:=
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34163:z
34157:k
34153:x
34147:k
34143:m
34137:N
34132:1
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34107:y
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34097:x
34091:k
34087:m
34081:N
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34045:k
34041:z
34037:+
34032:2
34027:k
34023:y
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34002:N
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33994:=
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33980:[
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33962:z
33959:z
33955:I
33947:y
33944:z
33940:I
33929:x
33926:z
33922:I
33909:z
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33856:z
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33838:y
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33831:I
33820:x
33817:x
33813:I
33806:[
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33780:I
33768:I
33756:I
33742:I
33730:I
33718:I
33704:I
33692:I
33680:I
33673:[
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33656:,
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33647:z
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33637:y
33631:k
33627:m
33621:N
33616:1
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33588:d
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33555:=
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33535:,
33530:k
33526:z
33520:k
33516:x
33510:k
33506:m
33500:N
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33409:k
33405:y
33399:k
33395:x
33389:k
33385:m
33379:N
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33352:f
33349:e
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33340:=
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33313:=
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33265:y
33262:x
33258:I
33235:x
33232:x
33228:I
33207:z
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33145:.
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33100:x
33095:(
33089:k
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33079:N
33074:1
33071:=
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33052:z
33046:k
33042:y
33036:k
33032:m
33026:N
33021:1
33018:=
33015:k
33000:k
32996:z
32990:k
32986:x
32980:k
32976:m
32970:N
32965:1
32962:=
32959:k
32942:k
32938:z
32932:k
32928:y
32922:k
32918:m
32912:N
32907:1
32904:=
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32881:2
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32872:z
32868:+
32863:2
32858:k
32854:x
32849:(
32843:k
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32833:N
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32825:=
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32806:y
32800:k
32796:x
32790:k
32786:m
32780:N
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32772:=
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32748:z
32742:k
32738:x
32732:k
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32714:=
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32696:k
32692:y
32686:k
32682:x
32676:k
32672:m
32666:N
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32635:2
32630:k
32626:z
32622:+
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32612:k
32608:y
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32597:k
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32587:N
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32579:=
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32544:z
32540:I
32532:y
32529:z
32525:I
32517:x
32514:z
32510:I
32500:z
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32485:y
32482:y
32478:I
32470:x
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32463:I
32453:z
32450:x
32446:I
32438:y
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32431:I
32423:x
32420:x
32416:I
32409:[
32404:=
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32387:I
32375:I
32363:I
32349:I
32337:I
32325:I
32311:I
32299:I
32287:I
32280:[
32275:=
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32247:y
32227:x
32207:,
32202:2
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32179:1
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32169:=
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32133:I
32106:,
32102:n
32094:I
32086:n
32082:=
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32051:n
32028:V
32025:d
32019:2
32015:]
32010:r
32006:[
32003:)
31999:r
31995:(
31987:Q
31976:=
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31970:d
31966:]
31962:r
31958:[
31952:T
31947:]
31942:r
31938:[
31935:)
31931:r
31927:(
31919:V
31911:=
31907:I
31885:x
31877:r
31873:=
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31865:]
31861:r
31857:[
31841:V
31833:3
31830:E
31811:r
31803:r
31780:,
31777:z
31774:d
31770:y
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31763:x
31760:d
31755:)
31750:r
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31733:3
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31704:(
31700:)
31697:z
31694:,
31691:y
31688:,
31685:x
31682:(
31674:V
31666:=
31662:I
31638:x
31618:y
31596:y
31593:x
31589:I
31568:x
31546:x
31543:x
31539:I
31512:.
31507:k
31503:z
31497:k
31493:y
31487:k
31483:m
31477:N
31472:1
31469:=
31466:k
31447:f
31444:e
31441:d
31435:=
31419:y
31416:z
31412:I
31408:=
31403:z
31400:y
31396:I
31388:,
31383:k
31379:z
31373:k
31369:x
31363:k
31359:m
31353:N
31348:1
31345:=
31342:k
31323:f
31320:e
31317:d
31311:=
31295:x
31292:z
31288:I
31284:=
31279:z
31276:x
31272:I
31264:,
31259:k
31255:y
31249:k
31245:x
31239:k
31235:m
31229:N
31224:1
31221:=
31218:k
31199:f
31196:e
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31187:=
31171:x
31168:y
31164:I
31160:=
31155:y
31152:x
31148:I
31113:,
31109:)
31103:2
31098:k
31094:y
31090:+
31085:2
31080:k
31076:x
31071:(
31065:k
31061:m
31055:N
31050:1
31047:=
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31028:f
31025:e
31022:d
31016:=
31000:z
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30993:I
30985:,
30981:)
30975:2
30970:k
30966:z
30962:+
30957:2
30952:k
30948:x
30943:(
30937:k
30933:m
30927:N
30922:1
30919:=
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30900:f
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30888:=
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30865:I
30857:,
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30847:2
30842:k
30838:z
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30809:k
30805:m
30799:N
30794:1
30791:=
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30760:=
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30741:x
30737:I
30704:I
30691:.
