Rotational Motion

Angular Quantities

根据弧度制的定义：\(\theta = \frac{l}{r}\)。

新单位，revolution(rev)，\(1 rev = 2\pi rad\)。

定义角速度，\(\omega = \frac{\Delta \theta}{\Delta t}\)。

定义角加速度，\(\alpha = \frac{\Delta \omega}{\Delta t}\)。

所有转动的量乘以 \(r\) 都会变成直线运动的量。\(v = r\omega\) \(a_{tan} = r\alpha\)。

向心加速度，\(a_R = \frac{v^2}{r} = \omega^2 r\)。

频率和周期，\(\omega = 2\pi f\)，\(T = \frac{1}{f}\)。

Constant Angular Acceleration

Angular	Linear
\(\omega = \omega_0+\alpha t\)	\(v=v_0+at\)
\(\theta=\omega_0t+\frac{1}{2}\alpha t^2\)	\(x=v_0t+\frac{1}{2}at^2\)

Rolling Motion (Without Slipping)

\(v=r\omega\)

Torque

\(\tau = rFsin\theta\)

\(r\): 旋转半径; \(\theta\): 力的夹角

Rotational Dynamics; Torque and Rotational Inertia

\(\tau=rF=r(mr\alpha)=mr^2\alpha\)

\(I\) 是转动惯量，以下是不同类型物体的：

Rotational Kinetic Energy

\({k.e.}_{rotational} = \frac{1}{2}I\omega^2\)

总动能：\(E=\frac{1}{2}Mv_{CM}^2+\frac{1}{2}I\omega^2\)

Angular Momentum and Conservation

动量：\(L = I\omega\)

\(\sum \tau=\frac{\bigtriangleup L}{\bigtriangleup t}\)

The total angular momentum of a rotating object remains constant if the net torque acting on it is zero.