📚 Circular Motion Exam Essentials for AQA A-Level Maths | A-Level AQA 数学：圆周运动 考点精讲
Circular motion is a key topic in AQA A-Level Mathematics, appearing within the mechanics strand – typically in the second year or in Further Mathematics modules. It extends the ideas of kinematics and forces to objects moving along a circular path, requiring a solid grasp of vectors, trigonometry, and Newton’s laws. Examiners test your ability to model horizontal and vertical circles, conical pendulums, and banked tracks, often combining these with energy methods. This article unpacks the core principles, essential formulas, and common problem types you must master to succeed.
圆周运动是 AQA A-Level 数学力学部分的核心课题，通常在第二年或进阶数学模块中出现。它将运动学和受力分析拓展到沿圆形轨迹运动的物体，要求扎实掌握向量、三角知识和牛顿定律。考官会重点考查你对水平圆周、竖直圆周、锥摆和倾斜轨道等模型的建模能力，并经常结合能量方法出题。本文梳理了必须掌握的核心原理、关键公式和常见题型，助你高效备考。
1. Angular Displacement and Angular Velocity | 角位移与角速度
When an object moves along a circular path, we describe its position using the angle θ (in radians) swept out from a fixed radius. The arc length s travelled along the circumference is given by s = rθ, where r is the radius of the circle. Radian measure is essential because it connects arc length and radius directly, and all standard formulas in circular motion rely on radians.
当物体沿圆形轨迹运动时，我们用从某一固定半径开始扫过的角度 θ（单位为弧度）来描述其位置。沿圆周走过的弧长 s 由公式 s = rθ 给出，其中 r 是圆的半径。弧度制至关重要，因为它将弧长与半径直接联系起来，圆周运动的所有标准公式都依赖弧度。
Angular velocity ω (omega) measures the rate of change of angular displacement: ω = dθ/dt. For uniform circular motion, ω is constant and related to the period T (time for one complete revolution) and frequency f by ω = 2π/T = 2πf. The units are rad s⁻¹.
角速度 ω 衡量角位移的变化率：ω = dθ/dt。对于匀速圆周运动，ω 恒定，并与周期 T（完成一整圈所需时间）和频率 f 的关系为 ω = 2π/T = 2πf。单位为弧度每秒（rad s⁻¹）。
2. Relating Linear and Angular Quantities | 线量与角量的关系
The linear speed v of a particle moving around a circle of radius r is directly linked to the angular velocity by the formula v = rω. This arises from differentiating s = rθ with respect to time, since r is constant. The direction of the velocity vector is always tangential to the circle, changing continuously even if the speed is constant.
绕半径为 r 的圆运动的质点，其线速率 v 与角速度直接相关，公式为 v = rω。这是通过对 s = rθ 关于时间求导得出的，因为 r 为常量。速度矢量的方向始终沿圆的切线方向，即使速率恒定，方向也在不断变化。
It is crucial to remember that for a given angular velocity, linear speed increases with radius. In many exam problems, you will need to convert between angular and linear quantities, especially when dealing with pulleys, wheels, or points on a rotating body.
务必牢记，在角速度一定时，线速率随半径增大而增大。在许多考题中，你需要在线量与角量之间进行转换，尤其是在处理滑轮、轮子或旋转体上的点时。
3. Centripetal Acceleration and Force | 向心加速度与向心力
An object moving in a circle experiences an acceleration directed towards the centre, known as centripetal acceleration. Its magnitude is given by a = v²/r or, using v = rω, a = rω². Although the speed may be uniform, the velocity vector is changing direction; this acceleration is responsible for that change and always acts radially inward.
做圆周运动的物体具有指向圆心的加速度，称为向心加速度。其大小为 a = v²/r，或利用 v = rω 可得 a = rω²。虽然速率可能是均匀的，但速度矢量在改变方向；此加速度就是引起这种变化的原因，且始终沿半径指向圆心。
By Newton’s second law, there must be a net force acting towards the centre, the centripetal force: F = ma = mv²/r = mrω². This is not a new type of force but the resultant of actual forces (tension, gravity, friction, normal reaction) pointing towards the centre.
