IB Physics: Circular Motion - Key Concepts and Exam Tips | IB 物理：圆周运动考点精讲

📚 IB Physics: Circular Motion – Key Concepts and Exam Tips | IB 物理：圆周运动考点精讲

Circular motion is a cornerstone of IB Physics, appearing in both Mechanics and Fields topics. Understanding angular quantities, centripetal force, and applications from banked curves to satellite orbits is essential for success. This guide distills the key concepts, formulas, and common pitfalls, equipping you with the knowledge to tackle any circular motion problem confidently.

圆周运动是 IB 物理的基石，贯穿力学与场的主题。掌握角量、向心力以及从倾斜弯道到卫星轨道的应用，是取得高分的关键。本文精炼核心概念、公式和常见误区，助你从容应对各类圆周运动试题。

1. Angular Displacement, Angular Velocity & Angular Acceleration | 角位移、角速度与角加速度

Angular displacement (Δθ) is the angle swept by a radius vector, measured in radians (rad). One complete revolution equals 2π radians. Angular velocity (ω) is the rate of change of angular displacement: ω = Δθ/Δt, with units of rad s⁻¹. For uniform circular motion, ω is constant in magnitude. Angular acceleration (α) describes how ω changes over time: α = Δω/Δt, and it only appears in non-uniform circular motion.

角位移（Δθ）是半径矢量扫过的角度，单位为弧度 (rad)。一整圈等于 2π 弧度。角速度（ω）是角位移的变化率：ω = Δθ/Δt，单位是 rad s⁻¹。匀速圆周运动中，ω 的大小恒定。角加速度（α）描述 ω 随时间的变化：α = Δω/Δt，仅出现在非匀速圆周运动中。

ω = 2π / T α = Δω / Δt

2. Period, Frequency, and Their Link to ω | 周期、频率及其与角速度的关系

The period (T) is the time taken for one complete revolution, measured in seconds. Frequency (f) is the number of revolutions per second, with unit hertz (Hz). They are inverses: T = 1/f. Angular velocity links directly to these: ω = 2π/T = 2πf. For an object moving in a circle of radius r, the speed remains constant only in uniform circular motion.

周期（T）是物体完成一圈所需的时间，单位为秒。频率（f）是每秒转过的圈数，单位为赫兹 (Hz)。二者互为倒数：T = 1/f。角速度与它们直接相关：ω = 2π/T = 2πf。对于半径为 r 的圆周运动，仅当运动为匀速时速率才保持不变。

Quantity	Symbol	Relation
Period	T	T = 1/f = 2π/ω
Frequency	f	f = 1/T = ω/2π

3. Linear Velocity and Its Relation to Angular Velocity | 线速度及其与角速度的关系

The linear speed v of an object in circular motion is the magnitude of its tangential velocity. Although speed may be constant in uniform circular motion, the velocity vector changes direction continuously. The relationship between linear speed v and angular speed ω is v = ωr. The tangential velocity vector is always perpendicular to the radius.

物体做圆周运动时的线速率 v 是其切线速度的大小。匀速圆周运动中速率可能恒定，但速度矢量方向时刻改变。线速率 v 与角速度 ω 的关系为 v = ωr。切线速度矢量始终垂直于半径。

v = ωr

Note that r must be measured from the centre of the circle, which is not always the length of a string or the radius of a wheel – be careful with geometry.

注意 r 必须从圆心测量，并非总是绳长或轮子的半径——需留意几何关系。

4. Centripetal Acceleration: Derivation and Forms | 向心加速度：推导与表达式

Centripetal acceleration (a_c) is directed towards the centre of the circle and is responsible for changing the direction of velocity. For uniform circular motion, its magnitude is a_c = v²/r. Substituting v = ωr gives the alternative form a_c = ω²r. These two expressions are mathematically equivalent and must be memorised.

向心加速度（a_c）指向圆心，负责改变速度的方向。在匀速圆周运动中，其大小为 a_c = v²/r。代入 v = ωr 得到另一形式 a_c = ω²r。这两个表达式在数学上等价，必须熟记。

a_c = v² / r = ω²r

5. Centripetal Force: The Net Force Towards the Centre | 向心力：指向圆心的合力

Centripetal force (F_c) is not a new type of force; it is the net force directed towards the centre of the circle. By Newton’s second law, F_c = m a_c, so F_c = m v²/r = m ω²r. Common sources include tension, gravity, friction, or the normal component of a contact force. Always identify which real forces provide the centripetal resultant.

向心力（F_c）并非一种新的力，而是指向圆心的合力。根据牛顿第二定律，F_c = m a_c，因此 F_c = m v²/r = m ω²r。常见的来源包括张力、重力、摩擦力或接触力的法向分量。务必明确哪些实际力提供了向心合力。

F_c = mv² / r = mω²r

6. Conical Pendulum and Horizontal Circles | 锥摆与水平圆周运动

A conical pendulum consists of a mass whirled in a horizontal circle at the end of a string. The string traces a cone. Resolving forces vertically and radially: T cosθ = mg and T sinθ = mω²r, where r = L sinθ (L is string length). Eliminating T gives cosθ = g/(ω²L). This links angular speed to the angle θ.

