📚 Circular Motion: Key Exam Points for IB & CCEA Physics | IB CCEA 物理：圆周运动 考点精讲
Circular motion is a fundamental topic in both the IB and CCEA A‑level Physics specifications, bridging kinematics, dynamics, and energy. A solid grasp of angular quantities, centripetal force, and the interplay between linear and rotational motion is essential for tackling exam questions on planetary orbits, banked curves, and vertical loops. This revision guide unpacks the key concepts, derivations, and common pitfalls, equipping you with the tools to approach any circular motion problem confidently.
圆周运动是 IB 和 CCEA A‑level 物理课程中的基础课题，它连接了运动学、动力学和能量三大模块。牢固掌握角量、向心力以及线运动与转动之间的关联，对解决行星轨道、弯道倾斜和竖直回环等考题至关重要。本复习指南将梳理核心概念、推导过程和常见误区，助你从容应对各类圆周运动问题。
1. Defining Circular Motion | 圆周运动的定义
An object moves in circular motion when its path is a circle of constant radius r. Even if the speed is constant, the velocity continuously changes direction, giving rise to an acceleration towards the centre — centripetal acceleration.
当物体的轨迹是一个半径为 r 的圆时，我们说它在做圆周运动。即使速率恒定，速度的方向也在持续改变，从而产生指向圆心的加速度——向心加速度。
The motion can be uniform (constant speed) or non‑uniform (changing speed). In both cases, there is always a radial component of acceleration. IB and CCEA exams frequently test your ability to distinguish between the two and to apply the correct equations.
运动可以是匀速的（速率不变）或非匀速的（速率改变）。两种情况下都存在径向加速度分量。IB 和 CCEA 考试常考查你区分两者的能力，并正确应用公式。
2. Angular Displacement, Velocity, and Acceleration | 角位移、角速度和角加速度
Angular displacement θ (measured in radians) is the angle swept out by the radius vector. One revolution equals 2π rad. The radian is the natural unit for rotational quantities because it makes the simple relationship s = rθ hold for arc length s.
角位移 θ (单位为弧度) 是半径矢量扫过的角度。一周等于 2π rad。弧度是转动量的自然单位，因为它保证弧长 s = rθ 的简洁关系成立。
Angular velocity ω is the rate of change of angular displacement: ω = Δθ / Δt. For uniform circular motion, ω is constant and related to the period T and frequency f by ω = 2π / T = 2πf.
角速度 ω 是角位移的变化率：ω = Δθ / Δt。对匀速圆周运动，ω 为常量，与周期 T 和频率 f 的关系为 ω = 2π / T = 2πf。
Angular acceleration α is the rate of change of angular velocity, α = Δω / Δt. It becomes relevant when the speed of rotation changes, such as in a spinning turntable speeding up.
角加速度 α 是角速度的变化率，α = Δω / Δt。当转动速率变化时（如转盘加速）就会涉及角加速度。
3. Linking Linear and Angular Quantities | 线量与角量的关联
The instantaneous linear velocity v of a point moving on a circle is tangent to the path and given by v = ω r. This relation is pivotal — it connects the angular description to the linear one and appears in almost every circular motion problem.
圆周上一点运动的瞬时线速度 v 沿切线方向，且满足 v = ω r。这一关系枢纽般地将角量描述与线量描述联系起来，几乎出现在每一道圆周运动题中。
The centripetal acceleration ac can be expressed in two equivalent forms using this link: ac = v² / r or ac = ω² r. Both are provided on the IB data booklet and CCEA formula sheet; exam success depends on choosing the right one based on the given information.
向心加速度 ac 可利用此关联表示成两种等价形式：ac = v² / r 或 ac = ω² r。两者都出现在 IB 数据手册和 CCEA 公式表中；考试成功取决于根据给定信息选用恰当的表达式。
v = ω r ac = v² / r = ω² r
4. Centripetal Force and Its Origin | 向心力及其来源
From Newton’s second law, any body experiencing centripetal acceleration must be subjected to a net force directed towards the centre: ΣF = m ac. This resultant force is called the centripetal force, Fc = m v² / r = m ω² r. It is not a new type of force but the name given to the net radial force causing circular motion.