根据牛顿第二定律，必须存在指向圆心的净力，即向心力：F = ma = mv²/r = mrω²。这不是一种新的力，而是真实存在的力（张力、重力、摩擦力、法向反力）在指向圆心方向上的合力。
Many students mistakenly treat ‘centrifugal force’ as a real force acting outward; in AQA mechanics, frame of reference is inertial, so only centripetal force towards the centre is considered. Always resolve forces radially to find the net inward force and equate it to mv²/r or mrω².
许多同学误将“离心力”当作真实的向外作用的力；在 AQA 力学中，参考系是惯性系，故只考虑指向圆心的向心力。务必沿径向分解力，求出向内的净力，并令其等于 mv²/r 或 mrω²。
4. Key Formulas and Units | 关键公式与单位
Below is a summary of the most important equations you need to memorise. They form the backbone of almost every circular motion question.
以下是你需要记忆的最重要的公式汇总，它们构成了几乎每道圆周运动题目的主干。
| Quantity | Formula(s) | Units |
| Angular displacement θ | θ = s/r | rad (dimensionless) |
| Angular velocity ω | ω = θ/t, ω = 2π/T, ω = 2πf | rad s⁻¹ |
| Linear speed v | v = rω | m s⁻¹ |
| Centripetal acceleration a | a = v²/r = rω² | m s⁻² |
| Centripetal force F | F = mv²/r = mrω² | N |
| Conical pendulum: T cosθ = mg; T sinθ = mrω² | tanθ = rω²/g; r = L sinθ | — |
Always ensure θ is in radians when using these formulas, except where degrees are specifically required in geometric reasoning. In calculations, set your calculator to radian mode.
使用这些公式时，务必确保 θ 用弧度表示，除非在几何推理中明确要求使用角度。计算时，请将计算器设置为弧度模式。
5. Horizontal Circular Motion | 水平圆周运动
In horizontal circular motion, the circle lies in a horizontal plane. A common scenario is a particle attached to a string moving in a horizontal circle, or a car cornering on a flat track. The weight acts vertically downward and is balanced by the vertical component of another force (e.g., tension or normal reaction), while the horizontal component of the forces provides the centripetal force.
在水平圆周运动中，圆处于水平面内。常见情景有：系在绳上的质点在水平面内做圆周运动，或汽车在平坦路面上转弯。重力竖直向下，被其他力的竖直分量（如张力或法向反力）平衡，而力的水平分量提供向心力。
For a particle on a smooth table connected to a string through a hole, the tension in the string is the only horizontal force and equals mrω². If hanging mass provides tension, then T = Mg, leading to Mg = mrω². Such set-ups allow you to relate the angular speed to the hanging mass and geometry.
对于放在光滑桌面上的质点，通过小孔用绳连接，绳的张力是唯一的水平力，且等于 mrω²。若悬挂的重物提供张力，则 T = Mg，从而得出 Mg = mrω²。此类装置能将角速度与悬挂质量和几何参数联系起来。
In car cornering on a flat road, friction between tyres and road supplies the centripetal force: f = mv²/r. The maximum frictional force is μmg, so the maximum speed without slipping is v = √(μgr).
在平坦路面上汽车转弯时，轮胎与路面的摩擦力提供向心力：f = mv²/r。最大静摩擦力为 μmg，因此不侧滑的最大车速为 v = √(μgr)。
6. Vertical Circular Motion – General | 竖直圆周运动总述
When a particle moves in a vertical circle, both the speed and the direction of the net force vary with position. Gravity now plays an active role in changing the speed: the particle speeds up as it descends and slows down as it ascends, unless external forces maintain constant speed. The centripetal force required at any point is still mv²/r, but v differs at different points.
