IGCSE Physics: Circular Motion – Key Concepts and Exam Questions | IGCSE 物理：圆周运动考点精讲

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

Circular motion is a fundamental topic in IGCSE Physics, describing the motion of objects moving along a circular path at constant speed. While the speed is constant, the velocity changes continuously due to the changing direction, which requires a resultant force acting towards the centre – the centripetal force. Understanding this concept is crucial for tackling exam questions on satellites, vehicles on banked tracks, and everyday phenomena like spinning a bucket of water. This guide covers all the key points, formulas, and common pitfalls you need to achieve top marks.

圆周运动是IGCSE物理中的一个基础主题，描述了物体沿圆形路径以恒定速率运动的情形。虽然速率不变，但由于方向不断改变，速度矢量一直在变化，这需要一个指向圆心的合力——向心力。理解这一概念对于解决关于卫星、弯道上的车辆以及诸如旋转水桶等日常现象的考题至关重要。本指南涵盖了所有关键点、公式和常见错误，助你拿下高分。

1. What is Circular Motion? | 什么是圆周运动？

When an object travels around a circular path with constant speed, it is performing uniform circular motion. The distance from the centre remains constant (radius r). Although the speed is steady, the direction of the velocity is continuously changing – the velocity vector is always tangent to the circle.

当物体以恒定速率沿圆形路径运动时，它做的是匀速圆周运动。到中心的距离保持不变（半径 r）。尽管速率恒定，但速度的方向不断变化——速度矢量总是沿着圆的切线方向。

Because velocity is a vector, any change in direction means the velocity is changing. Therefore, an object in uniform circular motion is accelerating, even if its speed never alters. This acceleration is always towards the centre of the circle.

由于速度是矢量，方向的任何改变都意味着速度在变化。因此，做匀速圆周运动的物体即使在速率不变的情况下也在加速。这个加速度始终指向圆心。

2. Speed and Velocity in a Circle | 圆周运动中的速率与速度

The speed v of an object in uniform circular motion can be found from the circumference of the circle (2πr) and the time for one complete revolution, the period T. This gives the formula v = 2πr / T. Speed is a scalar quantity; it tells us how fast the object is moving along the circle, but not the direction.

匀速圆周运动的速率 v 可以通过圆的周长（2πr）和完成一整圈所需的时间（周期 T）求出。由此得出公式 v = 2πr / T。速率是标量，它告诉我们物体沿圆周运动有多快，但不涉及方向。

The instantaneous velocity, however, is a vector. Its magnitude equals the speed, but its direction is always along the tangent to the circle at that point. As the object moves, the velocity direction changes continuously; hence the velocity is not constant, despite constant speed.

然而，瞬时速度是矢量。其大小等于速率，但方向始终沿着该点处的切线。随着物体运动，速度方向不断改变；因此，即使速率恒定，速度也不是恒定的。

3. Angular Velocity (ω) and Period (T) | 角速度与周期

Angular velocity (symbol ω, measured in radians per second, rad/s) describes how quickly the angle swept out by the radius changes. For one complete revolution, the angle is 2π radians, so ω = 2π / T. It links neatly to linear speed: since distance = radius × angle, we get v = ωr.

角速度（符号 ω，单位是弧度每秒，rad/s）描述半径扫过的角度变化快慢。对于一整圈，角度为 2π 弧度，因此 ω = 2π / T。它与线速率有简洁的联系：因为弧长 = 半径 × 角度，可得 v = ωr。

The frequency f (in hertz, Hz) is the number of revolutions per second, and is related to period by f = 1/T. So we can also write ω = 2πf. These relationships are essential when you need to switch between rotational and linear quantities in exam calculations.

频率 f（单位赫兹，Hz）是每秒转动的圈数，与周期关系为 f = 1/T。因此我们也可写作 ω = 2πf。在考试计算中，当需要在转动量和线量之间切换时，这些关系至关重要。

4. Centripetal Acceleration | 向心加速度

Since the velocity vector changes direction, there must be an acceleration. This acceleration is called centripetal acceleration, always directed towards the centre of the circle. For an object moving at speed v in a circle of radius r, the magnitude is given by a = v² / r.

由于速度矢量方向改变，必然存在加速度。这个加速度称为向心加速度，方向始终指向圆心。对于以速率 v 在半径为 r 的圆上运动的物体，其大小由 a = v² / r 给出。

a = v² / r and a = ω² r

Using the relation v = ωr, we obtain an alternative form: a = ω² r. Notice that even if the speed is constant, a is non‑zero because v is not zero and r is fixed. The direction of acceleration is always radially inward, perpendicular to the velocity.

利用关系式 v = ωr，我们得到另一种形式：a = ω² r。注意即使速率恒定，a 也不为零，因为 v 不为零且 r 固定。加速度的方向始终沿径向向内，垂直于速度。

5. Centripetal Force | 向心力

According to Newton’s second law, a resultant force is needed to produce the centripetal acceleration. This force is called the centripetal force, directed towards the centre of the circle. Its magnitude is given by F = mv² / r or, using angular velocity, F = mω² r.