30673:j
30670:i
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30637:m
30615:)
30609:)
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30603:(
30598:3
30594:x
30590:,
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30579:(
30574:2
30570:x
30566:,
30561:)
30558:k
30555:(
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30546:x
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30537:=
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30527:r
30503:z
30483:y
30463:x
30443:j
30423:i
30397:)
30391:)
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30318:r
30307:(
30301:k
30297:m
30291:N
30286:1
30283:=
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30261:e
30258:d
30252:=
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30237:i
30233:I
30210:.
30205:]
30193:I
30181:I
30169:I
30155:I
30143:I
30131:I
30117:I
30105:I
30093:I
30086:[
30081:=
30077:I
30050:k
30046:m
30025:N
29976:R
29952:R
29948:I
29922:,
29913:k
29904:R
29898:I
29890:T
29879:k
29872:=
29863:k
29854:R
29848:I
29834:k
29828:=
29812:k
29805:)
29799:2
29794:]
29788:i
29783:r
29774:[
29767:i
29763:m
29757:N
29752:1
29749:=
29746:i
29734:(
29721:k
29715:=
29706:k
29698:2
29693:]
29687:i
29682:r
29673:[
29659:k
29651:i
29647:m
29641:N
29636:1
29633:=
29630:i
29619:=
29607:2
29602:|
29591:i
29586:r
29577:|
29570:i
29566:m
29560:N
29555:1
29552:=
29549:i
29541:=
29532:L
29528:I
29497:k
29473:R
29452:L
29426:.
29417:k
29409:2
29404:]
29398:i
29393:r
29384:[
29370:k
29355:=
29347:)
29337:k
29326:i
29321:r
29308:i
29303:r
29294:(
29281:k
29266:=
29258:)
29248:k
29237:i
29232:r
29220:(
29212:)
29206:i
29201:r
29184:k
29177:(
29169:=
29161:)
29155:i
29150:r
29133:k
29126:(
29118:)
29108:k
29098:)
29093:)
29087:i
29082:r
29065:k
29058:(
29045:k
29038:(
29033:(
29025:=
29017:)
29012:)
29006:i
29001:r
28984:k
28977:(
28964:k
28957:(
28949:)
28944:)
28938:i
28933:r
28916:k
28909:(
28896:k
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28857:k
28832:i
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28802:,
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28755:)
28745:k
28735:)
28730:)
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28682:k
28675:(
28670:(
28662:)
28657:)
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28629:k
28622:(
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28602:(
28594:)
28589:)
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28541:k
28534:(
28506:)
28501:)
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28466:(
28453:k
28446:(
28438:)
28433:)
28427:i
28422:r
28405:k
28398:(
28385:k
28378:(
28374:=
28363:)
28357:i
28352:r
28342:2
28337:]
28328:k
28322:[
28313:(
28305:)
28299:i
28294:r
28284:2
28279:]
28270:k
28264:[
28255:(
28251:=
28242:2
28237:|
28226:i
28221:r
28212:|
28177:k
28154:,
28148:T
28137:k
28124:k
28114:E
28110:=
28106:)
28099:T
28088:k
28075:k
28065:E
28060:(
28054:2
28049:|
28040:k
28034:|
28024:2
28019:]
28010:k
28004:[
27978:y
27965:k
27959:=
27955:y
27950:]