当质点在竖直面内做圆周运动时，速度和净力的方向都随位置变化。重力此时积极参与改变速率：质点下降时加速，上升时减速，除非有外力维持恒定速率。任一点所需的向心力依然为 mv²/r，但不同位置的 v 不同。
The general approach is to draw a clear free-body diagram at the position of interest, resolve forces radially towards the centre, and set the net inward force equal to mv²/r. Tangential forces cause the change in speed and can be analysed using energy conservation or the work-energy principle.
一般方法是：在感兴趣的位置画出清晰的受力图，沿径向指向圆心分解力，令净向内力等于 mv²/r。切向力引起速率变化，可通过能量守恒或功能原理进行分析。
Vertical circle problems often ask for the minimum speed at the highest point for a particle to complete a full circle, or the tension in the string at the top and bottom positions.
竖直圆周问题常会问：质点要完成完整圆周运动，在最高点的最小速度是多少；或求最高点和最低点时绳的张力。
7. Vertical Circular Motion – Top and Bottom Tension | 竖直圆周运动顶端与底端张力
Consider a particle of mass m attached to a light inextensible string of length r, moving in a vertical circle. At the top of the circle, both weight mg and tension Ttop act downward towards the centre. The radial equation is Ttop + mg = mv²/r. The minimum speed to keep the string taut (Ttop ≥ 0) occurs when Ttop = 0, giving vmin = √(gr).
考虑一个质量为 m 的质点系在一根长为 r 的轻质不可伸长绳上，在竖直面内做圆周运动。在圆最高点，重力 mg 和张力 T顶 都沿半径向下指向圆心。径向方程为 T顶 + mg = mv²/r。保持绳子张紧（T顶 ≥ 0）的最小速率发生在 T顶 = 0 时，得出 vmin = √(gr)。
At the bottom of the circle, tension acts upward towards the centre while weight acts downward. The radial equation is Tbot − mg = mv²/r, so Tbot = mg + mv²/r. Tension is greatest at the bottom, where the speed is highest if only gravity acts.
在圆最低点，张力向上指向圆心，重力向下。径向方程为 T底 − mg = mv²/r，故 T底 = mg + mv²/r。如果只有重力做功，最低点速率最大，此时张力也最大。
If the particle is on a light rod instead of a string, the rod can support compression; thus the minimum speed at the top is zero because the rod can push outward. But the condition for string and rod differ – always check which one you have.
如果质点连在轻杆而非绳子上，杆可以承受压力；因此最高点的最小速率为零，因为杆可以向外推。但绳和杆的条件不同——务必确认题目给出的是哪种约束。
8. Conical Pendulum | 锥摆
A conical pendulum consists of a particle attached to a string of length L, moving in a horizontal circle with constant angular velocity ω. The string traces out a cone, making a constant angle θ with the vertical. The weight mg is balanced by the vertical component of tension: T cosθ = mg. The horizontal component provides the centripetal force: T sinθ = m r ω².
锥摆由一个质量为 m 的质点系在长度为 L 的绳上组成，质点以恒定角速度 ω 在水平面内做圆周运动。绳划出一个圆锥面，与竖直方向保持恒定的夹角 θ。重力 mg 由张力的竖直分量平衡：T cosθ = mg。水平分量提供向心力：T sinθ = m r ω²。
Dividing the two equations gives tanθ = r ω² / g. Since the radius of the circle is r = L sinθ, we can substitute to get an expression linking ω, L and θ. A typical exam question asks to find ω or θ given the other quantities.
两式相除得到 tanθ = r ω² / g。由于圆周半径为 r = L sinθ，代入后可得 ω、L 和 θ 的关系式。典型考题会要求根据已知量求出 ω 或 θ。
The period T of the conical pendulum can be expressed as T = 2π √(L cosθ / g), which reduces to the simple pendulum formula T = 2π √(L/g) when θ is small. This reveals the close connection to simple harmonic motion.
