Circular Motion: Key Exam Points | 圆周运动考点精讲

📚 Circular Motion: Key Exam Points | 圆周运动考点精讲

Circular motion is a cornerstone topic in IB and OCR A-Level Physics, bridging kinematics, dynamics, gravitation, and even electromagnetism. Grasping the links between angular and linear quantities, understanding centripetal acceleration and force, and applying Newton’s laws in circular contexts are essential skills for top exam performance. This guide breaks down every critical concept, derivation, and common pitfall to help you master circular motion with confidence.

圆周运动是 IB 和 OCR A-Level 物理的核心主题，它连接了运动学、动力学、万有引力乃至电磁学。掌握角量与线量的关系、理解向心加速度和向心力、并能在圆周情境中熟练运用牛顿定律，是取得高分的必备技能。本文梳理了所有关键概念、推导过程和常见误区，助你自信攻克圆周运动。

1. Angular Displacement and the Radian | 角位移与弧度制

Angular displacement θ describes the angle through which an object moves on a circular path. It is measured in radians (rad), where one radian is the angle subtended at the centre of a circle by an arc length equal to the radius. For one complete revolution, θ = 2π rad = 360°.

角位移 θ 描述物体在圆周路径上转过的角度，单位为弧度（rad）。1 弧度是弧长等于半径时所对的圆心角。物体完成一整圈时，θ = 2π rad = 360°。

Using radians simplifies the relation between arc length s and radius r: s = r θ. This linear relationship is vital for connecting angular and linear kinematics.

采用弧度制可以简化弧长 s 与半径 r 的关系：s = r θ。这种线性关系是联系角运动量和线运动量的关键。

s = rθ (θ in radians)

s = rθ （θ 使用弧度）

2. Angular Velocity and Linear Velocity | 角速度与线速度

Angular velocity ω is the rate of change of angular displacement. For uniform circular motion, ω = Δθ / Δt = 2π / T, where T is the period. Its unit is rad s⁻¹.

角速度 ω 是角位移的变化率。对匀速圆周运动，ω = Δθ / Δt = 2π / T，其中 T 为周期。单位是 rad s⁻¹。

Linear (tangential) speed v is related to ω by v = ω r. Even though speed may be constant, the direction of velocity changes continuously, giving rise to acceleration.

线速度（切向速率）v 与 ω 的关系为 v = ω r。即使速率恒定，速度方向持续变化，必然产生加速度。

v = ω r

Always remember that in uniform circular motion the speed is constant but velocity is not. This is a classic exam trap.

务必牢记：匀速圆周运动中速率不变，但速度是变化的——这是经典考题陷阱。

3. Period, Frequency and Angular Speed | 周期、频率与角速度

Period T is the time taken for one complete revolution (unit: s). Frequency f is the number of revolutions per second (unit: Hz or s⁻¹). They are related by f = 1/T.

周期 T 是物体完成一整圈所需的时间（单位：s）。频率 f 是每秒转动的圈数（单位：Hz 或 s⁻¹）。两者关系为 f = 1/T。

Angular velocity can be expressed in terms of period or frequency: ω = 2π / T = 2π f. This gives an alternative expression for linear speed: v = 2π r / T = 2π r f.

角速度可用周期或频率表示：ω = 2π / T = 2π f。由此可得到线速度的另一种表达式：v = 2π r / T = 2π r f。

ω = 2π / T = 2π f

In exam questions, you are often given rpm (revolutions per minute). Convert to Hz by dividing by 60.

考试中常给出转速（rpm），记得除以 60 换算为 Hz。

4. Centripetal Acceleration | 向心加速度

Any object moving in a circle experiences an acceleration directed towards the centre, called the centripetal acceleration. Its magnitude is a = v² / r = ω² r. This acceleration changes the direction of velocity but not the speed (in uniform circular motion).

任何做圆周运动的物体都具有指向圆心的加速度，称为向心加速度。其大小为 a = v² / r = ω² r。该加速度仅改变速度方向，不改变速率（匀速圆周运动）。

a = v² / r = ω² r = 4π² r / T²

The direction of centripetal acceleration is always perpendicular to the velocity and points radially inward. When solving problems, always indicate this direction clearly.

向心加速度的方向始终与速度垂直并指向圆心。解题时务必清晰标示该方向。

5. Derivation of a = v² / r | 向心加速度公式推导

Consider an object moving at constant speed v along a circle of radius r. In a short time Δt, it moves from point A to B through angle Δθ. The velocity vectors at A and B have equal magnitudes but different directions. The change in velocity Δv points approximately toward the centre.

考虑物体以恒定速率 v 沿半径为 r 的圆周运动。在短时间 Δt 内，它从 A 运动到 B，转过角度 Δθ。A 点和 B 点的速度矢量大小相等但方向不同，速度变化量 Δv 近似指向圆心。

From the geometry of the velocity triangle, the magnitude Δv ≈ v Δθ (for small Δθ). The distance travelled is v Δt = r Δθ. Hence Δθ = v Δt / r. Substituting gives Δv ≈ v² Δt / r.

