📚 AP Physics: Circular Motion – Exam Point Analysis | AP 物理：圆周运动考点解析
Circular motion is a cornerstone topic in AP Physics, appearing in both conceptual multiple-choice questions and complex free-response problems. Mastering the relationship between centripetal force, speed, radius, and period – and applying Newton’s laws in radial and tangential directions – is essential for a high exam score. This article breaks down every key point, common pitfalls, and proven problem-solving strategies.
圆周运动是 AP 物理的核心考点，出现在概念选择题和复杂的自由回答题中。掌握向心力、速率、半径和周期之间的关系，并能在径向和切向灵活运用牛顿定律，是取得高分的关键。这篇文章将逐一解析重要考点、常见错误和实用的解题策略。
1. Basic Quantities: Angular Displacement, Velocity & Linear Speed | 基本量：角位移、角速度与线速度
An object moving in a circle can be described using angular displacement θ measured in radians. One complete revolution corresponds to 2π radians. The angular velocity ω (omega) is the rate of change of θ: ω = Δθ/Δt, with units rad/s.
圆周运动可用角位移 θ 描述，单位是弧度。一整圈对应 2π 弧度。角速度 ω 是角位移的变化率：ω = Δθ/Δt，单位为 rad/s。
The linear speed v of a particle moving along a circular path of radius r is linked to angular velocity by v = rω. This relation is fundamental: when a rotating object speeds up or increases radius, v must change proportionally.
质点沿半径为 r 的圆周运动的线速率 v 与角速度的关系为 v = rω。这个关系至关重要：转速提高或半径增大时，v 必然按比例变化。
2. Centripetal Acceleration: Direction and Magnitude | 向心加速度：方向与大小
Even if speed is constant, the velocity vector changes direction continuously, producing an acceleration directed toward the centre of the circle. This centripetal acceleration has magnitude ac = v²/r = ω²r.
即使速率恒定，速度矢量方向不断改变，产生始终指向圆心的加速度。向心加速度大小为 ac = v²/r = ω²r。
Because ac is always perpendicular to velocity, it changes the direction of motion but not the speed. In uniform circular motion, the acceleration is purely radial.
由于 ac 始终垂直于速度，它只改变运动方向，不改变速率大小。在均匀圆周运动中，加速度完全是径向的。
3. Centripetal Force: It’s Not a New Force! | 向心力：并非新力种！
Many students make the mistake of adding a “centripetal force” to the free-body diagram. Centripetal force is the net force in the radial direction, not an additional force. It is always provided by real forces such as tension, gravity, friction, or the normal force.
很多学生会在受力图中额外添加一个“向心力”。向心力是沿径向的合力，而不是某种新的力。它总是由真实的力（如张力、重力、摩擦力、支持力）来充当。
A related misconception is inventing a “centrifugal force” acting outward. The sensation of being pushed outward in a turning car is due to inertia resisting the change in direction, not a real outward force.
另一个常见误区是捏造一个向外的“离心力”。转弯时感觉被向外甩，其实是惯性抵抗方向改变的体现，并非存在真实的向外作用力。
4. Period, Frequency, and Uniform Circular Motion Equations | 周期、频率与均匀圆周运动方程
For uniform circular motion (constant speed), the period T is the time for one revolution, and frequency f is the number of revolutions per second (f = 1/T). The angular velocity can be expressed as ω = 2π/T = 2πf.
均匀圆周运动（恒定速率）中，周期 T 是运动一周的时间，频率 f 为每秒圈数（f = 1/T）。角速度可表示为 ω = 2π/T = 2πf。
The linear speed v is related to period by v = 2πr/T. Combining with ac = v²/r gives alternative forms ac = 4π²r/T². These equations are frequently tested.