根据牛顿第二定律，需要一个合力来产生向心加速度。这个力称为向心力，指向圆心。其大小由 F = mv² / r 给出，或使用角速度表示为 F = mω² r。

F = mv² / r = mω² r

Crucially, centripetal force is not a new type of force. It is simply the name we give to the resultant force that acts towards the centre. It can be provided by tension (in a string), gravitational attraction (for a satellite), friction (for a car turning), or the normal reaction and weight (for a roller coaster loop).

关键的是，向心力不是一种新的力。它只是我们对指向圆心的合力的称呼。它可以由张力（绳子）、万有引力（卫星）、摩擦力（转弯的汽车）或支持力与重力的合力（过山车回环）提供。

6. Examples of Centripetal Force | 向心力的实例

In IGCSE problems, you will encounter several real‑world scenarios. A common one is a rubber bung whirled on a string: the tension in the string supplies the centripetal force. If the string breaks, the bung flies off tangent to the circle, not radially outward, because the centripetal force disappears and the bung continues with the velocity it had at that instant (inertia).

在IGCSE问题中，你会遇到若干实际场景。一个常见例子是用绳子旋转的橡胶塞：绳子中的张力提供向心力。如果绳子断裂，塞子会沿切线方向飞出，而不是径向朝外，因为向心力消失，塞子将以其在那一瞬间的速度继续运动（惯性）。

Vehicle on a flat curve: Friction between tyres and road provides the sideways centripetal force. If the road is icy, friction is insufficient, and the car skids outward (continues straight while the road curves).

水平弯道上的车辆：轮胎与路面之间的摩擦力提供侧向的向心力。若路面结冰，摩擦力不足，车辆就会沿直线滑出（道路弯曲但车辆保持直线）。
Satellite in orbit: The Earth’s gravitational pull acts as the centripetal force, keeping the satellite in a circular (or nearly circular) path.

轨道上的卫星：地球的引力充当向心力，使卫星保持在圆形（或近似圆形）路径上。
Roller coaster loop: At the top of a loop, weight and the normal reaction both act downwards, together providing the centripetal force. At the bottom, the normal reaction is upward, greater than the weight, to supply the centripetal demand.

过山车回环：在回环顶部，重力和支持力都向下，共同提供向心力。在底部，支持力向上且大于重力，以满足向心力的要求。

Always draw a free‑body diagram showing all real forces, then identify which component(s) point towards the centre – that is your centripetal force.

始终画受力分析图展示所有真实力，然后确定哪些分量指向圆心——那就是你的向心力。

7. Centrifugal Force: A Common Misconception | 离心力：常见误解

Many learners talk about an outward ‘centrifugal’ force pushing them against the car door when going round a corner. In reality, no outward force acts on you. Your body, due to inertia, tends to continue in a straight line while the car turns inwards. The car door then exerts an inward force (centripetal) on you, causing you to follow the circular path. What you feel is the reaction of the door pushing inward, not a mysterious outward pull.

很多学习者会谈到转弯时将他们推向车门的、向外的“离心力”。实际上，并没有向外的力作用在你身上。你的身体由于惯性倾向于沿直线前进，而汽车向内转弯。然后车门对你施加一个向内的力（向心力），使你做圆周运动。你感觉到的是车门向内推的反作用力，而非神秘的向外拉力。

Examiners frequently penalise mentions of ‘centrifugal force’ as a real force. Always use ‘centripetal force’ and explain the sensation as inertia. If you must mention centrifugal effects, clarify that they are felt in a rotating reference frame, but for IGCSE stick to inertial (laboratory) frame and real forces only.

考官经常扣分如果提到“离心力”是真实力。始终使用“向心力”并用惯性解释那种感觉。如果必须提及离心效应，需说明那是在旋转参考系中感觉到的，但IGCSE考试中坚持用惯性（实验室）参考系和只考虑真实力。

8. Calculations and Equations | 计算与公式应用

Here is a summary table of the key equations, with typical situations where each is useful.

下表汇总了关键方程，以及每个方程适用的典型情境。

Formula	When to Use
v = 2πr / T	Given radius and period, find speed
ω = 2π / T or ω = 2πf	Find angular velocity from period or frequency
v = ωr	Link linear speed and angular velocity
a = v² / r	Centripetal acceleration when speed and radius known
a = ω² r	Centripetal acceleration from angular velocity
F = mv² / r	Centripetal force from speed, mass and radius
F = mω² r	Centripetal force from angular velocity

Example: A car of mass 1200 kg rounds a roundabout of radius 25 m at a speed of 8.0 m/s. Calculate the frictional force required to keep the car moving in a circle.

