📚 A.4 Rigid Body Mechanics (HL only) | 刚体力学（仅高级课程）
In IB Physics HL, Topic A.4 explores the mechanics of rigid bodies—objects that do not deform under applied forces. This topic extends translational concepts to rotational motion, introducing torque, moment of inertia, angular momentum, and the conditions for equilibrium. Understanding rigid body mechanics is crucial for analysing everything from spinning tops to planetary motion.
在IB物理高级课程中，主题A.4探讨刚体力学——刚体是在外力作用下不发生形变的物体。这一主题将平动概念扩展到转动，引入了力矩、转动惯量、角动量和平衡条件。理解刚体力学对于分析从旋转陀螺到行星运动的各种现象至关重要。
1. Introduction to Rigid Bodies | 刚体简介
A rigid body is an idealisation where the distance between any two particles remains constant, regardless of external forces. This allows us to treat rotational motion separately from translational motion of the centre of mass. For a rigid body, we consider both its linear motion and its rotation about an axis.
刚体是一种理想化模型，其中任何两个质点之间的距离保持不变，无论外力如何。这使我们能够将转动与质心的平动分开处理。对于刚体，我们同时考虑其直线运动及其绕轴的转动。
The motion of a rigid body can be described by the translation of its centre of mass plus rotation about the centre of mass. This separation simplifies complex motion such as a rolling wheel.
刚体的运动可以描述为其质心的平动加上绕质心的转动。这种分离简化了例如滚轮等复杂运动。
2. Torque and Equilibrium | 力矩与平衡
Torque (τ) is the rotational analogue of force. It measures the tendency of a force to rotate an object about an axis. Mathematically, torque is the cross product of the position vector (r) and the force (F): τ = r × F. Its magnitude is τ = rF sin θ, where θ is the angle between r and F.
力矩（τ）是力的转动类比。它衡量力使物体绕轴转动的趋势。数学上，力矩是位置矢量（r）与力（F）的叉积：τ = r × F。其大小为 τ = rF sin θ，其中θ是r与F之间的夹角。
τ = r × F and τ = r F sin θ
ce must be zero (ΣF = 0) and the net external torque about any axis must be zero (Στ = 0).
刚体达到静力平衡必须满足两个条件：合外力为零（ΣF = 0）且对任意轴的合外力矩为零（Στ = 0）。
3. Moment of Inertia | 转动惯量
rᵢ². The greater the moment of inertia, the harder it is to change the rotational motion.
转动惯量（I）是质量的转动等价量。它取决于质量相对于转轴的分布。对于质点，I = mr²。对于质点系，I = Σ mᵢ rᵢ²。转动惯量越大，改变其转动状态越困难。
I = Σ mᵢ rᵢ²
lid cylinder about its central axis: I = ½ MR²; thin rod about its centre: I = (1/12) ML².
标准形状的常见转动惯量在数据手册中给出：例如，实心圆柱绕中心轴的转动惯量 I = ½ MR²；细杆绕中心的转动惯量 I = (1/12) ML²。
4. Angular Velocity and Acceleration | 角速度与角加速度
Δω/Δt. These quantities are analogous to linear displacement, velocity, and acceleration.
角位移（θ）以弧度为单位。角速度（ω）是角位移的变化率：ω = Δθ/Δt。角加速度（α）是角速度的变化率：α = Δω/Δt。这些量与线位移、线速度和线加速度类似。
ω = Δθ/Δt and α = Δω/Δt
kinematic equations mirror the linear ones: ω = ω₀ + αt, θ = ω₀t + ½ αt², ω² = ω₀² + 2αθ.
如果角加速度恒定，转动运动学方程与线性方程类似：ω = ω₀ + αt，θ = ω₀t + ½ αt²，ω² = ω₀² + 2αθ。
5. Rotational Kinetic Energy | 转动动能
A rotating rigid body possesses kinetic energy due to its rotation: Krot of a rolling body is the sum of translational and rotational parts: K = ½ mv_cm² + ½ I ω².
