Circular Motion for IGCSE OCR Physics: Key Exam Points | IGCSE OCR 物理：圆周运动考点精讲

📚 Circular Motion for IGCSE OCR Physics: Key Exam Points | IGCSE OCR 物理：圆周运动考点精讲

Circular motion is a fundamental topic in IGCSE OCR Physics, describing the movement of objects along a circular path at constant speed. Even though the speed does not change, the velocity is constantly changing due to the continuous change in direction, which means the object is accelerating. This centripetal acceleration points toward the centre of the circle and requires a net inward force – the centripetal force – to keep the object moving in a circle. Understanding these principles, along with the relationships between linear speed, angular speed, radius, period and frequency, is essential for tackling exam questions on everything from satellites to roller coasters.

圆周运动是IGCSE OCR物理中的基础课题，描述物体沿圆形轨迹以恒定速率运动的规律。虽然速率不变，但由于方向时刻改变，速度矢量始终在变化，因此物体具有加速度。这种向心加速度指向圆心，需要有一个指向圆心的净力——向心力——来维持圆周运动。掌握这些原理，以及线速度、角速度、半径、周期和频率之间的关系，对于解答从人造卫星到过山车等各类考题至关重要。

1. Defining Circular Motion: Speed vs Velocity | 圆周运动的定义：速率与速度的区别

In uniform circular motion, an object travels in a circle at a constant speed. However, because velocity is a vector quantity that depends on both magnitude and direction, the velocity is always changing as the object turns. This changing velocity means the object experiences acceleration, even if its speedometer reading never changes. In IGCSE OCR exam questions, you must be able to state that the speed is constant but the velocity is not, and you must link this to the presence of a resultant force.

在匀速圆周运动中，物体以恒定速率绕圆运动。但速度是矢量，既包含大小也包含方向，因此当物体转弯时速度不断变化。这种速度的变化意味着物体在加速，即使速率保持不变。在IGCSE OCR考题中，你需要明确指出速率恒定但速度不恒定，并将其与合外力的存在联系起来。

Speed: scalar, constant magnitude in uniform circular motion.
速率：标量，匀速圆周运动中大小恒定。
Velocity: vector, continuously changing direction.
速度：矢量，方向不断改变。
Acceleration: present because velocity changes, directed toward centre.
加速度：由于速度变化而存在，方向指向圆心。

2. Centripetal Acceleration and Its Direction | 向心加速度及其方向

Centripetal acceleration is the acceleration experienced by an object moving in a circle, always pointing toward the centre of the circle. Its magnitude is given by a = v² / r, where v is the linear speed and r is the radius of the circular path. This acceleration is responsible for changing the direction of the velocity without changing its magnitude. In the exam, you may be asked to explain why an object moving in a circle is accelerating even if the speed is uniform, and you must draw or identify the direction of the acceleration vector.

向心加速度是物体做圆周运动时经历的一种加速度，始终指向圆心。其大小由公式 a = v² / r 给出，其中 v 是线速度，r 是圆周半径。该加速度负责改变速度的方向而不改变其大小。考试中可能会要求你解释为什么匀速圆周运动的物体仍在加速，并画出或标识出加速度矢量的方向。

a = v² / r

When speed doubles, centripetal acceleration quadruples for the same radius.
当速度加倍时，在相同半径下向心加速度变为原来的四倍。
If radius doubles at the same speed, acceleration halves.
若半径加倍而速度不变，加速度减半。

3. Centripetal Force: The Net Force Causing Circular Motion | 向心力：产生圆周运动的净力

Centripetal force is not a new type of force; it is simply the resultant force directed toward the centre of the circle. It can be provided by tension, gravity, friction, or the normal reaction, depending on the situation. The magnitude of the centripetal force is F = mv² / r, where m is the mass of the object. In IGCSE OCR Physics, you must be able to identify which force acts as the centripetal force in various scenarios and apply this formula correctly.

