📚 Circular Motion | 圆周运动
Circular motion is a core topic in CIE A-Level Physics that describes the movement of an object along a circular path. Even when the speed is constant, the velocity changes continuously due to the changing direction. This change gives rise to a centripetal acceleration directed towards the centre of the circle. Understanding angular displacement, angular velocity, centripetal force and their applications in horizontal and vertical circles is essential for solving mechanics problems successfully.
圆周运动是CIE A-Level物理的核心课题，描述物体沿圆形路径的运动。即使速率恒定，由于方向不断变化，速度也在持续改变。这种变化产生了指向圆心的向心加速度。理解角位移、角速度、向心力及其在水平和竖直圆周中的应用，对于成功解决力学问题至关重要。
1. Angular Displacement and Radians | 角位移与弧度
Angular displacement θ is the angle through which an object moves on a circular path. It is measured in radians (rad), not degrees, for most formulas to work directly. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.
角位移 θ 是物体在圆形路径上转过的角度。为使大多数公式直接使用，它以弧度（rad）而非度来度量。一弧度是圆心处由长度等于半径的圆弧所张的角。
The arc length s travelled along the circle is related to the angular displacement by s = r θ ans. A full circle corresponds to an angle of 2π radians, meaning 360° = 2π rad.
沿圆周移动的弧长 s 与角位移的关系为 s = r θ，其中 r 为半径。此关系仅在 θ 以弧度为单位时成立。完整的圆对应角度为 2π 弧度，即 360° = 2π rad。
a; = π rad and s = π r. Converting from degrees to radians uses the factor π/180.
使用弧度可简化线量与角量的联系。例如，质点转过半圈，θ = π rad，且 s = π r。从度转换为弧度需乘以因子 π/180。
2. Angular Velocity | 角速度
Angular velocity ω is the rate of change of angular displacement. The average angular velocity is ω = Δθ / Δt. For uniform circular motion, ω is constant and measured in rad s−1.
角速度 ω 是角位移的变化率。平均角速度为 ω = Δθ / Δt。对于匀速圆周运动，ω 恒定，单位为 rad s−1。
revolutions per second, so f = 1/T. Angular velocity is linked to period and frequency by ω = 2π / T = 2π f.
物体以恒定速率做圆周运动，完成一圈所需时间为周期 T。频率 f 为每秒转数，故 f = 1/T。角速度与周期和频率的关系为 ω = 2π / T = 2π f。
minute (rpm), its angular velocity is ω = (3000 × 2π) / 60 = 100π rad s−1. The concept of angular velocity is fundamental when analysing rotating systems and is independent of the radius.
若圆盘以每分钟3000转（rpm）旋转，其角速度为 ω = (3000 × 2π) / 60 = 100π rad s−1。角速度概念是分析转动系统的基础，且与半径无关。
3. Linking Linear and Angular Speed | 线速度与角速度的关联
nce from the centre. The relationship is derived from s = rθ: dividing by time gives v = r ω, where v is the instantaneous linear speed tangent to the circle.
尽管刚体上所有点的角速度相同，线速度 v 却随到中心的距离而变化。此关系从 s = rθ 导出：除以时间得 v = r ω，其中 v 是沿圆周切向的瞬时线速率。
For uniform circular motion, the magnitude of v stays constant, but its direction continuously changes. The velocity vector is always tangent to the circular path. Therefore, even if speed is constant, the object is accelerating because the velocity vector changes direction.
在匀速圆周运动中，v 的大小保持不变，但其方向不断改变。速度矢量始终与圆形路径相切。因此，即使速率恒定，物体也在加速，因为速度矢量的方向在变化。
v = r ω
The equation shows that for a given angular velocity, points farther from the centre have higher linear speeds. This explains why the outer edge of a spinning wheel moves faster than points near the axle.
该方程表明，对于给定的角速度，离中心越远的点具有越大的线速率。这解释了为何旋转车轮的外缘比靠近轴的点移动得更快。
4. Centripetal Acceleration – Concept and Direction | 向心加速度 —— 概念与方向
An object in uniform circular motion experiences an acceleration directed towards the centre of the circle. This acceleration is called centripetal acceleration (ac). It is responsible for changing the direction of velocity without altering its magnitude.
做匀速圆周运动的物体经历指向圆心的加速度，称为向心加速度 (ac)。它负责改变速度的方向而不改变其大小。
The magnitude of centripetal acceleration is given by two equivalent expressions:
向心加速度的大小由两个等价的表达式给出：
ac = v2 / r = r ω2
These formulas show that ac increases with the square of speed for a fixed radius, or increases with radius for a fixed angular velocity. The direction is always radially inward, perpendicular to the velocity.