示例：一辆质量为1200 kg的汽车以8.0 m/s的速率绕半径25 m的环形交叉行驶。计算保持汽车做圆周运动所需的摩擦力。

F = mv² / r = (1200 kg × (8.0 m/s)²) / 25 m = (1200 × 64) / 25 = 3072 N ≈ 3100 N

Always check that units are consistent: mass in kg, speed in m/s, radius in m, force in N. Convert cm to m and g to kg before substituting.

始终检查单位统一：质量用 kg，速率用 m/s，半径用 m，力用 N。代入前将 cm 换算成 m，g 换算成 kg。

9. Investigating Circular Motion Experimentally | 圆周运动实验探究

A classic IGCSE experiment verifies the relationship F = mv² / r. A rubber bung of known mass m is attached to a string threaded through a glass tube. A weight hanger with slotted masses hangs from the other end, providing a known centripetal force F (equal to the weight of the hanging masses). By whirling the bung in a horizontal circle at a constant radius (measured by a marker on the string), the time for, say, 20 revolutions is recorded to find the period T and hence speed v = 2πr / T.

一个经典的IGCSE实验用来验证关系式 F = mv² / r。已知质量 m 的橡胶塞系在一根穿过玻璃管的绳子上。绳的另一端挂着一个带有槽码的砝码架，它提供已知的向心力 F（等于悬挂砝码的重量）。通过使塞子在水平面内做圆周运动并保持半径恒定（由绳子上的标记测量），记录比如20圈的时间以求出周期 T，进而算出速率 v = 2πr / T。

By varying the hanging mass (and thus F), the radius r, or the mass of the bung m, while keeping other factors constant, you can explore how F depends on m, v, and r. Typical findings confirm that F increases with mass, increases with the square of speed, and decreases with radius for constant speed – fully consistent with F = mv² / r.

通过改变悬挂砝码质量（从而改变 F）、半径 r 或塞子质量 m，并保持其他因素不变，你可以探究 F 与 m、v 及 r 的关系。典型结果证实：F 随质量增大而增大，随速度的平方增大而增大，且当速度不变时随半径增大而减小——完全符合 F = mv² / r。

10. Exam Tips and Common Pitfalls | 考试技巧与常见错误

1. Direction matters: When asked to draw an arrow for velocity or acceleration, always draw velocity tangential to the circle and acceleration (or net force) pointing towards the centre. Labelling them correctly can gain easy marks.

1. 方向很重要：当要求画出速度或加速度的箭头时，始终将速度画在切线方向，加速度（或合力）指向圆心。正确标注可轻松得分。

2. Don’t invent centrifugal force: Avoid using the word ‘centrifugal’ as a real force. Describe the outward sensation as inertia. Many mark schemes specifically penalise the term.

2. 不要发明离心力：避免将“离心”用作真实力。将向外的感觉描述为惯性。许多评分方案专门对该词扣分。

3. Identify the source of centripetal force: In any problem, clearly state which real force (or component) acts centripetally. For a car on a flat bend, it’s friction; for a conical pendulum, it’s the horizontal component of tension.

3. 明确向心力的来源：在任何问题中，清晰说明哪个真实力（或分量）充当向心力。对于水平弯道上的汽车，是摩擦力；对于锥摆，是张力的水平分量。

4. Formula manipulation: Practice rearranging the centripetal force and acceleration equations. You may need to find v from F and r, or r from a and ω. Know how to switch between v, ω, T and f seamlessly.

4. 公式变形：练习对向心力和向心加速度方程进行变形。你可能需要从 F 和 r 求 v，或从 a 和 ω 求 r。熟练在 v、ω、T 和 f 之间无缝切换。

5. Units and conversions: Always convert distance to metres, mass to kilograms, and time to seconds. A common mistake is using cm or minutes directly in the formulas.

5. 单位与换算：始终将距离换算为米，质量换算为千克，时间换算为秒。常见错误是直接在公式中使用厘米或分钟。

6. Vector diagrams: In explanations, mention that though speed is constant, velocity changes because of direction change, resulting in acceleration. This is a favourite definition question.

6. 矢量图：在解释中，提到尽管速率恒定，但因方向改变而导致速度变化，从而产生加速度。这是常考的定义题。

11. Summary | 总结

Uniform circular motion involves constant speed but continuously changing velocity, requiring a centripetal acceleration a = v² / r directed towards the centre. The corresponding centripetal force F = mv² / r is provided by real forces such as tension, gravity or friction. Key relationships include v = ωr, ω = 2π / T, and a = ω² r. Always draw tangent velocity and inward acceleration. Banish the thought of ‘centrifugal force’; it is inertia. With these fundamentals and careful unit handling, you will master IGCSE circular motion questions.

匀速圆周运动涉及恒定速率但不断变化的速度，需要指向圆心的向心加速度 a = v² / r。相应的向心力 F = mv² / r 由真实力（如张力、重力或摩擦力）提供。关键关系包括 v = ωr，ω = 2π / T 和 a = ω² r。始终画出切向速度与向内的加速度。摒弃“离心力”的想法；那是惯性。掌握这些基础并小心处理单位，你就能攻克IGCSE圆周运动考题。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

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