向心力并不是一种新的力，而是指向圆心的净合力。它可以由张力、重力、摩擦力或法向支持力提供，具体取决于情境。其大小由 F = mv² / r 给出，其中 m 是物体的质量。在IGCSE OCR物理中，你必须能够辨别不同场景下哪种力充当向心力，并正确应用该公式。

F = mv² / r

If mass doubles, centripetal force doubles (for same v and r).
质量加倍，向心力加倍（在v和r不变时）。
If speed increases, force increases proportionally to the square of speed – a key exam point.
速度增加，力按速度的平方增加——这是考试重点。

4. Common Sources of Centripetal Force | 向心力的常见来源

Examiners love asking students to identify the centripetal force in real‑world situations. A few classic examples include: a car rounding a bend – friction between the tyres and the road; a satellite orbiting Earth – gravitational attraction; an object swung on a string – tension in the string; and a roller coaster car at the top of a loop – a combination of weight and normal reaction. Always state specifically which force provides the centripetal requirement, rather than simply saying ‘centripetal force’.

考官喜欢考查学生在现实场景中辨认向心力的能力。一些经典例子包括：汽车转弯——轮胎与地面的摩擦力；绕地运行的人造卫星——万有引力；绳子末端旋转的物体——绳子的张力；环形轨道顶部的过山车——重力与法向支持力的合力。务必具体指明哪种力提供向心力，而不要只说“向心力”一词。

Situation / 情境	Centripetal Force / 向心力
Car on a roundabout	Friction / 摩擦力
Moon orbiting Earth	Gravity / 万有引力
Conical pendulum	Horizontal component of tension / 张力的水平分量
Electron around nucleus (simple model)	Electrostatic attraction / 静电力

5. Angular Velocity: A New Way to Describe Rotation | 角速度：描述转动的新方式

Angular velocity (ω) is the rate of change of angular displacement, measured in radians per second (rad/s). It links directly to linear speed through the relationship v = ωr. Many IGCSE OCR questions involve converting between linear and angular quantities, especially when dealing with objects moving on circular paths of different radii. One full rotation corresponds to an angle of 2π radians, which ties period and angular velocity together: ω = 2π / T, where T is the period.

角速度（ω）是角位移变化的快慢，单位为弧度每秒（rad/s）。它通过关系式 v = ωr 与线速度直接相关。许多IGCSE OCR考题涉及线量与角量之间的转换，特别是在处理不同半径圆周运动时。一整圈对应 2π 弧度，由此可将周期与角速度联系起来：ω = 2π / T，其中 T 为周期。

v = ωr

ω = 2π / T

If two objects are on the same rotating disc, they have the same angular velocity but different linear speeds (v ∝ r).
同一转盘上的两个物体具有相同的角速度，但线速度不同（v ∝ r）。

6. Period, Frequency and Their Connection to Speed | 周期、频率及其与速度的关系

The period (T) is the time taken for one complete revolution, and frequency (f) is the number of revolutions per second, related by f = 1/T. In circular motion, linear speed v can also be expressed using circumference and period: v = 2πr / T. Similarly, v = 2πrf. These formulas are frequently used in exam calculations, especially for objects like satellites or spinning laboratory masses, where time and rotation counts are given.

周期（T）是完成一整圈所需的时间，频率（f）是每秒转动的圈数，两者满足 f = 1/T。在圆周运动中，线速度 v 也可用周长和周期表示：v = 2πr / T；同样，v = 2πrf。这些公式在考试计算中经常使用，尤其是针对人造卫星或实验室中旋转重物的情形，题目会给出时间和转数。

v = 2πr / T = 2πrf

Frequency in hertz (Hz) is simply the reciprocal of the period in seconds.
频率的单位是赫兹（Hz），等于周期（秒）的倒数。
Count how many rotations occur in a given time to find frequency or period for use in speed calculations.
数出给定时间内的转数即可求出频率或周期，用于速度计算。

7. Experimental Investigation of Circular Motion | 圆周运动的实验探究

In the lab, circular motion can be studied by swinging a rubber bung attached to a string that passes through a glass tube, with a known mass hanging on the other end. The weight of the hanging mass provides the centripetal force, which can be changed by adding or removing masses. By measuring the time for a number of revolutions and the radius of the circle, you can verify the relationship F = mv²/r or explore how angular speed varies with radius. In the IGCSE OCR exam, you could be asked to describe such an experiment, identify safety precautions, or interpret data from it.

