📚 AP Physics B: Translational and Rotational Motion Knowledge System | AP物理B平动与转动知识体系梳理
In AP Physics B, motion can be separated into two fundamental types: translational motion, where an object moves through space without spinning, and rotational motion, where an object spins about a fixed axis. Although they appear different, every translational concept has a rotational counterpart, linked by radius or geometry. Mastering this duality is key to solving problems involving pulleys, rolling objects, collisions with rotation, and static equilibrium. This guide provides a systematic overview of the core principles, equations, and problem-solving strategies for both types of motion.
在AP物理B中，运动可分为两大基本类型：平动（物体在空间中移动而不旋转）和转动（物体绕固定轴旋转）。虽然看起来不同，但每一个平动概念都有对应的转动量，通过半径或几何关系联系起来。掌握这种对偶性是解决滑轮、滚动、含转动的碰撞以及静态平衡问题的关键。本指南系统梳理平动与转动的核心原理、方程和解题策略。
1. Translational vs Rotational Motion: Key Quantities | 平动与转动：关键物理量
The following table lists the fundamental quantities that describe translational motion and their direct rotational analogs. Note how each angular quantity is obtained by dividing the linear displacement along a circular arc by the radius r, or by multiplying the linear quantity by r. This mapping is essential for understanding rolling constraints and converting between the two frameworks.
下表列出了描述平动的基本物理量及其对应的转动量。注意每个角量要么是沿圆弧的线位移除以半径 r，要么是线量乘以 r。这种映射对于理解纯滚动约束及在两种框架间转换至关重要。
| Translational Quantity | Symbol | Rotational Analog | Symbol | Relation |
|---|---|---|---|---|
| Displacement | s | Angular displacement | θ | s = rθ |
| Velocity | v | Angular velocity | ω | v = rω |
| Acceleration | a | Angular acceleration | α | a = rα (tangential) |
| Mass | m | Moment of inertia | I | I = Σ m r² |
| Force | F | Torque | τ | τ = r F sin φ |
Angular displacement θ is measured in radians (rad), angular velocity ω in rad/s, and angular acceleration α in rad/s². Radian is the standard unit because it makes the arc-length relationship s = rθ exact. Always convert revolutions or degrees to radians when using rotational formulas.
角位移 θ 用弧度 (rad) 度量，角速度 ω 用 rad/s，角加速度 α 用 rad/s²。弧度是标准单位，因为它是弧长关系 s = rθ 精确成立的前提。使用转动公式时务必将圈数或度数转换为弧度。
2. Kinematics Equations: Constant Acceleration | 运动学方程：恒定加速度
For constant linear acceleration, we have four kinematic equations. Their rotational twins are obtained by substituting each linear variable with its angular counterpart. This parallelism holds only when angular acceleration α is constant, just as the linear equations require constant a.
对于恒定线加速度，我们有四个运动学方程。将其中的每个线变量替换为对应的角量，就得到了转动方程。这种平行关系仅在角加速度 α 恒定时成立，就像线方程要求 a 恒定一样。
v = u + a t → ω = ω₀ + α t
s = u t + ½ a t² → θ = ω₀ t + ½ α t²
s = ½ (u + v) t → θ = ½ (ω₀ + ω) t
v² = u² + 2 a s → ω² = ω₀² + 2 α θ
In these equations, u and ω₀ are initial velocities, v and ω are final velocities, t is time, s is linear displacement, and θ is angular displacement. Note that the angular equations assume the initial angular velocity is ω₀ and the final is ω after time t. These formulas allow you to solve for any unknown when three of the five quantities are given.
在这些方程中，u 和 ω₀ 是初速度，v 和 ω 是末速度，t 是时间，s 是线位移，θ 是角位移。注意角运动方程假定初角速度为 ω₀，经过时间 t 后的末角速度为 ω。当五个量中已知三个时，就可以用这些公式求解未知量。
3. Dynamics: Force, Mass, and Acceleration vs Torque, Inertia, and Angular Acceleration | 动力学：力、质量与加速度 vs 力矩、惯量与角加速度
Newton’s second law F = m a is the cornerstone of translational dynamics. Its rotational equivalent is τ = I α, where τ is the net external torque about a fixed axis, I is the moment of inertia (rotational mass), and α is the angular acceleration. Torque causes angular acceleration just as force causes linear acceleration. Torque is a vector quantity defined by τ = r × F, with magnitude τ = r F sinΦ, where Φ is the angle between the radius vector and the force.