由速度矢量构成的几何三角形可得，Δv ≈ v Δθ（当 Δθ 很小时）。物体经过的弧长为 v Δt = r Δθ，因此 Δθ = v Δt / r。代入得 Δv ≈ v² Δt / r。

The magnitude of acceleration is a = Δv / Δt ≈ v² / r. In the limit Δt → 0, the approximation becomes exact, and the direction is radially inward.

加速度大小为 a = Δv / Δt ≈ v² / r。取 Δt → 0 的极限后，近似变为精确，方向沿径向向内。

a = v² / r

This derivation is frequently examined in IB and OCR. Understand the vector subtraction and the small-angle approximation.

这一推导在 IB 和 OCR 考试中经常出现。要理解矢量相减和小角度近似。

6. Centripetal Force and Newton’s Second Law | 向心力与牛顿第二定律

According to Newton’s second law, a net force is required to produce centripetal acceleration. This force is called the centripetal force, F = m a = m v² / r = m ω² r, and is always directed towards the centre.

根据牛顿第二定律，产生向心加速度需要净外力，这个力称为向心力，F = m a = m v² / r = m ω² r，方向始终指向圆心。

Centripetal force is not a new type of force; it is the resultant of real forces such as tension, gravity, friction, or the normal reaction. Always identify the physical force(s) providing the centripetal component.

向心力不是一种新类型的力，而是真实力（如张力、重力、摩擦力、法向反作用力）的合力。解题时务必指明提供向心力的实际施力物体。

F = m v² / r = m ω² r

Common mistake: adding a separate ‘centripetal force’ on free-body diagrams. Do not do this. Draw only real forces, and then equate their net radial component to m v² / r.

常见错误：在受力图中单独添加“向心力”。切勿如此，只画真实力，再将其径向合力设为 m v² / r。

7. Horizontal Circular Motion Examples | 水平圆周运动实例

Examples include a car rounding a flat curve, a mass on a string whirled horizontally, and a conical pendulum. In each case, resolve forces horizontally toward the centre and set the net radial force equal to m v² / r.

典型实例有汽车在水平弯道转弯、水平旋转的绳系小球、圆锥摆等。每种情况均需将力沿水平方向分解，令径向合力等于 m v² / r。

For a car on a flat curve, the centripetal force is provided by static friction: f = μₛ N = m v² / r. The maximum safe speed is v_max = √(μₛ g r).

汽车在水平弯道上时，向心力由静摩擦力提供：f = μₛ N = m v² / r。安全通过的最大速率为 v_max = √(μₛ g r)。

f_friction = m v² / r ⇒ μₛ m g = m v² / r

f_摩擦 = m v² / r ⇒ μₛ m g = m v² / r

In a conical pendulum, the string traces a cone. The vertical component of tension balances weight, while the horizontal component provides centripetal force: T sinθ = m v² / r and T cosθ = m g.

圆锥摆中，细绳扫出圆锥面，张力的竖直分量平衡重力，水平分量提供向心力：T sinθ = m v² / r，T cosθ = m g。

8. Banked Tracks | 斜面弯道

To reduce reliance on friction, roads and railway tracks are banked at an angle θ. For a given speed, the horizontal component of the normal reaction can supply the needed centripetal force without friction.

为减少对摩擦的依赖，公路和铁路弯道会设置倾角 θ。对某一特定车速，法向反作用力的水平分量可恰好提供所需向心力，无需摩擦力。

Resolving the normal reaction N, we have N sinθ = m v² / r and N cosθ = m g. Dividing gives tanθ = v² / (r g). This is the ideal banking equation.

分解支持力 N 得 N sinθ = m v² / r，N cosθ = m g。两式相除得 tanθ = v² / (r g)，即为理想斜面倾角方程。

tanθ = v² / (r g)

If the speed exceeds the ideal value, friction acts down the slope to prevent skidding outward; if slower, friction acts up the slope. Exam questions often ask for the direction of friction.

车速大于理想值时，摩擦力沿斜面向下防止向外侧滑；车速较小时，摩擦力沿斜面向上防止向内滑动。考试常要求判断摩擦力方向。

9. Vertical Circular Motion | 竖直平面圆周运动

When an object moves in a vertical circle (e.g., a bucket of water, a roller coaster loop), speed is not constant because gravity does work. At the top and bottom, the net radial force still equals m v² / r, but v varies.

物体在竖直平面内做圆周运动（如水桶、过山车回环）时，由于重力做功，速率并不恒定。在最高点和最低点，径向合力仍然等于 m v² / r，但 v 随位置变化。

At the top, the net force towards the centre is mg + N = m v_top² / r. For the object to just maintain contact, N ≥ 0, giving the critical speed v_top,min = √(g r).