在实验室中，可以通过旋转一个绑在绳子上的橡皮塞来研究圆周运动，绳子穿过一根玻璃管，另一端悬挂已知质量的重物。悬挂物的重力提供向心力，可通过增减质量来改变。通过测量多圈所用的时间和圆的半径，可以验证 F = mv²/r 或探究角速度如何随半径变化。在IGCSE OCR考试中，可能会要求描述此类实验、指出安全注意事项或解析实验数据。

Safety: wear goggles, keep hands clear of rotating parts, use a strong string.
安全事项：佩戴护目镜，手部远离旋转部件，使用结实的绳子。
Independent variable: typically hanging mass (centripetal force) or radius.
自变量：通常是悬挂质量（向心力）或半径。
Dependent variable: period or speed, measured with a stopwatch.
因变量：周期或速度，用秒表测量。
Control variables: length of string, mass of bung (unless testing it).
控制变量：绳长、橡皮塞质量（除非作为测试变量）。

8. Banking and Bends: Applying Circular Motion to Vehicles | 倾斜弯道：圆周运动在车辆中的应用

When a car travels around a bend, the centripetal force is usually provided by friction between the tires and the road. If the road is banked (tilted), a component of the normal reaction can contribute to the centripetal force, reducing the reliance on friction and allowing higher speeds safely. IGCSE OCR exam questions may ask you to explain why roads are banked on sharp turns, or to calculate the ideal speed for a banked curve without friction. At this level, qualitative understanding and simple force resolutions are expected.

当汽车驶过弯道时，向心力通常由轮胎与地面间的摩擦力提供。若路面设有倾斜（超高），法向支持力的一个水平分量可以贡献向心力，从而减少对摩擦的依赖，使更高速度过弯成为可能。IGCSE OCR考题可能会问为什么急弯处要修成倾斜路面，或者在不考虑摩擦时计算理想的倾斜弯道车速。本阶段要求定性理解和简单的力分解。

On a flat bend, centripetal force = friction; skidding occurs if friction limit is exceeded.
在平弯道上，向心力 = 摩擦力；若超过最大静摩擦会打滑。
On a banked track, a portion of the normal force acts horizontally inward, assisting the turn.
在倾斜轨道上，支持力的水平分量指向内侧，辅助转弯。

9. Vertical Circular Motion and “Loop-the-Loop” | 竖直面内的圆周运动与“翻筋斗”

Vertical circular motion is more complex because the speed and the required centripetal force change with position, and gravity plays a direct role. A classic example is a roller coaster car going through a loop: at the top, both weight and the normal reaction from the track act downward, together providing the centripetal force. The minimum speed at the top is achieved when the normal reaction becomes zero, so weight alone provides the necessary centripetal force: mg = mv² / r, giving v = √(gr). In exams, you may be asked to calculate this minimum speed or explain why it is necessary.

竖直面内的圆周运动更为复杂，因为速度和所需的向心力随位置变化，且重力直接参与作用。一个经典的例子是过山车通过环形轨道：在顶部，重力与轨道的法向支持力均向下，共同提供向心力。顶部的最小速度发生在支持力减小到零时，此时仅由重力提供向心力：mg = mv² / r，得出 v = √(gr)。考试中可能要求计算这一最小速度，或解释其必要性。

At the top, minimum speed: v = √(gr)

在顶部，最小速度：v = √(gr)

At the bottom of a loop, the upward normal force must be greater than the weight to provide the extra centripetal force.
在环底部，向上的支持力必须大于重力，以提供额外的向心力。
Always draw a free-body diagram to see which forces act toward the centre at each point.
务必画出受力分析图，看在每一点哪些力指向圆心。

10. Satellites and Gravitational Orbits | 人造卫星与引力轨道

Satellites stay in orbit because the gravitational force between the satellite and the planet acts as the centripetal force. For a satellite in a stable circular orbit, we have GMm / r² = mv² / r, where M is the planet’s mass, m is the satellite’s mass, and r is the orbital radius. Simplifying gives v² = GM / r, showing that orbital speed decreases with increasing radius. IGCSE OCR questions often ask you to compare speeds of different satellites or explain why a satellite does not fall back to Earth even though gravity pulls it inward.