根据牛顿第二定律，任何具有向心加速度的物体必然受到一个指向圆心的合外力：ΣF = m ac。这个合外力称为向心力，Fc = m v² / r = m ω² r。它不是一种新型的力，而是导致圆周运动的径向合力之名。
Typical sources of centripetal force include tension (mass on a string), gravitational attraction (orbiting satellite), friction (car rounding a bend), and the normal component of a reaction force (banked track or roller coaster loop). IB mark schemes award marks for correctly identifying the physical force providing the centripetal force in a given scenario.
向心力的典型来源包括：张力（绳上的物体）、万有引力（轨道卫星）、摩擦力（转弯的汽车）以及支持力的法向分量（倾斜轨道或过山车回环）。IB 评分方案会奖励能够正确指认具体情境中提供向心力的物理力的考生。
5. Deriving a = v² / r | 推导 a = v² / r
Both IB and CCEA syllabuses expect students to understand the derivation of centripetal acceleration for uniform circular motion. Consider a particle moving from point A to B in a short time Δt, sweeping an angle Δθ. The change in velocity Δv forms an isosceles triangle with sides v. For small Δθ, the magnitude |Δv| ≈ v Δθ. Since Δθ = ω Δt = (v/r) Δt, we have |Δv| ≈ (v² / r) Δt. The magnitude of acceleration a = |Δv| / Δt thus becomes a = v² / r.
IB 和 CCEA 大纲均要求学生理解匀速圆周运动向心加速度的推导。考虑质点从 A 点运动到 B 点经历短时间 Δt，扫过角度 Δθ。速度的改变量 Δv 与 v 构成等腰三角形。当 Δθ 很小时，|Δv| ≈ v Δθ。又因 Δθ = ω Δt = (v/r) Δt，可得 |Δv| ≈ (v² / r) Δt。因此加速度的大小 a = |Δv| / Δt 即为 a = v² / r。
The direction of Δv in the limit Δt→0 points towards the centre of the circle, making the acceleration centripetal. Knowing this derivation helps avoid common mistakes such as thinking that v²/r is the result of outward (centrifugal) effects.
当 Δt→0 时 Δv 的方向指向圆心，因此加速度为向心加速度。掌握该推导有助于避免常见错误，例如误认为 v²/r 源于向外（离心）效应。
6. Period, Frequency, and Rotational Equivalents | 周期、频率与转动对应量
Circular motion is inherently periodic. The period T is the time taken for one complete revolution; frequency f is the number of revolutions per second (unit: Hz). The relations are f = 1/T, ω = 2πf = 2π/T.
圆周运动本质上是周期性的。周期 T 是完成一整圈所需时间；频率 f 是每秒转动的圈数（单位：Hz）。关系为 f = 1/T，ω = 2πf = 2π/T。
For a satellite orbiting a planet, the period can be linked to the orbital radius using Kepler’s third law or by equating gravitational force to m ω² r. In the CCEA specification, students may be asked to relate T and r for a conical pendulum or a whirling bung on a string.
对于绕行星运转的卫星，周期可通过开普勒第三定律或通过令万有引力等于 m ω² r 与轨道半径关联。在 CCEA 大纲中，学生可能被要求推导锥摆或旋转橡皮塞的 T 与 r 关系。
7. Worked Examples: Horizontal Circular Motion | 实例分析：水平圆周运动
Example 1: Car on a flat curve. A car of mass 1200 kg travels at 15 m s⁻¹ around a bend of radius 50 m. Find the frictional force needed. Using Fc = m v² / r, we get F = 1200 × (15)² / 50 = 5400 N. If the coefficient of friction is too low, the car will skid outward, which is why speed limits on curves exist.