转动的刚体具有转动动能：Krot = ½ I ω²。这与平动动能½ mv²类似。滚动体的总动能是平动部分与转动部分之和：K = ½ mv_cm² + ½ I ω²。
Krot = ½ I ω²
For rolling without slipping, the condition vcm and angular velocity. This allows us to express kinetic energy purely in terms of ω or v.
对于无滑滚动，条件 vcm = ωR 关联线速度与角速度。这允许我们单用ω或v表达动能。
6. Angular Momentum | 角动量
a fixed axis, L = I ω. The direction of angular momentum is given by the right-hand rule.
角动量（L）是矢量，对质点定义为 L = r × p，其中p是线动量。对于绕固定轴转动的刚体，L = I ω。角动量的方向由右手定则确定。
L = I ω
Angular momentum is conserved if no external torque acts on the system. This principle has profound implications in physics, from spinning ice skaters to the formation of galaxies.
如果系统不受外力矩作用，角动量守恒。这一原理在物理中影响深远，从旋转的滑冰者到星系的形成。
7. Conservation of Angular Momentum | 角动量守恒
moment of inertia decreases, angular velocity increases proportionally, as seen when a skater pulls in their arms.
角动量守恒定律指出，孤立系统总角动量保持不变。因此，I₁ ω₁ = I₂ ω₂。如果转动惯量减小，角速度会成比例增大，如滑冰者收回手臂时所观察到的。
I₁ ω₁ = I₂ ω₂
This conservation is a direct consequence of rotational symmetry in space, as described by Noether's theorem, but at the IB level we treat it as an empirical law verified by countless experiments.
该守恒是空间旋转对称性的直接结果，由诺特定理描述，但在IB层面我们将其视为经无数实验验证的经验定律。
8. Newton's Second Law for Rotation | 转动中的牛顿第二定律
The rotational analogue of Newton' cting on a body equals the product of its moment of inertia and its angular acceleration. This equation is fundamental for analysing rotational dynamics.
牛顿第二定律的转动形式为 Στ = I α。作用在物体上的合力矩等于其转动惯量与角加速度的乘积。该方程是分析转动动力学的基础。
Στ = I α
When solving problems, it is often useful to draw a free-body diagram showing all forces, then compute torques about a chosen pivot. Remember that the torque from a force acting through the pivot is zero.
解题时，绘制显示所有力的隔离图常常很有帮助，然后计算相对于选定支点的力矩。注意力的作用线通过支点时力矩为零。
9. Work and Power in Rotational Motion | 转动中的功和功率
is P = τ ω. These relationships are valuable in analysing rotating machinery and engines.
力矩所做的功为 W = τ Δθ，类似于直线运动中的 W = F Δx。转动中的功率为 P = τ ω。这些关系在分析旋转机械和发动机时非常有用。
W = τ Δθ and P = τ ω
If a constant torque acts through an angular displacement, the work-energy theorem for rotation applies: Wnet = ΔKrot = ½ I ω_f² - ½ I ω_i².
若恒定力矩作用经过一段角位移，转动中的动能定理适用：Wnet = ΔKrot = ½ I ω_f² - ½ I ω_i²。
10. Translational and Rotational Analogies | 平动与转动类比
It is extremely helpful to compare linear and rotational quantities side by side. The deep symmetry between them simplifies memorisation and problem solving. The table below summarises the key analogies.
将线量和角量并排比较极有帮助。它们之间的深刻对称性简化了记忆和解题。下表总结了关键类比。
| Concept | Linear / Translational | Rotational |
|---|---|---|
| Displacement | x | θ |
| Velocity | v | ω |
| Acceleration | a | α |
| Inertia | m | I = Σmr² |
| Cause of motion | Force F | Torque τ = r × F |
| Newton's 2nd law | F = ma | Στ = I α |
| Kinetic energy | ½ mv² | ½ I ω² |
| Momentum | p = mv | L = I ω |