27941:k
27935:[
27913:]
27904:k
27898:[
27874:L
27848:k
27822:T
27811:k
27798:k
27774:E
27753:,
27748:i
27743:r
27734:)
27727:T
27716:k
27703:k
27693:E
27688:(
27684:=
27675:k
27668:)
27662:i
27657:r
27640:k
27633:(
27624:i
27619:r
27611:=
27601:i
27596:r
27565:k
27540:i
27535:r
27508:i
27504:P
27477:k
27471:t
27468:+
27464:R
27460:=
27457:)
27454:t
27451:(
27447:L
27425:R
27398:k
27374:R
27351:i
27346:r
27324:n
27321:,
27315:,
27312:1
27309:=
27306:i
27303:,
27298:i
27294:P
27273:n
27249:R
27240:i
27235:r
27230:=
27225:i
27220:r
27195:]
27190:i
27185:r
27177:[
27156:R
27132:R
27128:I
27106:,
27097:k
27088:R
27082:I
27074:T
27063:k
27056:=
27047:k
27038:R
27032:I
27018:k
27012:=
27003:k
26996:)
26990:2
26985:]
26979:i
26974:r
26965:[
26958:i
26954:m
26948:N
26943:1
26940:=
26937:i
26925:(
26912:k
26906:=
26901:L
26897:I
26875:R
26848:k
26823:L
26819:I
26790:]
26786:r
26782:[
26761:]
26757:r
26753:[
26746:T
26740:]
26736:r
26732:[
26727:m
26705:2
26701:r
26697:m
26675:2
26670:]
26666:r
26662:[
26657:m
26627:R
26605:C
26583:d
26562:,
26557:2
26553:]
26548:d
26544:[
26541:M
26532:C
26526:I
26521:=
26515:R
26509:I
26483:d
26462:]
26458:d
26454:[
26433:C
26409:C
26405:I
26381:.
26376:2
26372:]
26367:d
26363:[
26359:)
26353:i
26349:m
26343:n
26338:1
26335:=
26332:i
26323:(
26315:)
26311:]
26307:C
26298:i
26293:r
26288:[
26283:i
26279:m
26273:n
26268:1
26265:=
26262:i
26253:(
26249:]
26245:d
26241:[
26238:+
26235:]
26231:d
26227:[
26223:)
26219:]
26215:C
26206:i
26201:r
26196:[
26191:i
26187:m
26181:n
26176:1
26173:=
26170:i
26161:(
26157:+
26153:)
26147:2
26143:]
26138:C
26129:i
26124:r
26119:[
26114:i
26110:m
26104:n
26099:1
26096:=
26093:i
26084:(
26077:=
26071:R
26065:I
26041:.
26036:2
26032:]
26027:d
26019:)
26014:C
26005:i
26000:r
25994:(
25990:[
25985:i
25981:m
25975:n
25970:1
25967:=
25964:i
25953:=
25948:2
25944:]
25939:)
25934:d
25930:+
25926:C
25921:(
25912:i
25907:r
25902:[
25897:i
25893:m
25887:n
25882:1
25879:=
25876:i
25865:=
25859:R
25853:I
25830:R
25808:C
25786:d
25765:,
25761:C
25757:+
25753:d
25749:=
25745:C
25741:+
25738:)
25734:C
25726:R
25722:(
25719:=
25715:R
25693:C
25670:.
25665:2
25660:]
25655:R
25646:i
25641:r
25635:[
25628:i
25624:m
25618:n
25613:1
25610:=
25607:i
25596:=
25590:R
25584:I
25561:R
25537:R
25533:I
25507:R
25485:C
25447:C
25443:I
25421:,
25410:C
25404:I
25391:+
25380:C
25374:I
25369:=
25331:)
25325:2
25321:]
25315:i
25311:r
25304:[
25299:i
25295:m
25289:n
25284:1
25281:=
25278:i
25266:(
25254:+
25245:)
25239:2
25235:]
25229:i
25225:r
25218:[
25213:i
25209:m
25203:n
25198:1
25195:=
25192:i
25180:(
25176:=
25166:]
25163:)
25150:i
25145:r
25137:(
25129:i
25124:r
25114:i
25110:m
25104:n
25099:1
25096:=
25093:i
25074:+
25071:]
25068:)
25055:i
25050:r
25042:(
25034:i
25029:r
25021:[
25016:i
25012:m
25006:n
25001:1
24998:=
24995:i
24984:=
24943:i
24939:P
24929:u
24919:u
24914:)
24908:2
24904:]
24898:i
24894:r
24887:[
24882:i
24878:m
24872:n
24867:1
24864:=
24861:i
24849:(
24845:=
24838:]
24835:)