锥摆由系在绳端、在水平面内做圆周运动的质量构成，绳子扫出一个圆锥面。对力进行竖直和径向分解：T cosθ = mg，T sinθ = mω²r，其中 r = L sinθ（L 为绳长）。消去 T 得到 cosθ = g/(ω²L)。该式将角速度与摆角 θ 联系起来。

If ω increases, cosθ decreases → θ increases.
若 ω 增大，cosθ 减小 → θ 增大。
Period T = 2π √(L cosθ / g).
周期 T = 2π √(L cosθ / g)。

7. Banked Curves and Vehicle Turning | 倾斜弯道与车辆转弯

When a vehicle moves around a banked curve, the horizontal component of the normal reaction contributes to centripetal force, reducing reliance on friction. For an ideal banking angle θ with no friction, N sinθ = mv²/r and N cosθ = mg, giving tanθ = v²/(rg). At higher speeds, friction acts inward down the slope; at lower speeds, friction acts outward.

当车辆驶过倾斜弯道时，法向反力的水平分量提供向心力，减少了对摩擦的依赖。对于无摩擦的理想倾斜角 θ，有 N sinθ = mv²/r 且 N cosθ = mg，得出 tanθ = v²/(rg)。速度较高时，摩擦力沿斜面向内；速度较低时，摩擦力向外。

v_ideal = √(rg tanθ)

8. Vertical Circular Motion: Speed and Tension | 竖直圆周运动：速度与张力

In vertical circular motion (e.g., a mass on a string, roller coaster loop), speed varies due to gravity. At the bottom, tension is maximum: T_bottom – mg = mv²/r. At the top, tension plus weight provide centripetal force: T_top + mg = mv²/r. To maintain a circular path at the top with the string just taut, T_top ≥ 0, giving the minimum speed v_min = √(gr).

在竖直圆周运动中（如绳端小球、过山车环圈），速度因重力而变化。最低点处，张力最大：T_bottom – mg = mv²/r。最高点处，张力与重力共同提供向心力：T_top + mg = mv²/r。为使最高点处绳子刚拉紧且不松脱，需 T_top ≥ 0，得到最小速率 v_min = √(gr)。

v_min (top) = √(gr)

9. Energy in Circular Motion | 圆周运动中的能量

If only conservative forces (e.g., gravity) do work in a vertical circle, mechanical energy is conserved. For a mass released from rest at a height, ½mv² + mgh = constant. This helps find speed at any point, which can then be used in force equations. For instance, the speed at the bottom after falling from height 2r above the bottom is v = √(4gr).

若竖直圆周运动中只有保守力（如重力）做功，则机械能守恒。对于从静止释放的质量，½mv² + mgh = 常数。据此可求出任意点的速度，进而代入力的方程。例如，从最低点上方 2r 处落下，到达最低点的速率为 v = √(4gr)。

Energy method often simplifies finding v before applying F_c = mv²/r.
能量法常能先简化求 v，再应用 F_c = mv²/r。

10. Gravitation as Centripetal Force | 万有引力作为向心力

For satellite and planetary orbits, the gravitational force provides the necessary centripetal force: GMm/r² = mω²r = mv²/r. This yields the orbital speed v = √(GM/r) and Kepler’s third law: T² = (4π²/GM) r³. For geostationary orbits, period T = 24 hours, and the orbital radius is fixed.

对于卫星和行星轨道，万有引力提供所需的向心力：GMm/r² = mω²r = mv²/r。由此得出轨道速率 v = √(GM/r) 以及开普勒第三定律：T² = (4π²/GM) r³。对于地球同步轨道，周期 T = 24 小时，轨道半径固定。

v_orbit = √(GM/r) T² ∝ r³

11. Exam Tips and Common Mistakes | 应试技巧与常见错误

Always draw a free-body diagram and label all forces. Determine the net force towards the centre – this is your centripetal force. Never include a ‘centrifugal force’ arrow; it does not exist in an inertial frame. Check that you use radius correctly: in conical pendulum, radius is L sinθ, not L. Convert revolutions per minute to rad s⁻¹ where needed. In vertical circles, speed is not constant; use energy conservation if necessary.

务必画受力图，标出所有作用力。确定指向圆心的合力——那就是向心力。切勿画出“离心力”箭头，它在惯性系中并不存在。注意正确使用半径：锥摆中，半径是 L sinθ 而非 L。必要时将转每分转换为 rad s⁻¹。竖直圆周运动中，速率不恒定，必要时使用能量守恒。

Be careful with square roots and squaring when manipulating v²/r and ω²r. Always check units: ω in rad s⁻¹, v in m s⁻¹, r in m. If a problem involves banking or a rotating space station, the normal force may not equal mg.

在运用 v²/r 和 ω²r 时注意平方和开方运算。务必检查单位：ω 用 rad s⁻¹，v 用 m s⁻¹，r 用 m。若涉及弯道倾斜或旋转空间站，法向力未必等于 mg。

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IB Physics: Circular Motion – Key Concepts and Exam Tips | IB 物理：圆周运动 考点精讲