牛顿第二定律 F = m a 是平动动力学的基石。其转动对应形式为 τ = I α，其中 τ 是绕定轴的合外力矩，I 是转动惯量（旋转质量），α 是角加速度。力矩引起角加速度，正如力引起线加速度。力矩是矢量，定义为 τ = r × F，大小为 τ = r F sinΦ，其中 Φ 是径向矢量与力之间的夹角。
Moment of inertia I measures how mass is distributed relative to the rotation axis. Objects with mass farther from the axis have larger I, and thus require more torque to achieve the same angular acceleration. The parallel axis theorem, I = I_cm + M d², lets you find I about any axis parallel to an axis through the center of mass.
转动惯量 I 衡量质量相对于转轴的分布。质量离轴越远，I 越大，因此需要更大的力矩才能产生相同的角加速度。平行轴定理 I = I_cm + M d² 可用来计算绕任意平行于质心轴的转动惯量。
4. Moment of Inertia: Mass Distribution and Rotational Inertia | 转动惯量：质量分布与旋转惯性
For common homogeneous shapes, the moment of inertia about a symmetry axis can be looked up or derived. Below are some standard results often tested in AP Physics B. All assume uniform density and rotation about the axis indicated.
对于常见的均质形状，绕对称轴的转动惯量可以查表或导出。以下是一些AP物理B常考的标准结果，均假设密度均匀且绕所指轴旋转。
| Shape | Axis | Moment of Inertia I |
|---|---|---|
| Point mass m at distance r | Through axis perpendicular to radius | m r² |
| Thin rod, length L, mass M | Through center, perpendicular to rod | (1/12) M L² |
| Thin rod, length L, mass M | Through end, perpendicular to rod | (1/3) M L² |
| Solid cylinder/disk, radius R | Central axis of cylinder | ½ M R² |
| Hollow cylinder/hoop, radius R | Central axis | M R² |
| Solid sphere, radius R | Through center | (2/5) M R² |
| Hollow thin spherical shell | Through center | (2/3) M R² |
Note that a hoop has the largest I for a given mass and radius because all its mass sits at the maximum distance R. A solid sphere, with mass concentrated closer to the center, has a smaller I. This explains why a solid sphere rolls down an incline faster than a hoop of the same mass and radius.
注意，给定质量和半径下，圆环的 I 最大，因为其所有质量都位于最大距离 R 处。实心球的质量更集中于中心，故 I 较小。这就解释了为什么相同质量、相同半径的实心球比圆环更快滚下斜面。
5. Work, Kinetic Energy, and Power | 功、动能与功率
Translational kinetic energy K_trans = ½ m v². Rotational kinetic energy K_rot = ½ I ω². A rigid body that is both translating and rotating (like a rolling ball) has total kinetic energy K = ½ m v² + ½ I ω². Work done by a constant force F over displacement d is W = F d cosθ; work done by a constant torque τ through an angular displacement θ is W = τ θ. The work–energy theorem holds for both: W_net = ΔK.
平动动能 K_trans = ½ m v²。转动动能 K_rot = ½ I ω²。同时平动和转动的刚体（如滚动的球）的总动能为 K = ½ m v² + ½ I ω²。恒力 F 在位移 d 上做的功为 W = F d cosθ；恒力矩 τ 经过角位移 θ 做的功为 W = τ θ。功能定理对两者都成立：W_net = ΔK。
Power is the rate of doing work. For translational motion, P = F v (if force and velocity are parallel) and for rotational motion, P = τ ω. In problems involving a rolling object descending a ramp, gravitational potential energy mgh is converted into both translational and rotational kinetic energy, assuming no slipping and no non-conservative forces.
