📚 Centripetal Forces | 向心力
Centripetal force is the resultant force directed towards the centre of a circular path, required to keep an object moving in that circle. In A-Level Physics, understanding centripetal force brings together Newton’s laws, vector analysis, and practical applications such as vehicle cornering, satellites, and the conical pendulum. This revision guide covers the key definitions, equations, sources, and exam skills for the CIE A-Level syllabus.
向心力是指向圆周运动路径中心的合力，是维持物体做圆周运动所必需的。在A-Level物理中，理解向心力需要结合牛顿定律、矢量分析以及车辆转弯、卫星和圆锥摆等实际应用。本复习指南涵盖CIE A-Level大纲中的关键定义、公式、来源和考试技巧。
1. What is Centripetal Force? | 什么是向心力？
When an object moves in a circle at constant speed, its velocity is still changing because the direction of motion changes continuously. Since velocity is a vector, a change in direction means the object is accelerating. Newton’s second law therefore requires a net force acting towards the centre of the circle. This net force is called the centripetal force.
物体以恒定速率做圆周运动时，由于运动方向不断改变，速度矢量仍在变化。速度是矢量，方向改变意味着物体在加速。因此牛顿第二定律要求有一个指向圆心的净力。这个净力就叫做向心力。
It is essential to understand that centripetal force is not a new type of force. It is simply the name given to the resultant of real forces such as tension, friction, gravity, or the normal reaction when that resultant is directed towards the centre of a circular path.
必须理解向心力不是一种新的力。它只是对指向圆心的实际力（如张力、摩擦力、重力或支持力）的合力所起的名称。
F = mv² / r = mω²r
Here m is the mass, v is the linear speed, r is the radius, and ω is the angular speed.
其中 m 是质量，v 是线速度，r 是半径，ω 是角速度。
2. Direction and Work Done | 方向与做功
The centripetal force always points towards the centre of the circle and is always perpendicular to the instantaneous velocity. Since the velocity is tangent to the circle, the angle between the centripetal force and the displacement at any instant is 90°.
向心力始终指向圆心，并且始终垂直于瞬时速度。由于速度沿圆周切线方向，向心力与任一瞬间位移的夹角为90°。
Work done is given by W = F s cos θ. Because θ = 90°, we have cos 90° = 0, so W = 0. This means the centripetal force does no work on the object. In uniform circular motion, the speed remains constant and only the direction of motion changes.
功的公式为 W = F s cos θ。因为 θ = 90°，cos 90° = 0，所以 W = 0。这意味着向心力对物体不做功。在匀速圆周运动中，速率保持不变，只有运动方向改变。
If a tangential force is also present, the speed will change and the motion is non-uniform circular motion. In that case the total force has both a tangential component and a radial centripetal component.
如果还存在切向力，速率就会改变，形成非匀速圆周运动。此时总力既有切向分量，也有径向的向心分量。
3. Centripetal Acceleration | 向心加速度
An object moving in a circle of radius r at constant speed v has an acceleration directed towards the centre. This acceleration is called centripetal acceleration. Its magnitude is given by:
物体以恒定速率 v 绕半径 r 的圆周运动时，具有指向圆心的加速度，称为向心加速度。其大小为：
a = v² / r = rω²
Since v = rω, the two forms are equivalent. Angular speed ω is measured in rad s⁻¹, which ensures the relationships are dimensionally consistent.
由于 v = rω，这两种形式是等价的。角速度 ω 的单位是 rad s⁻¹，这保证了公式在量纲上的一致性。
The direction of the acceleration is the same as the direction of the net force: towards the centre. Although the object ‘accelerates’ towards the centre, it never reaches the centre because its tangential velocity keeps it moving around the circle.
加速度的方向与净力方向相同：指向圆心。虽然物体持续指向圆心加速，但由于存在切向速度，它永远不会到达圆心，而是围绕圆周运动。
4. Key Equations and Relationships | 核心公式与关系
The following table summarises the essential quantities, symbols, and equations for circular motion. You should be able to rearrange these confidently under exam conditions.
下表总结了圆周运动中的基本物理量、符号和公式。你应当能够在考试条件下熟练地改写这些公式。
| Quantity | Symbol | Equation |
| Linear speed | v | v = rω |
| Angular speed | ω | ω = 2π / T = 2πf |
| Centripetal acceleration | a | a = v² / r = rω² |
| Centripetal force | F | F = mv² / r = mω²r |
| Period and frequency | T, f | T = 1 / f |
Remember that one full revolution corresponds to an angular displacement of 2π radians. Therefore the period T is the time for one revolution, and the frequency f is the number of revolutions per second.