24831:u
24822:i
24817:r
24809:(
24801:i
24796:r
24788:[
24783:i
24779:m
24773:n
24768:1
24765:=
24762:i
24746:u
24740:2
24736:]
24730:i
24726:r
24719:[
24714:i
24710:m
24704:n
24699:1
24696:=
24693:i
24682:=
24670:]
24662:3
24658:u
24648:2
24644:u
24634:1
24630:u
24623:[
24616:]
24610:)
24605:2
24600:i
24597:,
24594:2
24590:r
24583:+
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24573:i
24570:,
24567:1
24563:r
24556:(
24546:i
24543:,
24540:2
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24527:i
24524:,
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24517:r
24506:i
24503:,
24500:1
24496:r
24487:i
24484:,
24481:3
24477:r
24464:i
24461:,
24458:3
24454:r
24445:i
24442:,
24439:2
24435:r
24426:)
24421:2
24416:i
24413:,
24410:3
24406:r
24399:+
24394:2
24389:i
24386:,
24383:1
24379:r
24372:(
24362:i
24359:,
24356:1
24352:r
24343:i
24340:,
24337:2
24333:r
24320:i
24317:,
24314:3
24310:r
24301:i
24298:,
24295:1
24291:r
24280:i
24277:,
24274:2
24270:r
24261:i
24258:,
24255:1
24251:r
24242:)
24237:2
24232:i
24229:,
24226:3
24222:r
24215:+
24210:2
24205:i
24202:,
24199:2
24195:r
24188:(
24179:[
24172:i
24168:m
24162:n
24157:1
24154:=
24151:i
24140:=
24128:]
24120:3
24116:u
24111:)
24106:2
24101:i
24098:,
24095:2
24091:r
24084:+
24079:2
24074:i
24071:,
24068:1
24064:r
24057:(
24049:2
24045:u
24038:i
24035:,
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24028:r
24019:i
24016:,
24013:3
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24002:+
23997:1
23993:u
23986:i
23983:,
23980:1
23976:r
23967:i
23964:,
23961:3
23957:r
23950:+
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23937:u
23930:i
23927:,
23924:3
23920:r
23911:i
23908:,
23905:2
23901:r
23894:+
23889:2
23885:u
23880:)
23875:2
23870:i
23867:,
23864:3
23860:r
23853:+
23848:2
23843:i
23840:,
23837:1
23833:r
23826:(
23818:1
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14100:V
14095:M
14090:2
14087:1
14082:+
14071:C
14065:I
14050:2
14047:1
14042:=
14037:K
14033:E
14006:.
14000:C
13994:V
13983:C
13977:V
13971:)
13965:i
13961:m
13955:n
13950:1
13947:=
13944:i
13935:(
13929:2
13926:1
13921:+
13912:)
13906:2
13901:]
13895:i
13890:r
13881:[
13874:i
13870:m
13864:n
13859:1
13856:=
13853:i
13841:(
13827:2
13824:1
13819:=
13806:C
13800:V
13789:C
13783:V
13777:)
13771:i
13767:m
13761:n
13756:1
13753:=
13750:i
13741:(
13735:2
13732:1
13727:+
13723:)
13718:)
13708:]
13702:i
13697:r
13688:[
13681:T
13675:]
13669:i
13664:r
13655:[
13647:T
13635:(
13629:i
13625:m
13619:n
13614:1
13611:=
13608:i
13599:(
13593:2
13590:1
13585:=
13572:C
13566:V
13555:C
13549:V
13543:)
13537:i
13533:m
13527:n
13522:1
13519:=
13516:i
13507:(
13501:2
13498:1
13493:+
13489:)
13484:)
13474:]
13468:i
13463:r
13454:[
13449:(
13441:)
13431:]
13425:i
13420:r
13411:[
13406:(
13400:i
13396:m
13390:n
13385:1
13382:=
13379:i
13370:(
13364:2
13361:1
13356:=
13347:K
13343:E
13318:]
13313:i
13308:r
13300:[
13280:0
13277:=
13272:i
13267:r
13257:i
13253:m
13247:n
13242:1
13239:=
13236:i
13209:.