例 1：水平弯道上的汽车。一辆质量为 1200 kg 的汽车以 15 m s⁻¹ 的速度驶过半径为 50 m 的弯道，求所需摩擦力。利用 Fc = m v² / r，得到 F = 1200 × (15)² / 50 = 5400 N。若摩擦系数过低，汽车将向外侧滑，因此弯道处有速度限制。
Example 2: Conical pendulum. A small mass m swings in a horizontal circle at the end of a string of length L that makes an angle θ with the vertical. Resolving forces vertically: T cosθ = mg. Horizontally: T sinθ = m v² / (L sinθ). Eliminating T gives v = √(g L sinθ tanθ). The period T = 2π √(L cosθ / g) is independent of m but depends on θ. IB exams often ask for the period expression in terms of the vertical height h = L cosθ.
例 2：锥摆。质量为 m 的小球系在长度为 L 的绳端，绳与竖直方向夹角为 θ，小球在水平面内做圆周运动。竖直方向：T cosθ = mg；水平方向：T sinθ = m v² / (L sinθ)。消去 T 得到 v = √(g L sinθ tanθ)。周期 T = 2π √(L cosθ / g)，与 m 无关，但取决于 θ。IB 常常要求将周期用竖直高度 h = L cosθ 表达。
8. Banking of Curves | 弯道倾斜
When a road or track is banked at an angle θ to the horizontal, a component of the normal reaction helps provide the centripetal force, reducing reliance on friction. For an ideally banked curve where no friction is needed, the design speed v satisfies tanθ = v² / (r g). This formula is derived by resolving normal force N into vertical (N cosθ = mg) and horizontal (N sinθ = m v² / r) components.
当道路或轨道与水平面倾斜 θ 角时，支持力的一个分量协助提供向心力，降低对摩擦的依赖。对于无需摩擦的理想倾斜弯道，设计车速满足 tanθ = v² / (r g)。这一公式由将支持力 N 分解为竖直 (N cosθ = mg) 和水平 (N sinθ = m v² / r) 分量得出。
In IB Data Analysis questions, students may be asked to plot v² versus tanθ or determine g from the gradient of a v²–tanθ graph. The CCEA practical endorsement may involve a similar investigation using a turntable and a mass suspended at an angle.
在 IB 数据分析题中，学生可能需要绘制 v² 对 tanθ 的图像，或通过 v²–tanθ 图的斜率求重力加速度 g。CCEA 的实验考核可能涉及用转盘和倾斜悬挂的砝码进行类似探究。
9. Vertical Circular Motion & Energy | 竖直圆周运动与能量
Motion in a vertical circle is almost always non‑uniform because gravitational potential energy converts to kinetic energy and vice versa. While speed changes, the centripetal force requirement at any instant is still Fc = m v² / r, where v is the instantaneous speed.
竖直圆周运动几乎总是非匀速的，因为重力势能与动能相互转化。虽然速率不断改变，但任一时刻所需的向心力仍为 Fc = m v² / r，其中 v 为瞬时速率。
Consider an object whirled on a string in a vertical circle. At the top, both tension T and weight mg act downwards, so T + mg = m vtop² / r. The minimum speed to maintain a circular path occurs when T=0, giving vmin,top = √(g r). At the bottom, T – mg = m vbot² / r, so tension is greatest there. Energy conservation between top and bottom yields vbot² = vtop² + 4 g r.
考虑用绳在竖直面内旋转的物体。在最高点，张力 T 和重力 mg 均向下，故 T + mg = m vtop² / r。保持圆周路径的最低速率出现在 T=0 时，得 vmin,top = √(g r)。在最低点，T – mg = m vbot² / r，张力最大。利用从最高点到最低点的能量守恒可得 vbot² = vtop² + 4 g r。
A classic CCEA structured question provides measurements of tension at the bottom and asks to deduce the speed, or combines this with projectiles when the string breaks. IB often frames these within the context of roller coasters or loop‑the‑loop experiments.
CCEA 的一道经典结构化题会提供最低点张力测量值，要求学生推算速率，或将其与绳断后的抛体运动相结合。IB 则常将其放置于过山车或回环实验的情境中。
10. Non‑uniform Circular Motion | 非匀速圆周运动
When the speed changes, there is both a radial (centripetal) component ar = v² / r and a tangential component at = α r. The net acceleration is the vector sum of these two. This appears in topics such as a pendulum bob swinging through its lowest point, or a flywheel speeding up.
