📚 AS Physics: Circular Motion Essentials | AS 物理：圆周运动考点精讲
Mastering circular motion is essential for AS Physics. It involves understanding angular displacement, angular velocity, centripetal acceleration and force, and applying these concepts to horizontal and vertical circles. This guide covers key exam points with clear explanations and bilingual notes.
掌握圆周运动对 AS 物理至关重要。需要理解角位移、角速度、向心加速度和向心力，并应用到水平和竖直圆周问题中。本文以中英双语精讲考点，助你冲刺高分。
1. Introduction to Circular Motion | 圆周运动简介
Uniform circular motion is motion in a circle at constant speed. Although the speed is constant, the velocity is continuously changing because the direction changes. This change in velocity implies an acceleration directed towards the centre of the circle, known as centripetal acceleration.
匀速圆周运动指速率不变的圆周运动。虽然速率恒定，但速度方向时刻改变，因此存在加速度，方向始终指向圆心，称为向心加速度。
The period T is the time taken for one complete revolution, and frequency f is the number of revolutions per second: f = 1/T. Both are useful for describing the cyclic nature of the motion.
周期 T 是完成一圈所需的时间，频率 f 是每秒转动的圈数：f = 1/T。两者可方便地描述运动的周期性。
2. Angular Displacement and Angular Velocity | 角位移与角速度
Angular displacement θ (in radians) is defined as the ratio of arc length s to radius r: θ = s / r. One complete revolution equals 2π radians, and these dimensionless units help simplify equations.
角位移 θ（弧度制）定义为弧长 s 与半径 r 之比：θ = s / r。一整圈对应 2π 弧度，使用弧度制可简化许多公式。
Angular velocity ω is the rate of change of angular displacement: ω = Δθ / Δt. Its unit is rad s⁻¹. For uniform circular motion, ω = 2π / T = 2π f.
角速度 ω 是角位移的变化率：ω = Δθ / Δt，单位是 rad s⁻¹。对于匀速圆周运动，ω = 2π / T = 2π f。
ω = 2πf = 2π / T
Always check that your calculator is in radian mode when using ω. Many exam mistakes arise from mixing degrees and radians.
使用角速度时务必确保计算器处于弧度模式，混淆角度和弧度是常见扣分点。
3. Relationship Between Linear and Angular Quantities | 线量与角量的关系
The instantaneous linear speed v along the circumference is related to angular velocity by v = rω. The direction of velocity is always tangential to the circle.
线速度 v 沿圆周切线方向，大小与角速度的关系为 v = rω。速度方向时刻沿着切线。
v = rω
Because the radius is constant, a larger angular velocity gives a larger linear speed. Inversely, for a fixed angular velocity, speed increases with radius. Remember that v is a scalar speed; the velocity vector changes continuously.
因为半径恒定，角速度越大线速度越大。反之，角速度一定时，半径越大线速度越大。注意 v 是速率，速度矢量方向是不断变化的。
4. Centripetal Acceleration | 向心加速度
The centripetal acceleration ac points towards the centre of the circle. Its magnitude is given by a = v² / r or a = rω². This result can be derived by considering the change in velocity vector over a small time interval as an object moves along a circular arc.
向心加速度 ac 指向圆心，大小为 a = v² / r 或 a = rω²。该结果可通过分析物体在短时间内的速度矢量变化推导得出。
a = v² / r = rω²
Even when speed is constant, this acceleration is non-zero and always perpendicular to the velocity. It is responsible for changing the direction of motion, not the speed.
即使速率不变，该加速度也不为零且始终与速度方向垂直。它的作用是改变速度方向而不是大小。
5. Centripetal Force | 向心力
According to Newton’s second law, a centripetal force Fc must act on the object to produce centripetal acceleration: F = ma = mv² / r = mrω². This net force always points towards the centre.
根据牛顿第二定律，必须有一个向心力 Fc 产生向心加速度：F = ma = mv² / r = mrω²。该合力始终指向圆心。
F = mv² / r = mrω²
Centripetal force is not a new type of force; it is provided by tension, gravity, friction, normal reaction, or a combination of forces. Its role is to keep the object on the circular path.
向心力不是某种新型的力，它可以由拉力、重力、摩擦力、支持力或它们的合力提供。它的作用是把物体维持在圆周路径上。
6. Horizontal Circular Motion Examples | 水平面圆周运动实例
In a conical pendulum, a bob swings in a horizontal circle. The string traces a cone. The tension and weight produce a net horizontal force towards the centre: T sinθ = mv² / r, and T cosθ = mg.