13205:)
13198:C
13192:V
13181:C
13175:V
13168:i
13164:m
13158:n
13153:1
13150:=
13147:i
13138:(
13132:2
13129:1
13124:+
13120:)
13115:)
13109:i
13104:r
13087:(
13077:C
13071:V
13064:i
13060:m
13054:n
13049:1
13046:=
13043:i
13034:(
13030:+
13026:)
13021:)
13015:i
13010:r
12993:(
12985:)
12979:i
12974:r
12957:(
12951:i
12947:m
12941:n
12936:1
12933:=
12930:i
12921:(
12915:2
12912:1
12907:=
12902:K
12898:E
12873:C
12864:i
12859:r
12854:=
12849:i
12844:r
12819:,
12815:)
12808:C
12802:V
12797:+
12792:i
12787:r
12770:(
12762:)
12755:C
12749:V
12744:+
12739:i
12734:r
12717:(
12711:i
12707:m
12701:n
12696:1
12693:=
12690:i
12680:2
12677:1
12672:=
12667:i
12662:v
12652:i
12647:v
12640:i
12636:m
12630:n
12625:1
12622:=
12619:i
12609:2
12606:1
12601:=
12596:K
12592:E
12569:i
12564:v
12540:i
12535:r
12513:n
12510:,
12504:,
12501:1
12498:=
12495:i
12492:,
12487:i
12483:P
12462:n
12429:C
12408:,
12403:2
12398:]
12392:i
12387:r
12378:[
12371:i
12367:m
12361:n
12356:1
12353:=
12350:i
12339:=
12333:C
12327:I
12302:C
12298:I
12276:,
12265:C
12259:I
12254:=
12245:)
12239:2
12234:]
12228:i
12223:r
12214:[
12207:i
12203:m
12197:n
12192:1
12189:=
12186:i
12174:(
12170:=
12166:L
12144:C
12135:i
12130:r
12125:=
12120:i
12115:r
12090:]
12085:i
12080:r
12072:[
12044:C
12040:=
12032:(
12017:R
12013:V
11987:,
11983:)
11976:R
11970:V
11960:i
11955:r
11944:i
11940:m
11934:n
11929:1
11926:=
11923:i
11914:(
11910:+
11906:)
11901:)
11887:i
11882:r
11873:(
11864:i
11859:r
11848:i
11844:m
11838:n
11833:1
11830:=
11827:i
11815:(
11811:=
11800:)
11793:R
11787:V
11782:+
11777:i
11772:r
11755:(
11746:i
11741:r
11730:i
11726:m
11720:n
11715:1
11712:=
11709:i
11701:=
11689:i
11684:v
11674:i
11669:r
11658:i
11654:m
11648:n
11643:1
11640:=
11637:i
11629:=
11621:L
11595:C
11573:R
11546:.
11541:]
11535:0
11528:x
11524:b
11516:y
11512:b
11499:x
11495:b
11486:0
11479:z
11475:b
11465:y
11461:b
11453:z
11449:b
11440:0
11434:[
11421:]
11417:b
11413:[
11404:y
11399:]
11395:b
11391:[
11379:y
11371:b
11346:)
11341:z
11337:b
11333:,
11328:y
11324:b
11320:,
11315:x
11311:b
11307:(
11304:=
11300:b
11278:]
11274:b
11270:[
11236:R
11212:R
11208:V
11161:R
11155:V
11150:+
11145:i
11140:r
11124:=
11119:i
11114:v
11091:R
11082:i
11077:r
11072:=
11067:i
11062:r
11036:R
11013:i
11008:v
10984:i
10979:r
10957:n
10954:,
10948:,
10945:1
10942:=
10939:i
10936:,
10931:i
10927:P
10906:n
10895:.
10864:.
10855:k
10843:C
10838:I
10834:=
10806:C
10801:I
10779:C
10750:i
10746:P
10719:k
10713:=
10708:i
10698:t
10686:i
10676:e
10651:0
10647:=
10642:i
10632:e
10620:i
10610:e
10580:,
10576:A
10568:)
10562:i
10552:e
10543:i
10539:r
10529:i
10525:m
10519:n
10514:1
10511:=
10508:i
10499:(
10495:+
10486:k
10476:)
10470:2
10465:i
10461:r
10451:i
10447:m
10441:n
10436:1
10433:=
10430:i
10421:(
10417:=
10406:)
10401:A
10397:+
10392:i
10382:e
10373:i
10369:r
10360:2
10347:i
10337:t
10328:i
10324:r
10313:(
10304:i
10294:e
10285:i
10281:r
10271:i
10267:m
10261:n
10256:1
10253:=
10250:i
10242:=
10203:.