在最高点，指向圆心的合力为 mg + N = m v_top² / r。若恰好仍与轨道接触，则 N ≥ 0，得出临界速率 v_top,min = √(g r)。

At the bottom, N – mg = m v_bot² / r, so apparent weight (normal force) is greater than mg. Energy conservation links speeds at different points: ½ m v_bot² = ½ m v_top² + 2 m g r.

在最低点，N – mg = m v_bot² / r，因此视重（支持力）大于 mg。能量守恒可联系不同位置的速率：½ m v_bot² = ½ m v_top² + 2 m g r。

Always draw free-body diagrams for top and bottom separately, and apply Newton’s second law in the radial direction.

务必分别画出最高点和最低点的受力图，并沿径向应用牛顿第二定律。

10. Conical Pendulum and the Period Expression | 圆锥摆与周期表达式

For a conical pendulum, combining T sinθ = m ω² r and T cosθ = m g, with r = L sinθ (L is string length), yields ω = √(g / (L cosθ)). The period is T_period = 2π √(L cosθ / g).

对圆锥摆，由 T sinθ = m ω² r 与 T cosθ = m g，结合 r = L sinθ（L 为绳长），可解得 ω = √(g / (L cosθ))。周期为 T_period = 2π √(L cosθ / g)。

ω = √(g / (L cosθ))

This is analogous to the simple pendulum but with an extra cosθ factor. Observe that as θ increases, period decreases, and the height h = L cosθ stays constant if ω is fixed.

这与单摆相似，但多了一个 cosθ 因子。可观察到随着 θ 增大，周期减小；当 ω 固定时，高度 h = L cosθ 保持不变。

11. Non-uniform Circular Motion | 非匀速圆周运动

If the speed along the circular path changes, there is both a radial (centripetal) acceleration aᵣ = v² / r and a tangential acceleration aₜ = Δv / Δt or aₜ = r α, where α = Δω / Δt is the angular acceleration.

若沿圆周的速率发生变化，则既有径向（向心）加速度 aᵣ = v² / r，又有切向加速度 aₜ = Δv / Δt 或 aₜ = r α，其中 α = Δω / Δt 为角加速度。

The total acceleration is the vector sum: a = √(aᵣ² + aₜ²). Net force must provide both components: radial force equals m v² / r, tangential force equals m aₜ.

总加速度为两者的矢量和：a = √(aᵣ² + aₜ²)。合外力必须同时提供这两个分量：径向力等于 m v² / r，切向力等于 m aₜ。

a = √(aᵣ² + aₜ²) where aᵣ = v²/r, aₜ = r α

a = √(aᵣ² + aₜ²) ，其中 aᵣ = v²/r, aₜ = r α

Non-uniform circular motion occurs in vertical loops and whenever a tangential force (e.g., tension with varying speed, descending a curved ramp) exists. Energy methods combined with radial dynamics are often needed.

非匀速圆周运动出现在竖直回环，以及任何存在切向力（如变速率张力、沿弯曲轨道下滑）的场合。通常需要将能量法与径向动力学相结合。

12. Common Pitfalls and Exam Tips | 常见错误与应试技巧

Misconception 1: ‘Centrifugal force pushes objects outward.’ In an inertial frame, there is no centrifugal force. The sensation of being thrown outward is due to inertia and the need for a centripetal force to change direction. Only introduce centrifugal force if working in a rotating reference frame, and then treat it as a fictitious force.

误区一：“离心力将物体向外推”。在惯性系中并不存在离心力。感觉被向外甩是因为惯性，以及需要向心力来改变方向。除非在转动参考系中分析，才可引入离心力作为假想力。

Misconception 2: ‘Velocity is constant in uniform circular motion.’ Speed is constant, velocity is not because direction changes.

误区二：“匀速圆周运动中速度不变。”速率不变，但速度矢量因方向变化而改变。

Misconception 3: ‘Centripetal force is an extra force.’ It is just the name for the net radial force. Never add it on top of tension, gravity, etc.

误区三：“向心力是一个额外的力。”它只是径向合力的别名。切勿在张力、重力等真实力之上再添加一个向心力。

Exam tip: Always start with a clear free-body diagram, choose a radial axis, and write F_net (towards centre) = m v² / r. Use energy conservation only when speed changes.

应试技巧：先画清晰的受力图，选定径向坐标轴，列出 F_合力（指向圆心）= m v² / r。仅当速率变化时才使用能量守恒。

When given angular data in degrees or revolutions, convert to radians or seconds immediately. Check units: for ω rad s⁻¹, for v m s⁻¹.

题目给出角度或转数时，立刻转换为弧度或秒。检查单位：ω 用 rad s⁻¹，v 用 m s⁻¹。

Practice deriving a = v² / r from vector diagrams, as this can appear in structured questions. Understand the limit process and small-angle approximation.

练习从矢量图推导 a = v² / r，这常在结构化问题中出现。理解极限过程和小角度近似。

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Circular Motion: Key Exam Points | 圆周运动考点精讲