来自卢德鑫的《大学物理》,想着期末还会用到索性就记个笔记,养成记笔记的好习惯
其实是期中考试之前补天
圆周运动
无限小角位移矢量\(d \vec{\theta} = d \theta \vec{k}\),并且是一个矢量
转动引起的无限小位移\(d \vec{R} = d \vec{\theta} \times \vec{R}\)
速度\(\vec{v} = \dfrac{d\vec{R}}{dt} = \dfrac{d\vec{\theta}}{dt} \times \vec{R} = \vec{\omega} \times \vec{R}\)
加速度\(\vec{a} = \dfrac{d\vec{v}}{dt} = \dfrac{d \vec{\omega}}{dt} \times \vec{R} + \vec{\omega}\times \dfrac{d\vec{R}}{dt} = \vec{\beta}\times \vec{R} + \vec{\omega} \times \vec{R}\) 其中前者为径向,后者为切向
极坐标系
\(d \vec{e_r} = d \theta \vec{e_{\theta}}\)
\(d \vec{e_{\theta}} = - d \theta \vec{e_{r}}\)
可以用直角坐标求出
则由\(\vec{r} = r \vec{e_r}\)等几式可得
速度\(\vec{v} = \dfrac{dr}{dt} \vec{e_r} + r\dfrac{d\theta}{dt} \vec{e_\theta} = \vec{v_r} + \vec{v_{\theta}}\)
加速度\(\vec{a} = \dfrac{d\vec{v}}{dt} = \dfrac{d}{dt} \left(\dfrac{dr}{dt} \vec{e_r} + r\dfrac{d\theta}{dt} \vec{e_\theta} \right) = \dfrac{d^2r}{dt^2} \vec{e_r} + \dfrac{dr}{dt} \dfrac{d\theta}{dt} \vec{e_\theta} + \dfrac{dr}{dt}\dfrac{d\theta}{dt}\vec{e_{\theta}}+ r\dfrac{d^2\theta}{dt^2} \vec{e_\theta} - r\dfrac{d\theta}{dt} \dfrac{d\vec{e_\theta}}{dt} = \vec{a_r} + \vec{a_{\theta}}\)
合并同类项,可得\(\vec{a} = \left[\dfrac{d^2r}{dt^2} - r\left(\dfrac{d\theta}{dt}\right)^2\right]\vec{e_r} + \left (2\dfrac{dr}{dt}\dfrac{d\theta}{dt} + r\dfrac{d^2\theta}{dt^2}\right )\vec{e_{\theta}}\)
用圆点表示对时间的导数,则上式可以写成\(\vec{a} = \left(\ddot{r} - r\dot{\theta}^2\right) \vec{e_r} + \left (2\dot{r}\dot{\theta} + r\ddot{\theta}\right )\vec{e_{\theta}}\)
转动参考系
速度的变换\(\vec{v} = \vec{v}' + \vec{\omega} \times \vec{r}'\)
加速度的变换\(\vec{a} = \vec{a}' + \vec{\omega} \times \left(\vec{\omega} \times \vec{r}'\right) + 2\vec{\omega} \times \vec{v}'\)
对于非匀速转动,情况更为复杂,需要在最后加入角速度对于时间的导数有\(\vec{a} = \vec{a}' + \vec{\omega} \times \left(\vec{\omega} \times \vec{r}'\right) + 2\vec{\omega} \times \vec{v}' + \dot{\vec{\omega}} \times \vec{r}'\)