当速率改变时，同时存在径向（向心）分量 ar = v² / r 和切向分量 at = α r。合加速度是两者的矢量和。这在单摆摆锤经过最低点或飞轮加速等课题中出现。
The angular equations for constant angular acceleration mirror the linear SUVAT equations: ω = ω₀ + α t, θ = ω₀ t + ½ α t², ω² = ω₀² + 2 α θ. These are extremely useful for CCEA’s rotational dynamics problems involving torques and moment of inertia, and for IB Option B Engineering Physics.
匀角加速度的转动方程与线性的 SUVAT 方程对偶：ω = ω₀ + α t，θ = ω₀ t + ½ α t²，ω² = ω₀² + 2 α θ。它们在涉及力矩和转动惯量的 CCEA 转动动力学问题以及 IB 选修 B 工程物理中极为有用。
11. Common Misconceptions and Pitfalls | 常见误区与陷阱
A persistent misconception is that there is an outward ‘centrifugal’ force acting on a body in circular motion. In reality, the net force points inward; the sensation of being thrown outward is due to inertia, not a real force. IB multiple‑choice questions frequently set distractors around this idea.
一个顽固的误区是认为做圆周运动的物体受到一个向外的“离心力”。事实上，合外力指向圆心；“被向外抛”的感觉源自惯性，而非一个真实的力。IB 选择题常围绕这一概念设置干扰项。
Another common error is forgetting that the centripetal force is a resultant, not an additional force to be added to a free‑body diagram. Always identify the real forces (gravity, normal, tension, friction) and then set their radial component equal to m v² / r.
另一个常见错误是忘记向心力是一个合力，而不是应添入受力图的一个额外力。正确的做法是：先标出真实力（重力、支持力、张力、摩擦力），然后令其径向分量等于 m v² / r。
Mixing up radians and degrees in calculations involving ω or s = rθ leads to incorrect results. Remember that whenever an angle appears in radians, the formula is simplest. Exam scripts often show candidates using v = ω r with ω in °/s, which is wrong.
在涉及 ω 或 s = rθ 的计算中混淆弧度与度会导致错误结果。切记：只要角度以弧度出现，公式就是最简形式。阅卷常发现考生用 v = ω r 时 ω 单位用 °/s，这完全是错误的。
12. Exam Tips and How to Score High | 考试技巧与高分策略
Read the question carefully: identify whether the motion is uniform or not, and isolate the specific position if it is vertical circular motion. Draw a clear free‑body diagram with all forces and indicate the positive direction towards the centre.
仔细审题：先判定运动是匀速还是非匀速，若是竖直圆周运动则需锁定具体位置。画清晰的受力图，标出所有力，并指定指向圆心为正方向。
Write down the fundamental equation ΣFradial = m v² / r or m ω² r before substituting numbers. In IB exams, this step alone often earns a mark. For CCEA longer questions, showing clear substitutions and unit conversions (e.g., cm→m, rev/min→rad/s) is essential.
在代入数值前先写下基本方程 ΣFradial = m v² / r 或 m ω² r。在 IB 考试中，这一步本身通常就能得分。对于 CCEA 的较长问答题，展示清晰的代换和单位换算（如 cm→m，rev/min→rad/s）至关重要。
When analysing energy in vertical circles, combine the radial force equation with conservation of mechanical energy. Combine them algebraically before plugging in numbers to minimise rounding errors. If asked about the minimum speed at the top, set the required force (e.g., tension or reaction) to zero.
在处理竖直圆周运动的能量分析时，将径向力方程与机械能守恒结合使用。先在代数上合并，再代入数值，以减小舍入误差。若题目问及最高点的最小速率，将所需的力（如张力或支持力）设为零。
For graph‑based questions, use the linearised equations: e.g., v² versus r for a constant ω, or tanθ versus v² for a banked curve. The gradient and intercept yield physical quantities, a favourite investigation in both IB Internal Assessment and CCEA practical exams.
对于图像类问题，使用线性化的方程：例如，匀速转动时 v² 对 r 的图像，或对于倾斜弯道 tanθ 对 v² 的图像。斜率和截距能给出物理量，这是 IB 内部评估和 CCEA 实验考试都青睐的探究方式。
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