IB & CIE Physics: Key Points on Circular Motion | IB CIE 物理：圆周运动考点精讲

📚 IB & CIE Physics: Key Points on Circular Motion | IB CIE 物理：圆周运动考点精讲

Circular motion is a fundamental topic in both IB and CIE A-Level Physics, examining how objects move along circular paths at constant speed (uniform circular motion) or varying speed. Understanding centripetal force, angular velocity, and the interplay between linear and angular quantities is essential for tackling a wide range of exam questions, from satellites to banked curves.

圆周运动是 IB 和 CIE A-Level 物理的核心内容，研究物体以恒定速率（匀速圆周运动）或变化速率沿圆周路径运动的规律。理解向心力、角速度以及线量与角量之间的关系，是解答从人造卫星到倾斜弯道等各类考试题的关键。

---

1. What is Uniform Circular Motion? | 什么是匀速圆周运动？

Uniform circular motion occurs when an object moves in a circle at a constant speed. Although the speed is unchanged, the velocity is not, because its direction continuously changes. This continuous change in velocity means there is an acceleration directed toward the centre of the circle, called centripetal acceleration.

匀速圆周运动是指物体以恒定的速率沿圆形路径运动。虽然速率不变，但由于速度方向时刻在变化，速度矢量并非恒定。这种速度的持续变化意味着存在一个指向圆心的加速度，即向心加速度。

---

2. Angular Displacement and Angular Velocity | 角位移与角速度

Angular displacement θ is the angle swept out by the radius, measured in radians. Angular velocity ω is defined as the rate of change of angular displacement: ω = Δθ/Δt. In uniform circular motion, ω is constant. One radian corresponds to the angle when the arc length equals the radius, and the conversion between revolutions and radians is 1 rev = 2π rad.

角位移 θ 是半径扫过的角度，以弧度为单位。角速度 ω 定义为角位移的变化率：ω = Δθ/Δt。在匀速圆周运动中，ω 恒定不变。一弧度对应于弧长等于半径时所张的角，转数与弧度的换算关系为 1 转 = 2π rad。

ω = Δθ/Δt , unit: rad s⁻¹

---

3. Linking Linear and Angular Quantities | 线量与角量的联系

The linear speed v of an object travelling in a circle of radius r is related to angular velocity by v = rω. While linear speed is constant in uniform circular motion, the continuous change in direction gives rise to centripetal acceleration: a_c = v²/r = rω². This acceleration is always perpendicular to the instantaneous velocity and points radially inward.

物体在半径为 r 的圆周上运动的线速率 v 与角速度的关系为 v = rω。在匀速圆周运动中，线速率恒定，但方向的持续变化导致了向心加速度：a_c = v²/r = rω²。这一加速度始终垂直于瞬时速度，沿半径指向圆心。

a_c = v² / r = rω²

---

4. Centripetal Force | 向心力

According to Newton’s second law, a net force is needed to produce the centripetal acceleration. This centripetal force has magnitude F_c = m v²/r = m r ω² and is directed towards the centre of the circle. It is crucial to note that centripetal force is not a new type of force; it is the resultant force acting towards the centre, often provided by tension, gravity, friction, or the normal reaction.

根据牛顿第二定律，必须有一个合力来产生向心加速度。这个向心力大小为 F_c = m v²/r = m r ω²，方向指向圆心。必须强调的是，向心力并非一种新的性质力，而是指向圆心的合力，通常由张力、重力、摩擦力或法向反作用力等充当。

F_c = m v² / r = m r ω²

---

5. Period and Frequency | 周期与频率

The period T is the time taken for one complete revolution, and frequency f is the number of revolutions per unit time. They are linked by f = 1/T. For uniform circular motion, the period can be expressed as T = 2πr/v = 2π/ω. These relationships are especially useful for calculating orbital periods of planets and satellites.

周期 T 是完成一整圈所需的时间，频率 f 是单位时间内转过的圈数。两者满足 f = 1/T。对于匀速圆周运动，周期可表示为 T = 2πr/v = 2π/ω。这些关系在计算行星和人造卫星的轨道周期时格外有用。

---

6. Horizontal Circular Motion: Conical Pendulum & Flat Curves | 水平面圆周运动：圆锥摆与水平弯道

A conical pendulum consists of a mass whirling in a horizontal circle at the end of a string. The string traces a cone; the centripetal force is the horizontal component of tension, while the vertical component balances the weight. For a car turning on a flat road, the necessary centripetal force comes from static friction between the tyres and the road. If the speed exceeds a critical value, the friction required cannot be supplied and the car will skid.

圆锥摆由系在绳子末端、在水平面内做圆周运动的摆球构成。绳子扫出一个圆锥，向心力由张力的水平分量提供，竖直分量则与重力平衡。对于在水平路面上转弯的汽车，所需的向心力由轮胎与路面间的静摩擦力提供。如果速度超过某个临界值，所需摩擦力无法满足，汽车将发生侧滑。

---

7. Banked Curves without Friction | 无摩擦倾斜弯道

Banking a curve at an angle θ allows a vehicle to negotiate the turn without relying on friction. The horizontal component of the normal reaction provides the centripetal force: N sinθ = m v²/r, while the vertical component supports the weight: N cosθ = m g. Dividing the two equations gives the design speed condition: tanθ = v²/(r g).