记住一整圈对应的角位移是 2π 弧度。因此周期 T 是完成一圈的时间，频率 f 是每秒的圈数。
When solving problems, first identify which physical force or forces provide the centripetal force. Then equate that resultant force to mv² / r or mω²r. Do not add a separate ‘centripetal force’ to the free-body diagram if the force has already been included as tension, friction, or another real force.
解题时，首先确定哪个或哪些实际力提供向心力。然后将该合力等于 mv² / r 或 mω²r。如果某个力已经作为张力、摩擦力或其他实际力出现在受力图中，不要重复添加一个单独的向心力。
5. Sources of Centripetal Force | 向心力的来源
Centripetal force can be provided by different types of forces depending on the situation. The table below gives common examples.
向心力可以根据具体情况由不同类型的力提供。下表给出一些常见例子。
| Situation | Force providing centripetal force |
| Rubber bung whirled on a string | Tension in the string |
| Car turning on a flat road | Friction between tyres and road |
| Satellite orbiting Earth | Gravitational attraction |
| Electron orbiting a nucleus | Electrostatic attraction |
| Roller coaster at the top of a loop | Weight plus normal reaction |
In each case, the named force is not an additional centripetal force; it is the actual physical force whose resultant acts towards the centre.
在每种情况下，所列的力都不是额外的向心力；它是实际存在的物理力，其合力指向圆心。
6. Horizontal Circular Motion Examples | 水平圆周运动实例
A classic experiment is a rubber bung tied to a string and whirled in a horizontal circle. The tension in the string provides the centripetal force. If the bung has mass m, speed v, and the radius of the circle is r, then the tension must satisfy T = mv² / r.
一个经典实验是用细绳系住橡皮塞并使其在水平面内旋转。绳中的张力提供向心力。如果橡皮塞质量为 m、速度为 v、圆周半径为 r，则张力必须满足 T = mv² / r。
When a car turns on a level road, the horizontal frictional force between the tyres and the road provides the centripetal force. The maximum speed for a safe turn is limited by the maximum static friction, given by F_max = μR, where R is the normal reaction and μ is the coefficient of friction.
当汽车在水平路面上转弯时，轮胎与路面之间的水平摩擦力提供向心力。安全转弯的最大速度受到最大静摩擦力的限制，即 F_max = μR，其中 R 是支持力，μ 是摩擦系数。
A coin placed on a rotating horizontal turntable will remain in circular motion as long as static friction can provide the required mv² / r. If the turntable rotates too fast, the required centripetal force exceeds the maximum friction and the coin slides outward relative to the turntable.
放在旋转水平转盘上的硬币，只要静摩擦力能够提供所需的 mv² / r，就能保持圆周运动。如果转盘旋转过快，所需向心力超过最大摩擦力，硬币就会相对于转盘向外滑动。
7. Vehicle Cornering and Banked Tracks | 车辆转弯与倾斜轨道
On a flat road, the centripetal force for a cornering vehicle comes entirely from sideways friction. This can be unsafe in wet or icy conditions because friction is reduced. To reduce the reliance on friction, roads and railway tracks are often banked at an angle θ to the horizontal.
在平路上，转弯车辆所需的向心力完全来自侧向摩擦力。在湿滑或结冰条件下，摩擦力减小，这可能不安全。为了减少对摩擦的依赖，道路和铁路轨道通常相对于水平面倾斜一个角度 θ。
On a banked track, the normal reaction R has a vertical component R cos θ that balances the weight mg, and a horizontal component R sin θ that acts towards the centre of the curve. At the ideal banking speed, no sideways friction is needed, and the horizontal component alone provides the centripetal force.
在倾斜轨道上，支持力 R 的竖直分量 R cos θ 平衡重力 mg，水平分量 R sin θ 指向弯道中心。在理想倾斜速度下，不需要侧向摩擦，仅水平分量就提供向心力。
R cos θ = mg
R sin θ = mv² / r
Dividing the second equation by the first gives the ideal banking condition:
用第二个方程除以第一个方程，得到理想倾斜条件：
tan θ = v² / (rg)
This equation is useful for calculating the design speed for a banked curve or the required banking angle for a given speed.
该方程可用于计算倾斜弯道的设计速度，或给定速度所需的倾斜角度。
8. The Conical Pendulum | 圆锥摆
A conical pendulum consists of a small mass attached to a string and moving in a horizontal circle so that the string traces out a cone
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