Chapter1 动力学
1.1 Newton's Laws
Drag（阻力）
$$R=bv$$
Note
This assumption is valid for objects falling slowly through a liquid and for very small objects, such as dust particles, moving through air.
$$R=cv^2$$
Note
This assumption is valid for large and fast moving objects, such as a skydiver moving through air in free fall.
Scaling
Center-of-mass Frame（质心系）
In the center-of-mass frame, the total momentum is always zero.
Proof
$\vec{v}{CM}=\frac{\sum m_i\vec{v}_i}{M}$
$\vec{p}''}=\sum m_i\vec{vi=\sum m_i(\vec{v}_i-\vec{v})$
$=\sum m_i\vec{v}i-\sum m_i\vec{v}$
$=M\vec{v}{CM}-M\vec{v}$
$=0$
König’s theorem（柯尼希定理）
$$E_k=\frac{1}{2}Mv_{CM}^2+\sum\frac{1}{2}m_iv_i'^2$$
The total kinetic energy equals the kinetic energy of center-of-mass plus the kinetic energy in the center-of-mass frame.（总动能等于质心的动能加质心系下的总动能）
Gravitational Potential Energy（引力势能）
总机械能：
$$E=\frac{1}{2}mv^2-\frac{GMm}{r}=-\frac{GMm}{2r}$$
如果是椭圆轨道：$E=-\frac{GMm}{2a}$，其中$a$为椭圆的半长轴
Thrust（推力）
对于水平火箭发射：
$$(M+\Delta m)v=M(v+\Delta v)+\Delta m(v-v_e)$$
$$M\Delta v=v_e\Delta m$$
$$Mdv=-v_edM$$
$$F_{Thrust}=M\frac{dv}{dt}=|v_e\frac{dM}{dt}|$$
1.2 Rotations
Angular displacement（角位移）
$$\Delta\theta=\theta_f-\theta_i$$
Note
Finite angular displacement is not a vector.
Angular speed（角速度）
$$\omega=\frac{d\theta}{dt}$$
Units: $rad/s$ or $s^{-1}$ Direction: positive for counterclockwise motion（逆时针）
Angular acceleration（角加速度）
$$\alpha=\frac{d\omega}{dt}$$
Units: $rad/s^2$ or $s^{-2}$ Direction: positive when the rate of counterclockwise rotation is increasing
Note
When rotating about a fixed axis, every particle on a rigid object rotates through the same angle and has the same angular speed and the same angular acceleration.
For rotation about a fixed axis, the only direction that uniquely specifies the rotational motion is the direction along the axis of rotation. Therefore, the directions of $\omega$ and $\alpha$ are along this axis. They obey the Right-Hand Rule.
Rotational kinematics（旋转动力学）
$$\omega_f=\omega_i+\alpha t$$
$$\theta_f=\theta_i+\omega_i t+\frac{1}{2}\alpha t^2$$
$$\omega_f^2-\omega_i^2=2\alpha(\theta_f-\theta_i)$$
（Under constant angular acceleration）
Transformation between angular and linear vectors
$$v=\omega r~(\vec{v}=\vec{\omega}\times\vec{r})$$
$$a_{\bot}=r\omega^2$$
$$a_{//}=r\alpha$$
Torque（力矩）
$$\tau=Fd~(\vec{\tau}=\vec{r}\times\vec{F})
Note
$d$ is the perpendicular distance from the axis to the line of action of $F$; also called the moment arm（力臂） of F.
counterclockwise torque: positive; clockwise torque: negative
Moment of inertia（转动惯量）
$$\tau=I\alpha$$
$$I=mr^2$$
Rotational kinetic energy（转动动能）
$$K_R=\frac{1}{2}I\omega^2$$
Power in rotation（转动功率）
$$P=\tau\omega$$
Useful equations in rotational and linear motion
Parallel-Axis Theorem（平行轴定理）
$$I_P=I_{CM}+Md^2$$
Note
$I_P$: moment of inertia with respect to the axis crossing $P$.
$I_{CM}$:moment of inertia with respect to the axis crossing the center of mass.
$d$:the distance between $P$ and the center of mass.
Coriolis force（科里奥利力）
$$d\vec{r}|_I=d\vec{r}|_R+(\vec{\omega}\times\vec{r})dt$$
$$\frac{d\vec{r}}{dt}|_I=\frac{d\vec{r}}{dt}|_R+\vec{\omega}\times\vec{r}$$
$$\frac{d^2\vec{r}}{dt^2}|_I=[\frac{d}{dt}|_R+\vec{\omega}\times]^2\vec{r}=\frac{d^2\vec{r}}{dt^2}|_R+2\vec{\omega}\times\frac{d\vec{r}}{dt}|_R+\vec{\omega}\times(\vec{\omega}\times\vec{r})$$
$$\vec{F_R}=m\frac{d^2\vec{r}}{dt^2}|_R=m\frac{d^2\vec{r}}{dt^2}|_I-2m\vec{\omega}\times\frac{d\vec{r}}{dt}|_R-m\vec{\omega}\times(\vec{\omega}\times\vec{r})$$
$$=\vec{F_I}-2m\vec{\omega}\times\vec{v_R}-m\vec{\omega}\times(\vec{\omega}\times\vec{r})$$
If $\vec{r}\bot\vec{\omega}$:
$$\vec{F_R}=\vec{F_I}-2m\vec{\omega}\times\vec{v_R}+m\omega^2\vec{r}$$
Coriolis force（科里奥利力）：
$$-2m\vec{\omega}\times\vec{v_R}$$
Centrifugal force（离心力）：
$$m\omega^2\vec{r}$$
Pure rolling motion（纯滚动）
$$v_{CM}=R\omega$$
$$a_{CM}=r\alpha$$
$$K=\frac{1}{2}I_{CM}\omega^2+\frac{1}{2}Mv_{CM}^2=\frac{1}{2}I_P\omega^2$$
Note
$P$ is the point which is between the object and the surface, and it is at rest relative to the surface because slipping does not occur.
Angular momentum（角动量）
$$L=mvr\sin\theta=I\omega$$
Conservation of angular momentum（角动量守恒）
The total angular momentum of a system is constant in both magnitude and direction if the resultant external torque acting on the system is zero.