10199:A
10195:+
10190:i
10180:e
10171:i
10167:r
10158:2
10145:i
10135:t
10126:i
10122:r
10112:=
10101:A
10097:+
10092:i
10082:e
10073:i
10069:r
10053:k
10035:k
10021:i
10011:e
10002:i
9998:r
9982:k
9973:=
9964:i
9959:A
9931:i
9921:e
9905:k
9899:=
9894:i
9884:t
9858:i
9853:r
9830:R
9807:i
9797:e
9767:k
9742:.
9738:A
9734:+
9729:i
9724:r
9700:+
9695:i
9690:r
9674:=
9669:i
9664:A
9597:A
9575:R
9552:i
9548:P
9518:i
9513:r
9487:,
9482:i
9477:A
9470:i
9466:m
9457:i
9452:r
9442:n
9437:1
9434:=
9431:i
9423:=
9407:,
9402:i
9397:A
9390:i
9386:m
9380:n
9375:1
9372:=
9369:i
9361:=
9353:F
9327:R
9302:n
9299:,
9293:,
9290:1
9287:=
9284:i
9281:,
9276:i
9272:P
9251:n
9204:C
9199:I
9176:.
9172:V
9164:V
9160:M
9155:2
9152:1
9147:+
9142:2
9131:C
9126:I
9120:2
9117:1
9112:=
9107:K
9103:E
9079:C
9074:I
9052:C
9025:.
9021:V
9013:V
9008:)
9002:i
8998:m
8992:n
8987:1
8984:=
8981:i
8972:(
8966:2
8963:1
8958:+
8954:)
8948:i
8938:t
8929:i
8925:r
8915:i
8911:m
8905:n
8900:1
8897:=
8894:i
8885:(
8877:V
8870:+
8866:)
8860:i
8850:t
8838:i
8828:t
8819:2
8814:i
8810:r
8800:i
8796:m
8790:n
8785:1
8782:=
8779:i
8770:(
8764:2
8754:2
8751:1
8746:=
8736:,
8732:)
8727:V
8723:+
8718:i
8708:t
8699:i
8695:r
8683:(
8675:)
8670:V
8666:+
8661:i
8651:t
8642:i
8638:r
8626:(
8620:i
8616:m
8610:n
8605:1
8602:=
8599:i
8589:2
8586:1
8581:=
8571:,
8566:i
8561:v
8551:i
8546:v
8539:i
8535:m
8529:n
8524:1
8521:=
8518:i
8508:2
8505:1
8500:=
8491:K
8487:E
8431:C
8426:I
8403:.
8394:k
8382:C
8377:I
8373:=
8369:L
8346:,
8341:2
8336:i
8332:r
8322:i
8318:m
8312:i
8304:=
8298:C
8293:I
8269:C
8264:I
8237:,
8234:0
8231:=
8222:i
8212:e
8203:i
8199:r
8189:i
8185:m
8179:n
8174:1
8171:=
8168:i
8156:,
8152:C
8143:i
8138:r
8133:=
8124:i
8114:e
8105:i
8101:r
8072:C
8042:.
8038:V
8030:)
8024:i
8014:e
8005:i
8001:r
7991:i
7987:m
7981:n
7976:1
7973:=
7970:i
7961:(
7957:+
7948:k
7938:)
7932:2
7927:i
7923:r
7913:i
7909:m
7903:n
7898:1
7895:=
7892:i
7883:(
7879:=
7868:)
7863:V
7859:+
7854:i
7844:t
7835:i
7831:r
7819:(
7810:i
7800:e
7791:i
7787:r
7777:i
7773:m
7767:n
7762:1
7759:=
7756:i
7748:=
7736:i
7731:v
7721:i
7716:r
7706:i
7702:m
7696:n
7691:1
7688:=
7685:i
7677:=
7669:L
7623:V
7619:+
7614:i
7604:t
7595:i
7591:r
7580:=
7576:V
7572:+
7567:i
7557:e
7548:i
7544:r
7528:k
7519:=
7515:V
7511:+
7506:i
7501:r
7485:=
7476:i
7471:v
7462:,
7457:i
7447:e
7431:k
7425:=
7420:i
7410:t
7402:,
7389:=
7380:k
7373:,
7365:i
7361:r
7350:i
7345:r
7334:=
7325:i