功率是做功的速率。平动中 P = F v（力与速度平行时），转动中 P = τ ω。在物体沿斜面无滑动滚下的问题中，重力势能 mgh 同时转换为平动动能和转动动能，假设无滑动且无非保守力。
6. Impulse and Momentum vs Angular Impulse and Angular Momentum | 冲量与动量 vs 角冲量与角动量
Linear momentum p = m v is a vector; its change is caused by impulse J = ∫ F dt. For constant force, J = F Δt = Δp. Rotational analog: angular momentum L = I ω (for a rigid body about a fixed axis) or L = r × p for a particle. The angular impulse is ∫ τ dt, and for a constant torque, τ Δt = ΔL.
线动量 p = m v 是矢量；其变化由冲量 J = ∫ F dt 引起。恒力情况下 J = F Δt = Δp。转动对应量：刚体绕定轴的角动量 L = I ω，或对质点 L = r × p。角冲量为 ∫ τ dt，恒力矩时 τ Δt = ΔL。
Angular momentum is conserved if the net external torque on a system is zero. This is analogous to conservation of linear momentum when net external force is zero. Classic examples include a spinning ice skater pulling in her arms: I decreases, so ω increases to keep L constant. In planetary orbits, L = m v r sinφ is conserved, leading to Kepler’s second law.
如果系统所受合外力矩为零，则角动量守恒。这类似于合外力为零时线动量守恒。经典例子包括花样滑冰运动员收臂旋转：I 减小，ω 增大以保持 L 不变。在行星轨道中，L = m v r sinφ 守恒，导致开普勒第二定律。
7. Conservation Laws: Energy, Momentum, and Angular Momentum | 守恒定律：能量、动量与角动量
In AP Physics B, you will frequently combine these conservation principles. For an isolated system with no net external force, total linear momentum is conserved. For an isolated system with no net external torque, total angular momentum is conserved. Mechanical energy is conserved when only conservative forces (gravity, spring force) do work, but be careful to include rotational kinetic energy when rotation is present.
在AP物理B中，你会经常联合使用这些守恒原理。无合外力的孤立系统，总线动量守恒。无合外力矩的孤立系统，总角动量守恒。当只有保守力（重力、弹力）做功时机械能守恒，但注意当存在转动时须包含转动动能。
A common collision problem: a bullet embeds itself into a rod fixed at one end. Immediately after collision, angular momentum about the pivot is conserved, but linear momentum is not because the pivot exerts an external force. After the collision, energy conservation (or work–energy) determines the swing height. Always check which conservation law applies by identifying external forces and torques.
常见碰撞问题：子弹嵌入一端固定的杆中。碰撞瞬间，绕转轴的角动量守恒，但线动量不守恒，因为轴施加了外力。碰撞后，能量守恒（或功能关系）决定摆起高度。永远通过识别外力和外力矩来检验哪个守恒定律适用。
8. Rolling Without Slipping | 纯滚动
Rolling without slipping is a key motion where a round object rotates as it moves, and the point of contact with the surface is instantaneously at rest. The no-slip condition links translation and rotation: the translational velocity of the center of mass v_cm and the angular velocity ω about the center satisfy v_cm = R ω, where R is the radius. Similarly, tangential acceleration a_cm = R α.
纯滚动是一种关键运动，圆形物体边移动边旋转，与地面接触点瞬时静止。不滑动条件将平动与转动联系起来：质心平动速度 v_cm 与绕质心的角速度 ω 满足 v_cm = R ω，其中 R 是半径。同样地，切向加速度 a_cm = R α。
The total kinetic energy of a rolling object is K = ½ m v_cm² + ½ I_cm ω². Using the no-slip condition, this can be expressed entirely in terms of v_cm or ω. For an object rolling down an incline of height h, energy conservation gives mgh = ½ m v² + ½ I ω². Substituting I = β m R² (where β is a shape factor: 1 for hoop, 1/2 for disk, 2/5 for sphere) and ω = v/R yields v_cm = √(2gh / (1+β)). This shows that smaller β gives greater final speed.
