刚体动力学

弧度:

\[\begin{aligned} \Delta s &= ({\pi \over {180}}\Delta \theta ^\circ )R \newline \Delta \theta &= {\pi \over {180}}\Delta \theta ^\circ \end{aligned}\]

所以:

\[\begin{aligned} \Delta s &= \Delta \theta R \newline {360^\circ } &= 2\pi \end{aligned}\]

切线速度:

\[{v_T} = \frac{ds}{dt} = R\frac{d\theta}{dt} = \omega R\]

切线加速度:

\[{a_T} = \frac{d{v_T}}{dt} = R\frac{d\omega }{dt} = \alpha R\]

向心加速度:

\[{a_C} = \frac{v_T^2}{R} = {\omega ^2}R\]

转动刚体的动能公式:

\[K = \frac{1}{2}\sum\limits_i {m_iv_i^2} = \frac{1}{2}\sum\limits_i {m_iR_i^2\omega _i^2} = \frac{1}{2}I{\omega ^2}\]

从上式得出转动惯量:

\[I = \sum\limits_i {m_iR_i^2} \]

角动量:

\[L = I\omega \]

力矩:

\[\tau = \sum {F_iR_i} \sin {\theta _i} = \frac{dL}{dt} = I\alpha \]

向量:

\[\overrightarrow \tau = \overrightarrow r \times \overrightarrow F \]

转动刚体做功:

\[dW = \tau d\theta \]

物体的质心:

\[cm = \sum\limits_i {m_iR_i^2} \]

平衡轴定理:

\[I = {I_{cm}} + M{d^2}\]

\(I_{cm}\)为质心的转动惯量,\(M\)为物体的总质量,\(d\)为旋转轴到质心的距离。