锥摆的周期 T 可表示为 T = 2π √(L cosθ / g)，当 θ 很小时，此式退化为单摆公式 T = 2π √(L/g)。这揭示了它与简谐运动的紧密联系。
9. Banked Tracks and Circular Motion | 倾斜轨道与圆周运动
Banked tracks (or roads) are designed so that a vehicle can negotiate a curve without relying on friction, or with reduced friction. For a banked curve of radius r banked at angle θ to the horizontal, the normal reaction N has a horizontal component N sinθ pointing towards the centre, and a vertical component N cosθ = mg.
倾斜轨道（或路面）的设计使得车辆能在不依赖摩擦力（或减少摩擦力）的情况下转弯。对于半径为 r、倾角为 θ（与水平面夹角）的倾斜弯道，法向反力 N 的水平分量 N sinθ 指向圆心，竖直分量 N cosθ = mg。
For frictionless banking, the centripetal force is entirely provided by N sinθ, leading to tanθ = v²/(rg). This gives the design speed for a given bank angle. If the car travels faster or slower, friction acts to prevent sliding up or down the slope.
在无摩擦倾斜轨道上，向心力完全由 N sinθ 提供，从而导出 tanθ = v²/(rg)。这给出了给定倾角下的设计车速。若车辆行驶速度比设计值快或慢，摩擦力就会起作用，防止车辆沿斜坡上滑或下滑。
Problems often ask for the range of speeds possible with a given coefficient of friction μ, or for the angle that eliminates lateral friction. Resolve forces parallel and perpendicular to the slope, and apply F ≤ μN where appropriate.
题目常会要求计算在给定摩擦系数 μ 下的可能速度范围，或求出消除侧向摩擦所需的倾角。需要沿斜面与垂直斜面方向分解力，并在适当处应用 F ≤ μN。
10. Common Mistakes and Exam Tips | 常见错误与应试技巧
1. Forgetting to use radians: Every formula involving ω, θ, v = rω assumes θ in radians. If you are given degrees, convert to radians first. Double-check calculator mode.
1. 忘记使用弧度：所有含 ω、θ 的公式如 v = rω 都假设 θ 以弧度为单位。如果题目给的是度数，务必先转换为弧度。确认计算器处于弧度模式。
2. Mixing up centripetal force with a separate force: The centripetal force is the resultant of real forces. Never add it as an extra force on your diagram; simply set the net inward force equal to mv²/r.
2. 将向心力与实际力混淆：向心力是真实力的合力。切勿在受力图上将其画为一个额外的力；只需令净向内力等于 mv²/r。
3. Ignoring energy changes in vertical circles: The speed is not constant. Use conservation of energy to link speeds at different heights: ½mv² + mgh = constant, often combined with centripetal condition at top/bottom.
3. 忽视竖直圆周中的能量变化：速率并非常量。利用能量守恒联系不同高度处的速率：½mv² + mgh = 常量，常与最高点/最低点的向心力条件联用。
4. Wrong sign in radial equations: At the top, weight and tension both point towards the centre; at the bottom, weight points away from the centre. Write them carefully.
4. 径向方程符号错误：在最高点，重力和张力都指向圆心；在最低点，重力背离圆心。写方程时需格外小心。
5. Not using the correct radius: In conical pendulum, the circle radius is r = L sinθ, not the string length L. In banked tracks, the horizontal circle radius is r, given in the problem.
5. 半径使用错误：在锥摆中，圆半径是 r = L sinθ，而非绳长 L。在倾斜轨道中，水平圆周半径 r 通常是题目给出的。
6. Practice with past papers: AQA frequently mixes circular motion with projectiles or energy, so build fluency in connecting topics.
6. 练习真题：AQA 常将圆周运动与抛体运动或能量相结合考查，因此要熟练串联不同知识点。
By systematically applying the principle that the resultant force towards the centre equals mrω² or mv²/r, and combining it with energy or Newton’s laws along other axes, you can tackle any circular motion question confidently.
通过系统地应用“指向圆心的合力等于 mrω² 或 mv²/r”这一原则，并结合能量或沿其他轴的牛顿定律，你就能自信地应对任何圆周运动题目。
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