# Chapter 3

# 3.10 Relative Velocity

$$V_{AB}$$

Velocity of A with respect to B

# Chapter 4

Type of Forces: Contact Gravitational Electromagnetic Force

Newton's First Law: Inertial Law

Inertial Reference Frame: reference frame where First Law holds

Accelerating (non-inertial)

Mass: measure of inertial

Second Law: acceleration proportional to net force, inversely proportional to mass

Second Law's origin expression: The rate of change of momentum of an object is equal to the net force applied to it.

Third Law: F_{AB} = -F_

Dynamics in Non-inertial Frame: apply fictitious or pseudo forces

# Chapter 10

# Basic concept

rigid body: 刚体 object with a definite shape that does not change

motion of a rigid body: translation of center of mass + rotation relative to center of mass

angular velocity: $\omega$ 方向遵循右手法则

angular acceleration: $\alpha$

切向加速度 $a_t=\alpha r$

法向加速度 $a_n=\frac{v^2}{r}=\omega v=\omega^2r$

# 纯滚动 Rolling without slipping

静摩擦力 接触点静止

$v=\omega r$

# Torque 力矩

$\tau=FRsin\theta$

lever arm / moment arm: $R_{\perp}=Rsin\theta$

# Rotational dynamics 转动力学

类比平动

# rotational inertia 转动惯量

also named: moment of inertia

$F=ma\Rightarrow \tau=FR=mR^2\alpha=I\alpha$

取决于总质量与质量的分布

对于同一轴的转动惯量可以叠加 superimposed

# 常见几何形状的转动惯量

长细棒：$I=\frac{1}{3}ML^2$（绕端点）$I=\frac{1}{12}ML^2$（绕中点）

矩形薄板：$I=\frac{1}{12}M(a^2+b^2)$

球：$I=\frac{2}{5}MR^2$

薄球壳：$I=\frac{2}{3}MR^2$

圆盘 / 圆柱：$I=\frac{1}{2}MR^2$

# 测量转动惯量的方法

1. 积分（对于规则形状）
2. 实验

# parallel-axis theorem 平行轴定理

$I=I_c + Mh^2$

$I_c$ 转轴过质心的转动惯量

$I$ 转轴平行于$I_c$ 的转轴

$h$ 两条轴间的距离

# perpendicular-axis theorem 垂直轴定理

适用条件：薄片

$I_z=I_x+I_y$ $I_x,I_y$ 为两条薄片内互相垂直的轴的转动惯量 $I_z$ 为垂直于薄片过前两条轴交点的轴的转动惯量

# Angular Momentum 角动量

$L=I\omega$

\Sigma \tau=I\alpha=\frac{dL}

# 角动量守恒

$\Sigma \tau=0$

# Rotational Kinetic Energy 转动动能

$K=\Sigma\frac{1}{2}m_iv^2_{i}=\Sigma\frac{1}{2}m_i(\omega r_i)^2=\Sigma\frac{1}{2}m_ir_i^2\omega^2=\frac{1}{2}I\omega^2$

# 功 - 动能关系

$dW=\tau d\theta$

$W=\frac{1}{2}I\omega_f^2-\frac{1}{2}I\omega_i^2=\Delta K$

$P=\tau \omega$

# 滚动

法一 分解为 平动与转动

法二 $V_P=0$ P 点可以被视作瞬时的轴，此时运动为纯转动（无平动）

围绕 P 点转动的动能：$K=\frac{1}{2}I_P\omega^2$

平行轴定理：$I_P=I_{CM}+MR^2$

$\Rightarrow K=\frac{1}{2}I_{CM}\omega^2+\frac{1}{2}MV_{CM}^2$（两部分 围绕质心转动的动能 + 质心平动的动能）

(CM->center of mass)

# Rolling smoothly

纯滚动，无滑动（无摩擦力做功产生热能）

$v=\omega R$ 等关系式适用

# Chapter 12

# Oscillatory motion

# Simple Harmonic Motion

# Definition

$F=-kx$

Magnitude: The force acting on an object is proportional to the position of the object relative to some equilibrium position

Direction: Pointing to the equilibrium position

# Formula derivation

$F=ma=m\frac{d^2x}{dt^2}=-kx$

let $\omega=\sqrt{\frac{s}{m}}\Rightarrow \frac{d^2x}{dt^2}+\omega^2x=0$ Differential Equation of SHM

Solving: x(t)=Ae^

# Physical quantity

$x=Acos(\omega t+\phi)$

$v=-A\omega sin(\omega t+\phi)$ let $v_m=A\omega$

$a=-A\omega^2cos(\omega t+\phi)$ let $a_m=A\omega^2$

# Features

Period

Frequency

Angular Frequency

Phase difference: $\Delta\phi=\phi_2-\phi_1$

same phase versus opposite phase

Simple harmonic motion can be viewed as a projection of uniform circular motion on one dimension

# Simple Pendulum

# definition

Along the arc, $F=-mgsin\theta$

When $\theta$ is small ($\theta<15\degree$), F \approx -mg\theta=-mg\frac{x}

\omega=\frac{g}

# Physical Pendulum 复摆

# definition

A pendulum consisting of an actual object that is allowed to rotate freely around a horizontal axis.

Difference from Simple Pendulum:

• Simple pendulum typically consists of a weight suspended from a fixed point by a rod or string
• Physical Pendulum is a more complex system

# Formula derivation

$\tau=-F_ghsin\theta$ when $\theta$ is small, $\tau \approx -F_gh\theta=I\alpha$

let $\omega^2=\frac{F_gh}{I}\Rightarrow\frac{d^2\theta}{dt^2}+\omega^2\theta=0$

# Terminology

inertial 惯性

net force 净力，合力

uniform motion 匀速运动

free-body diagram 受力图