人造卫星能在轨道上运行，是因为卫星与行星之间的万有引力充当了向心力。对于稳定的圆轨道，有 GMm / r² = mv² / r，其中 M 为行星质量，m 为卫星质量，r 为轨道半径。化简得 v² = GM / r，表明轨道半径越大，运行速度越小。IGCSE OCR考题常要求学生比较不同卫星的速度，或解释为何卫星受到向内引力却不会坠回地球。

v² = GM / r

Geostationary satellites have a period of 24 hours and orbit above the equator; their radius is fixed.
地球同步卫星的周期为 24 小时，位于赤道上空，轨道半径固定。
Lower orbits have higher speeds and shorter periods – a key comparison to remember.
较低轨道有更高速度和更短周期——这是需要牢记的关键对比。

11. Solving Exam Problems: Tips and Common Pitfalls | 答题技巧与常见易错点

When tackling circular motion questions, always note whether the motion is horizontal or vertical, identify what provides the centripetal force, and sketch a diagram. Use consistent units: speed in m/s, radius in m, force in N. Beware of forgetting to square the speed in a = v²/r or F = mv²/r. If a question asks for the force exerted by one object on another (e.g., the force a seat exerts on a pilot at the top of a loop), do not simply give ‘mv²/r’; you must consider all forces and solve for the contact force.

解答圆周运动问题时，务必注意到运动是水平还是竖直面内的，确定向心力的来源，并画出示意图。单位要保持一致：速度用 m/s，半径用 m，力用 N。注意不要漏掉公式 a = v²/r 或 F = mv²/r 中速度的平方。如果考题问的是某一物体对另一物体的作用力（例如环形顶部座椅对飞行员的力），不要直接给出“mv²/r”；必须综合考虑所有力并求解接触力。

Common mistake: using diameter instead of radius – always halve given diameters.
常见错误：用直径代替半径——务必将已知直径除以 2。
When given rpm (revolutions per minute), convert to frequency in Hz or period in seconds first.
已知转速（转每分）时，先转换为赫兹频率或秒为单位的周期。
Label forces clearly in free-body diagrams: weight, tension, normal reaction, friction.
在受力图中清晰标注各力：重力、张力、支持力、摩擦力。

12. Key Equations Summary for Quick Revision | 核心公式速查表

Having a concise formula sheet in mind can save time in the exam. Below is a table of the most important equations for IGCSE OCR circular motion, along with brief notes on when to use each one. Always check which quantities are given and which are asked for before selecting the equation.

脑中存有一份简明的公式表可以在考试中节省时间。下表列出了IGCSE OCR圆周运动最重要的公式，并附有简要说明，提示何时使用。选择公式前，务必先检查已知量和待求量。

Equation / 公式	Use when / 适用情况
v = ωr	Converting between linear and angular speed / 线速度与角速度转换
a = v² / r	Finding centripetal acceleration / 求向心加速度
F = mv² / r	Calculating centripetal force / 计算向心力
ω = 2π / T = 2πf	Relating angular velocity to period/frequency / 角速度与周期/频率的关系
v = 2πr / T	Finding speed from circumference and period / 由周长和周期求速度
v = √(GM / r) (orbits)	Orbital speed of satellite (given M and r) / 卫星轨道速度（已知 M 和 r）

Remember: these equations assume a constant radius and uniform speed. If friction or air resistance is mentioned, you may need to consider energy changes, but for the core IGCSE OCR assessment, uniform circular motion is the focus.

请记住：这些公式假设半径不变且速率恒定。如果题目提到摩擦或空气阻力，可能需要考虑能量变化，但对于IGCSE OCR核心评价而言，匀速圆周运动才是重点。

Published by TutorHao | IGCSE Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Circular Motion for IGCSE OCR Physics: Key Exam Points | IGCSE OCR 物理：圆周运动 考点精讲