向心加速度推导

角加速度

我们假设质点进行匀速圆周运动,有:$$\left| {\vec {v_1} } \right| = \left| {\vec {v_2} } \right|$$

因为两个绿色区域相似,有:$$\frac{\left| {\vec {\Delta r} } \right|}{\left| {\vec r} \right|} = \frac{\left| {\vec {\Delta v} } \right|}{\left| {\vec v} \right|}$$ 因为路程=速度*时间,当$\theta \to 0$时,有:$$\left| \vec {\Delta r} \right| = \left| {\vec v} \right|\Delta t$$

$$\frac{\left| \overrightarrow {\Delta r} \right|}{\left| \overrightarrow r \right|} = \frac{\left| {\vec v} \right|\Delta t}{\left| {\vec r} \right|} = \frac{\left| {\overrightarrow {\Delta v} } \right|}{\left| {\vec v} \right|}$$

$$\therefore \frac{ {\vec v}^2 \Delta t }{\left| \vec r\right|} = \frac{\left| \vec {\Delta v} \right|}{\Delta t} = \vec a $$

当$t \to 0$时成立。最后有:

$$\vec a = \frac{ {\vec v}^2}{\left| {\vec r} \right|}\overrightarrow n$$

$\overrightarrow n$为法向量,指向圆心;$\vec a$即为向心加速度。