滚动体的总动能为 K = ½ m v_cm² + ½ I_cm ω²。利用不滑动条件，可全部用 v_cm 或 ω 表达。对于沿高度为 h 的斜面滚下的物体，能量守恒给出 mgh = ½ m v² + ½ I ω²。代入 I = β m R²（β 是形状系数：圆环 1，圆盘 1/2，球体 2/5）和 ω = v/R，得 v_cm = √(2gh / (1+β))。这表明 β 越小，末速度越大。
9. Static Equilibrium and Torque | 静态平衡与力矩
An object in static equilibrium has zero net force and zero net torque about any axis. The conditions are Σ F_x = 0, Σ F_y = 0, and Σ τ = 0. Torques must be computed with a consistent sign convention (e.g., clockwise negative, counterclockwise positive). You can choose any pivot point to sum torques; a clever choice (e.g., at the point of an unknown force) can eliminate that unknown from the torque equation.
处于静态平衡的物体，其合外力为零且绕任意轴的合外力矩为零。条件为 Σ F_x = 0，Σ F_y = 0 和 Σ τ = 0。计算力矩时须规定一致的符号规则（如顺时针为负、逆时针为正）。可选择任意转轴求力矩和；巧选支点（如某未知力作用点）可使该未知力在力矩方程中不出现。
Typical problems involve a beam supported by a rope and a hinge, or a ladder leaning against a wall. Draw an extended free-body diagram showing all forces at their points of application. The weight acts at the center of mass. Write torque equations about a strategically chosen pivot to solve for unknowns like tension or hinge force components.
典型问题包括由绳子和铰链支撑的梁，或斜靠在墙上的梯子。画出扩展受力图，标出所有力的作用点。重力作用在质心。绕策略性选择的支点写力矩方程，以求解未知量，如绳中张力或铰链力的分力。
10. Problem-Solving Strategies and Common Pitfalls | 解题策略与常见误区
Mastering translational and rotational motion requires a systematic approach. Here are key strategies:
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Identify the type of motion: pure translation, pure rotation, rolling without slipping, or general translation+rotation. Write down the constraints (e.g., v = Rω for rolling).
确定运动类型：纯平动、纯转动、纯滚动或一般平动加转动。写下约束条件（如滚动时 v = Rω）。
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Draw clear free-body diagrams and extended free-body diagrams showing where forces act. For rotation, include torques about a chosen axis.
画出清晰的受力图和扩展受力图，标明力作用位置。对转动，选取转轴标明力矩。
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Use conservation laws when applicable: energy (if only conservative forces do work), momentum (if net external force is zero), angular momentum (if net external torque is zero). Check for external impulses or external torques during collisions.
适用时使用守恒定律：能量（仅保守力做功时）、动量（合外力为零时）、角动量（合外力矩为零时）。碰撞过程中注意检查外冲量或外力矩。
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Convert all angular quantities to radians. Ensure s = rθ and v = rω are applied correctly; these hold only when θ is in radians and ω in rad/s.
将所有角量转换为弧度。确保 s = rθ 和 v = rω 正确应用；这仅在 θ 用弧度、ω 用 rad/s 时成立。
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Be mindful of the direction of torque and angular momentum: use the right-hand rule to determine vector directions. In problems where rotation direction matters (e.g., pulley accelerations), assign consistent positive directions for both translation and rotation.
注意力矩和角动量的方向：使用右手定则确定矢量方向。当转动方向关键时（如滑轮加速度），为平动和转动规定一致的正方向。
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Common pitfalls: forgetting that static friction can do no work in rolling without slipping; confusing tangential acceleration a_t = rα with centripetal acceleration a_c = v²/r; using moment of inertia about the wrong axis; and failing to include rotational kinetic energy in energy conservation for rolling bodies.
常见误区：忘记纯滚动中静摩擦不做功；混淆切向加速度 a_t = rα 与向心加速度 a_c = v²/r；绕错误轴使用转动惯量；对滚动体应用能量守恒时遗漏转动动能。
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