7315:e
7285:i
7275:e
7259:k
7253:=
7248:i
7238:t
7212:i
7207:r
7184:R
7161:i
7156:e
7133:k
7108:R
7086:V
7039:,
7035:V
7031:+
7026:i
7021:r
7005:=
7001:V
6997:+
6993:)
6988:R
6979:i
6974:r
6968:(
6956:=
6947:i
6942:v
6933:,
6929:R
6920:i
6915:r
6910:=
6901:i
6896:r
6865:i
6860:v
6837:R
6816:n
6813:,
6807:,
6804:1
6801:=
6798:i
6795:,
6790:i
6786:P
6765:n
6732:k
6684:3
6680:R
6671:3
6668:4
6663:=
6660:m
6636:,
6631:2
6627:R
6623:m
6617:5
6614:2
6608:=
6596:5
6592:R
6587:)
6580:5
6577:1
6571:+
6565:3
6562:2
6553:1
6549:(
6539:=
6527:R
6522:R
6514:]
6508:5
6504:z
6497:5
6494:1
6488:+
6483:3
6479:z
6473:2
6469:R
6462:3
6459:2
6450:z
6445:4
6441:R
6436:[
6422:2
6419:1
6413:=
6403:z
6400:d
6394:2
6389:)
6383:2
6379:z
6370:2
6366:R
6361:(
6347:2
6344:1
6336:R
6331:R
6320:=
6317:z
6314:d
6308:4
6304:)
6300:z
6297:(
6294:r
6282:2
6279:1
6271:R
6266:R
6255:=
6241:,
6238:C
6234:I
6209:z
6187:.
6182:2
6178:z
6169:2
6165:R
6161:=
6156:2
6152:y
6148:+
6143:2
6139:x
6135:=
6130:2
6126:)
6122:z
6119:(
6116:r
6096:z
6076:z
6056:r
6034:,
6029:2
6025:R
6021:=
6016:2
6012:z
6008:+
6003:2
5999:y
5995:+
5990:2
5986:x
5942:L
5922:,
5917:2
5913:)
5909:R
5906:+
5903:L
5900:(
5891:M
5887:+
5877:,
5874:C
5870:I
5866:+
5861:2
5856:)
5851:2
5848:L
5843:(
5832:M
5828:+
5818:,
5815:C
5811:I
5807:=
5802:P
5798:I
5777:P
5754:s
5746:2
5742:R
5735:=
5732:m
5712:,
5707:2
5703:R
5699:m
5694:2
5691:1
5686:=
5681:4
5676:4
5672:R
5666:s
5657:2
5654:=
5648:d
5644:r
5641:d
5637:r
5634:s
5629:2
5625:r
5616:R
5611:0
5598:2
5593:0
5585:=
5582:V
5579:d
5573:2
5569:r
5559:Q
5551:=
5541:,
5538:C
5534:I
5510:d
5506:r
5503:d
5499:r
5496:s
5493:=
5490:V
5487:d
5467:z
5423:R
5403:s
5373:s
5367:=
5364:m
5344:,
5333:2
5325:m
5319:=
5315:)
5309:8
5304:3
5294:+
5289:8
5284:3
5273:(
5267:3
5263:s
5254:=
5248:2
5237:2
5224:|
5218:3
5213:3
5209:x
5203:s
5192:=
5189:x
5186:d
5182:s
5177:2
5173:x
5162:2
5151:2
5135:=
5132:V
5129:d
5123:2
5119:x
5109:Q
5101:=
5091:,
5088:C
5084:I
5063:x
5003:s
4949:y
4929:x
4905:y
4885:x
4861:z
4831:Q
4811:V
4791:)
4788:z
4785:,
4782:y
4779:,
4776:x
4773:(
4752:r
4731:)
4728:z
4725:,
4722:y
4719:,
4716:x
4713:(
4668:V
4665:d
4660:2
4651:r
4642:)
4639:z
4636:,
4633:y
4630:,
4627:x
4624:(
4616:Q
4608:=
4603:P
4599:I
4570:2
4565:i
4561:r
4555:i
4551:m
4545:i
4537:=
4532:P
4528:I
4500:.
4495:2
4490:i
4486:r
4480:i
4476:m
4470:N
4465:1
4462:=
4459:i
4451:=
4446:P
4442:I
4419:2
4415:r
4411:m
4389:.