线速率 v 与周期的关系为 v = 2πr/T。结合 ac = v²/r 可得 ac = 4π²r/T²，这些变形公式经常在考题中出现。
5. Applying Newton’s Second Law to Circular Motion | 圆周运动中牛顿第二定律的应用
The key to solving circular motion problems is to choose the radial direction (pointing toward the centre) as positive and write ΣFr = m ac = m v²/r. If the speed changes, there is also a tangential component: ΣFt = m at.
解题的关键是设定指向圆心的径向为正方向，列出径向方程 ΣFr = m ac = m v²/r。如果速率变化，还需要切向方程 ΣFt = m at。
Always start with a clear free-body diagram, identify the forces acting on the object, and resolve them into radial and tangential components. Never pre-assume that a certain force equals m v²/r – derive it.
务必从清晰的受力图开始，确认物体受到的所有力，并将它们沿径向和切向分解。千万不要预先假定某个力就等于 m v²/r，一定要通过受力分析推导。
6. Horizontal Circular Motion: Conical Pendulum & Flat Curve | 水平面圆周运动：圆锥摆和水平弯道
A conical pendulum consists of a mass tied to a string of length L, moving in a horizontal circle. Resolving tension T: T cosθ = mg and T sinθ = m v²/(L sinθ). Solving yields ω = √(g/(L cosθ)).
圆锥摆由系在长 L 的绳上的摆球组成，在水平面内做圆周运动。分解张力 T：T cosθ = mg，T sinθ = m v²/(L sinθ)。解得 ω = √(g/(L cosθ))。
When a car rounds a flat, unbanked curve, the static friction between tires and road supplies the centripetal force: fs = m v²/r. With fs ≤ μs N and N = mg, the maximum safe speed is vmax = √(μs g r). If speed exceeds this, skidding occurs.
汽车在水平无倾斜弯道转弯时，轮胎与路面的静摩擦力提供向心力：fs = m v²/r。由 fs ≤ μs N 和 N = mg 可得最大安全车速 vmax = √(μs g r)。超过该速度将发生侧滑。
7. Banked Curves: Using the Normal Force | 倾斜弯道：利用支持力的分力
On an ideally banked curve without friction, the horizontal component of the normal force alone provides the centripetal force. From N sinθ = m v²/r and N cosθ = mg, we obtain tanθ = v²/(r g). This is the design speed for a given bank angle.
在理想的无摩擦倾斜弯道上，仅由支持力的水平分力提供向心力。由 N sinθ = m v²/r 和 N cosθ = mg 可得 tanθ = v²/(r g)，即为给定倾角下的设计速度。
When friction is present, both the friction force and the normal force contribute radially. The maximum and minimum safe speeds can be derived by adding a friction component parallel to the incline.
当存在摩擦时，摩擦力和支持力共同提供径向合力。通过加入沿斜面的摩擦分力，可以推导出车辆的最大和最小安全速率。
8. Vertical Circular Motion: Rope and Track Models | 垂直圆周运动：绳子和轨道模型
For an object whirled in a vertical circle with a string, tension changes with position. At the bottom: T – mg = m v²/r (tension maximum). At the top: T + mg = m v²/r. If the object is just able to complete the circle, T = 0 at the top.
用绳子将物体在竖直面内甩动时，张力随位置变化。最低点：T – mg = m v²/r（张力最大）；最高点：T + mg = m v²/r。若物体恰好能完成圆周运动，最高点处 T = 0。
For a roller-coaster car on a track, the normal force N replaces tension. At the top of a loop, N + mg = m v²/r (track above the car). At the bottom, N – mg = m v²/r. The physics is identical; just replace T with N.
对于过山车轨道，支持力 N 代替张力。环圈最高点（轨道在车上方）时 N + mg = m v²/r，最低点时 N – mg = m v²/r。物理原理完全一致，只需把 T 换成 N 即可。
9. Critical Speed and Apparent Weightlessness | 临界速度与表观失重
The minimum speed to maintain vertical circular motion with a string is reached at the top, where the tension drops to zero. Setting T = 0 gives vcrit = √(gr). Below this speed, the string goes slack and the object
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