这些公式表明，对于固定半径，ac 随速率的平方而增大；对于固定的角速度，则随半径增大而增大。其方向始终沿径向向内，与速度垂直。
5. Derivation of a = v2/r | a = v2/r 的推导
points approximately toward the centre. By geometry, the triangle formed by velocity vectors is similar to the triangle AOB. Thus, Δv / v = chord AB / r. For small Δt, chord AB ≈ arc AB = v Δt.
考虑物体以恒定速率 v 在半径 r 的圆上从 A 点移动到 B 点，历时短时间 Δt。速度变化 Δv 近似指向中心。由几何关系，速度矢量三角形与三角形 AOB 相似。因此 Δv / v = 弦 AB / r。对于小的 Δt，弦 AB ≈ 弧 AB = v Δt。
Substituting gives Δv / v = v Δt / r, so Δv / Δt = v2 / r. The instantaneous acceleration is the limit as Δt → 0, yielding a = v2/r. Using v = rω gives the alternative form a = rω2.
代入得 Δv / v = v Δt / r，故 Δv / Δt = v2 / r。当 Δt → 0 时取极限即得瞬时加速度 a = v2/r。利用 v = rω 可得另一形式 a = rω2。
This derivation shows that centripetal acceleration arises purely from the continuous change in direction of the velocity vector. There is no tangential acceleration in uniform circular motion.
此推导表明，向心加速度完全来自速度矢量方向的持续改变。匀速圆周运动中不存在切向加速度。
6. Centripetal Force | 向心力
According to Newton's second law, a net force must act towards the centre to produce centripetal acceleration. This net force is the centripetal force Fc:
根据牛顿第二定律，必须有一个指向中心的净力以产生向心加速度。这个净力就是向心力 Fc：
Fc = m ac = m v2 / r = m r ω2
Centripetal force is not a new type of force; it is the resultant of other forces such as tension, gravity, friction, or the normal reaction. In any circular motion problem, you must identify which physical force or combination provides the centripetal force.
向心力不是一种新型力；它是其他力（如张力、重力、摩擦力或法向反作用力）的合力。在任何圆周运动问题中，必须确定哪个物理力或哪些力的组合提供了向心力。
If the centripetal force suddenly disappears, the object will move off along a straight line tangent to the circle, in accordance with Newton's first law. This is because there is no longer a force to change its direction.
如果向心力突然消失，物体将沿圆的切线方向做直线运动，符合牛顿第一定律。这是因为不再有力改变其方向。
7. Horizontal Circular Motion Examples | 水平圆周运动实例
ss m moving at speed v around a bend of radius r, the required frictional force is f = m v2 / r. The maximum friction is μ m g, giving a maximum safe speed vmax = √(μ g r).
一个常见实例是汽车在平坦路面上转弯。向心力由轮胎与路面之间的摩擦力提供。质量为 m 的汽车以速度 v 绕半径 r 的弯道行驶，所需摩擦力为 f = m v2 / r。最大摩擦力为 μ m g，由此得出最大安全车速 vmax = √(μ g r)。
Another horizontal scenario is a mass on a frictionless table tied to a string passing through a hole, with a hanging weight providing tension. Here the tension equals the weight and supplies exactly the centripetal force for the rotating mass.
另一个水平场景是光滑桌面上的物体，系有一根穿过小孔的绳子，另一端悬挂重物提供张力。此时张力等于悬挂重量，正好为旋转物体提供向心力。
A coin placed on a rotating turntable stays on as long as static friction can supply m r ω2. When the turntable spins too fast, friction is insufficient and the coin slips off tangentially.
置于旋转转盘上的硬币，当静摩擦力能提供 m r ω2 时保持不动。转盘转得过快时，摩擦力不足，硬币沿切线方向滑出。
8. Vertical Circular Motion | 竖直面圆周运动
In vertical circular motion, gravity also plays a role. Consider a mass whirled in a vertical circle on a string. At the top of the circle, the tension Ttop and weight mg both act downwards, contributing to the centripetal force: Ttop + mg = m v2 / r. At the bottom, tension acts upwards and weight downwards: Tbot - mg = m v2 / r.
在竖直面圆周运动中，重力也起作用。考虑用绳子在竖直面内旋转的质量。在最高点，张力 Ttop 和重力 mg 均向下，贡献向心力：Ttop + mg = m v2 / r。在最低点，张力向上而重力向下：Tbot - mg = m v2 / r。
The minimum speed at the top for the mass to maintain a circular path occurs when the string just goes slack, Ttop = 0. Then mg = m vmin2 / r, giving vmin = √(g r). This is the critical speed at the top.