4384:2
4379:i
4375:r
4369:i
4365:m
4359:N
4354:1
4351:=
4348:i
4338:2
4327:2
4324:1
4319:=
4314:2
4309:)
4303:i
4299:r
4291:(
4284:i
4280:m
4273:2
4270:1
4263:N
4258:1
4255:=
4252:i
4244:=
4239:i
4234:v
4224:i
4219:v
4212:i
4208:m
4201:2
4198:1
4191:N
4186:1
4183:=
4180:i
4172:=
4167:K
4163:E
4140:P
4118:i
4114:r
4091:i
4087:m
4066:N
4042:r
4020:2
4016:r
4012:m
4000:)
3936:4
3932:d
3918:1
3913:=
3908:4
3904:r
3893:4
3890:1
3885:=
3880:y
3877:y
3873:I
3869:=
3864:x
3861:x
3857:I
3826:3
3822:h
3818:b
3812:=
3807:x
3804:x
3800:I
3769:3
3765:b
3761:h
3755:=
3750:y
3747:y
3743:I
3714:3
3710:h
3706:b
3700:=
3695:x
3692:x
3688:I
3658:4
3654:b
3648:=
3643:y
3640:y
3636:I
3632:=
3627:x
3624:x
3620:I
3605:]
3580:r
3570:b
3566:h
3562:.
3548:y
3545:y
3541:I
3518:x
3515:x
3511:I
3456:.
3452:m
3434:2
3430:)
3425:s
3421:/
3417:d
3414:a
3411:r
3401:(
3393:2
3389:s
3384:/
3380:m
3360:2
3355:n
3347:g
3342:=
3339:L
3319:L
3298:s
3294:/
3290:d
3287:a
3284:r
3251:.
3245:r
3242:m
3236:P
3232:I
3226:=
3219:2
3214:n
3206:g
3201:=
3198:L
3178:,
3170:P
3166:I
3161:r
3158:g
3155:m
3148:=
3142:L
3139:g
3133:=
3128:n
3103:L
3075:L
3047:t
3027:,
3019:2
3011:4
3004:2
3000:t
2996:r
2993:g
2990:m
2984:=
2977:2
2972:n
2963:r
2960:g
2957:m
2951:=
2946:P
2942:I
2921:t
2901:P
2877:r
2857:g
2837:m
2817:,
2809:P
2805:I
2800:r
2797:g
2794:m
2787:=
2782:n
2755:P
2751:I
2728:n
2715:(
2669:2
2665:r
2661:m
2639:2
2635:r
2631:m
2628:=
2625:I
2603:.
2598:2
2590:I
2585:2
2582:1
2577:=
2572:2
2563:)
2557:2
2553:r
2549:m
2545:(
2539:2
2536:1
2531:=
2527:v
2519:v
2515:m
2510:2
2507:1
2502:=
2497:K
2493:E
2465:,
2456:k
2447:I
2444:=
2434:2
2430:r
2426:m
2423:=
2412:)
2407:r
2402:)
2389:r
2384:(
2371:)
2366:r
2358:r
2353:(
2348:(
2344:m
2341:=
2330:)
2325:r
2313:m
2309:(
2301:r
2297:=
2293:p
2285:r
2281:=
2273:L
2221:r
2209:=
2205:v
2178:2
2174:r
2170:m
2167:=
2164:I
2135:2
2131:r
2127:m
2124:=
2121:I
2100:r
2047:k
2018:,
2009:k
2000:I
1997:=
1987:2
1983:r
1979:m
1976:=
1965:)
1960:r
1955:)
1942:r
1937:(
1924:)
1919:r
1911:r
1906:(
1901:(
1897:m
1894:=
1884:)
1880:r
1868:m
1865:(
1858:r
1854:=
1850:F
1842:r
1838:=
1804:a
1800:m
1797:=
1793:F
1771:r
1759:=
1755:a
1707:F
1685:r
1663:F
1655:r
1651:=
1624:.
1619:2
1615:r
1611:m
1608:=
1605:I
1585:r
1565:m
1531:k
1517:,
1512:2
1508:k
1504:m
1501:=
1498:I
1488:k
1484:m
1477:k
1473:r
1460:r
1456:m
1440:.
1435:2
1431:r
1427:m
1424:=
1421:I
1411:r
1407:m
1403:I
1382:.
1376:I
1373:=
1359:α
1352:τ
1313:.
1305:L
1300:=
1297:I
1286:ω
1279:L
1272:I
1249:.
1150:m
1130:r
1108:2
1104:r
1100:m
904:e
897:t
890:v
580:)
199:t
196:d
190:p
186:d
180:=
175:F
144:L
141:M
114:L
109:=
106:I
66:I
34:.
20:)
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