Hanjie's Blog

一只有理想的羊驼

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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。

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