将弯道倾斜一个角度 θ，可使车辆无需依赖摩擦力即可转弯。法向反作用力的水平分量提供向心力：N sinθ = m v²/r，竖直分量则平衡重力：N cosθ = m g。两式相除得到理想设计速度条件：tanθ = v²/(r g)。

tanθ = v² / (r g)

---

8. Vertical Circular Motion | 竖直平面内的圆周运动

When an object moves in a vertical circle (e.g., a bucket of water or a roller coaster loop), its speed often changes. At the very top, both weight m g and the normal reaction N act downward, so N + m g = m v²/r. For the object to just maintain contact, N ≥ 0, giving a minimum speed v_min = √(r g). At the bottom of the circle, N – m g = m v²/r, so the reaction force is larger than the weight.

当物体在竖直平面内做圆周运动（如水桶实验或过山车回环），速率通常会发生改变。在最高点，重力 m g 和法向反作用力 N 均向下，故 N + m g = m v²/r。物体刚好能保持接触的条件为 N ≥ 0，由此得到最小速率 v_min = √(r g)。在最低点，N – m g = m v²/r，因此反作用力大于重力。

v_min(top) = √(r g)

---

9. Centripetal Force vs. Centrifugal Force | 向心力与离心力

In an inertial frame of reference, only centripetal force exists, acting inward. The sensation of being ‘thrown outward’ is due to inertia, not a real outward force. In a rotating frame, a fictitious centrifugal force appears to balance the centripetal force. Exam solutions must not include centrifugal force in free-body diagrams drawn in an inertial frame.

在惯性参考系中，只存在指向圆心的向心力。感觉自己被“甩向外侧”是由于惯性，而非真实的向外力。在旋转参考系中，会引入一个虚拟的离心力来与向心力平衡。在惯性系下绘制受力图时，切勿在图中标出离心力。

---

10. Circular Motion and Gravitation: Satellites | 圆周运动与万有引力：人造卫星

For satellites orbiting a planet, the gravitational attraction provides the centripetal force: G M m / r² = m v²/r. This yields an orbital speed v = √(G M / r) and period T = 2πr/v = 2π√(r³/G M). These relationships explain why geostationary satellites must orbit at a specific radius above the equator with a period of exactly 24 hours.

对于绕行星运转的卫星，万有引力提供向心力：G M m / r² = m v²/r。由此可得轨道速率 v = √(G M / r) 以及周期 T = 2πr/v = 2π√(r³/G M)。这些关系解释了为何地球同步卫星必须以24小时为周期运行在赤道上方某一特定半径的轨道上。

v_orbit = √(G M / r)

---

11. Common Exam Pitfalls and How to Avoid Them | 常见失分点与避坑指南

A frequent mistake is drawing centripetal force as an extra force on a free-body diagram. Remember that it is the net force toward the centre. Another error is using degrees instead of radians when applying ω = Δθ/Δt or a = rω². When solving vertical circle problems, always check the direction of forces at each point and use conservation of energy if the speed varies. Also, carefully distinguish between v and ω when substituting into formulas.

常见错误是在受力图中将向心力画成一个单独的力。务必记住向心力是沿径向的合力。另一个错误是在使用 ω = Δθ/Δt 或 a = rω² 时用角度制而非弧度制。解竖直圆周运动问题时，务必要检查各点力的方向，若速率变化则需结合能量守恒。此外，代入公式时务必仔细区分线速度 v 和角速度 ω。

---

12. Summary and Key Equations | 总结与核心公式

Master these key relationships to excel in circular motion questions. The table below summarises the essential equations you must have at your fingertips for IB and CIE Physics exams.

掌握以下核心关系，可在圆周运动题目中脱颖而出。下表汇总了在 IB 和 CIE 物理考试中必须烂熟于心的关键公式。

Quantity | 物理量	Equation | 公式
Linear speed | 线速率	v = rω
Centripetal acceleration | 向心加速度	a = v²/r = rω²
Centripetal force | 向心力	F = m v²/r = m r ω²
Period | 周期	T = 2π/ω = 2π r / v
Banked curve (ideal) | 理想倾斜弯道	tanθ = v²/(r g)
Minimum speed at top of vertical circle | 竖直圆周顶部最小速率	v_min = √(r g)
Orbital speed (satellite) | 轨道速率（卫星）	v = √(G M / r)

Always begin by identifying the force(s) providing the centripetal component, resolve along the radial direction, and apply the appropriate equation. Practice a variety of scenarios, including loops, banked tracks, and conical systems, to build confidence.

解题时始终先找出提供向心力的力，沿径向分解，并准确选用合适的公式。通过反复练习回环、倾斜轨道和圆锥摆等不同情境，逐步建立起从容的解题信心。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）