在圆锥摆中，摆球在水平面内做圆周运动，绳子形成一个锥面。拉力和重力的合力提供水平向心力：T sinθ = mv² / r，并且 T cosθ = mg。
For a car turning on a flat road, friction between tyres and road provides the centripetal force: f = mv² / r. The maximum speed without skidding is vmax = √(μg r).
汽车在水平路上转弯时，摩擦力提供向心力：f = mv² / r。不侧滑的最大速度为 vmax = √(μg r)。
Banked curves reduce the reliance on friction. The horizontal component of the normal reaction helps supply the centripetal force: N sinθ = mv² / r, N cosθ = mg, giving the ideal speed v = √(gr tanθ) for no friction.
斜面弯道可以减少对摩擦的依赖。支持力的水平分量帮助提供向心力：N sinθ = mv² / r，N cosθ = mg，得出无摩擦时的理想速度 v = √(gr tanθ)。
7. Vertical Circular Motion | 竖直面圆周运动
In vertical circle problems (e.g., a mass on a string or a roller coaster loop), speed changes due to gravity. The centripetal force is still mv²/r, but tension or normal reaction varies with position.
在竖直面圆周运动中（如绳球模型、过山车圆环），速率会因重力而改变，但向心力公式仍为 mv²/r，拉力或支持力随位置变化。
At the lowest point, the net force towards the centre is T − mg = mv² / r, so T = mg + mv² / r (maximum tension).
在最低点，指向圆心的合力为 T − mg = mv² / r，因此 T = mg + mv² / r（拉力最大）。
At the highest point, the net force towards the centre is T + mg = mv² / r, so T = mv² / r − mg. T can be zero at the minimum possible speed.
在最高点，指向圆心的合力为 T + mg = mv² / r，因此 T = mv² / r − mg。在最小速度时 T 可以为零。
8. Critical Speeds and “Looping the Loop” | 临界速度与完整圆周条件
For a mass on a string to complete a vertical circle, the minimum speed at the top is when tension T = 0: mg = mv² / r ⇒ vtop, min = √(gr). From energy conservation, the required speed at the bottom is vbottom, min = √(5gr).
绳球模型中，完成完整圆周的顶部临界速度为 T = 0 时，mg = mv² / r ⇒ v顶, min = √(gr)。通过机械能守恒，底部所需最小速度 v底, min = √(5gr)。
For a rigid rod (or a roller coaster car on a track with both inner and outer wheels), the minimum speed at the top can be zero because the rod can support the mass. The limiting condition depends on whether the constraint is ‘string-like’ or ‘rod-like’.
对于杆球模型（或过山车车轮内外均有支撑），顶部速度可以为零，因为杆能提供向上的支持力。判断临界条件要看约束属于“绳模型”还是“杆模型”。
9. Centrifugal Force – Misconception | “离心力”——常见误区
In an inertial frame of reference, there is no outward ‘centrifugal force’. The sensation of being thrown outward is due to inertia: the object tends to move in a straight line tangent to the circle, while the centripetal force pulls it inward.
在惯性参考系中，不存在向外拉的“离心力”。感觉向外甩是由于惯性——物体有沿切线方向做匀速直线运动的趋势，而向心力将其拉向圆心。
In a rotating reference frame, a fictitious centrifugal force appears, but at AS Level, always analyse forces from an inertial frame: the net force towards the centre equals mv²/r. This will help you avoid conceptual mistakes.
在转动参考系中会出现假想的离心力，但 AS 阶段务必从惯性系分析：指向圆心的合力等于 mv²/r。这样可以避免概念性错误。
10. Key Equations Summary | 核心公式一览
| Quantity | Equation |
|---|---|
| Angular velocity | ω = Δθ/Δt = 2πf = 2π/T |
| Linear speed | v = rω |
| Centripetal acceleration | a = v²/r = rω² |
| Centripetal force | F = mv²/r = mrω² |
| Conical pendulum | T sinθ = mv²/r, T cosθ = mg |
| Vertical circle (lowest) | T = mg + mv²/r |
| Vertical circle (highest, string) | mg + T = mv²/r, with vmin = √(gr) |
Memorise these, and be able to derive them from first principles. Always identify the source of centripetal force in any context before plugging numbers into the formulas.
熟记这些公式，并能
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导