在最高点维持圆形路径的最小速率发生在绳子刚好松弛时，Ttop = 0。此时 mg = m vmin2 / r，得出 vmin = √(g r)。这就是最高点的临界速率。
ler coaster, the normal reaction from the track replaces tension. At the top, N + mg = m v2 and the condition for just maintaining contact is N = 0, yielding the same critical speed.
对于过山车的环形轨道，轨道法向反力取代了张力。在顶部，N + mg = m v2 / r，保持刚接触的条件为 N = 0，得出相同的临界速率。
9. Conical Pendulum | 圆锥摆
eta; to the vertical. Resolving forces vertically: T cos θ = mg. Resolving horizontally: T sin θ = m r ω2, where r = L sin θ and L is the string length.
圆锥摆由一个系在绳子上的质量组成，在水平面内绕圈，绳子扫出一个圆锥面。绳子与竖直方向夹角为 θ。竖直方向分解力：T cos θ = mg。水平方向分解力：T sin θ = m r ω2，其中 r = L sin θ，L 为绳长。
Dividing the two equations gives tan θ = (r ω2) / g = (L sin θ ω2) / g. This simplifies to ω = √(g / (L cos θ)). The period of rotation is therefore Tperiod = 2π √(L cos θ / g).
两式相除得 tan θ = (r ω2) / g = (L sin θ ω2) / g。简化后得 ω = √(g / (L cos θ))。因此旋转周期为 Tperiod = 2π √(L cos θ / g)。
Notice that the period depends on L and θ but not on the mass. As θ increases, the bob moves in a larger circle and the period decreases slightly. The conical pendulum is a classic problem for practising force resolution in circular motion.
注意周期取决于 L 和 θ，而与质量无关。随着 θ 增大，摆球在更大的圆周上运动，周期略有减小。圆锥摆是练习圆周运动受力分解的经典问题。
10. Banked Tracks | 倾斜弯道
, the horizontal component of the normal reaction can provide the centripetal force. In the ideal frictionless case: N sin θ = m v2 / r and N cos θ = mg.
公路和铁轨常设计成倾斜，以便无需完全依赖摩擦即可允许更高速度。对于倾角 θ、半径 r 的倾斜弯道，法向反力的水平分量可提供向心力。在理想无摩擦情况下：N sin θ = m v2 / r，N cos θ = mg。
Dividing yields tan θ = v2 / (r g). The design speed or ideal speed for a frictionless banked track is therefore videal = √(r g tan θ). At this speed, no frictional force is needed to stay on the curved path.
两式相除得 tan θ = v2 / (r g)。因此无摩擦倾斜轨道的设计速度或理想速度为 videal = √(r g tan θ)。以此速度行驶时，不需要摩擦力即可保持在弯道上。
If friction is present, extra forces act along the slope. For a car travelling faster than the ideal speed, friction acts down the slope to prevent sliding outward. At lower speeds, friction acts up the slope. Both cases can be analysed with careful resolution.
若存在摩擦，沿斜面会有额外力作用。当汽车速度超过理想速度时，摩擦力沿斜面向下以防止向外滑出。速度较低时，摩擦力沿斜面向上。两种情况都可通过认真分解力来分析。
11. Key Equations and Problem-Solving Tips | 关键方程与解题技巧
Mastering circular motion requires a firm grasp of the essential equations and the ability to identify the source of centripetal force. Below is a summary of the key formulas:
掌握圆周运动需要牢固掌握基本方程，并能识别向心力的来源。以下是关键公式总结：
| Equation | 中文 |
|---|---|
| s = r θ | 弧长与角位移 |
| ω = 2π/T = 2π f | 角速度与周期、频率 |
| v = r ω | 线速度与角速度 |
| a = v2/r = r ω2 | 向心加速度 |
| F = m v2/r = m r ω2 | 向心力 |
| vmin at top = √(g r) | 竖直圆最高点最小速率 |
| tan θ = v2/(r g) (banked, no friction) | 倾斜弯道无摩擦tan关系 |
Always draw a free-body diagram and clearly indicate the direction of centripetal acceleration (towards the centre). Equate the net force in the radial direction to m v2/r or m r ω2. Remember that centripetal force is the resultant; do not add it as a separate force on the diagram.
始终绘制受力图，清晰标出向心加速度方向（指向中心）。将径向净力与 m v2/r 或 m r ω2 等同起来。切记向心力是合力，不要在图上将其作为一个单独的力额外添加。
Check whether the motion is
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