📚 Circular Motion: Key Kinematic Descriptions | 圆周运动：运动学描述要点
Circular motion is one of the most important topics in A-Level physics. From a car going around a roundabout to a planet orbiting a star, objects often move along curved paths. To describe such motion precisely, we need angular quantities such as angular displacement, angular speed, period and centripetal acceleration. This article explains these ideas step by step, focusing on the kinematics of circular motion: the vocabulary and equations that describe motion without yet considering forces.
圆周运动是 A-Level 物理中最重要的主题之一。从汽车绕环岛行驶到行星绕恒星运行，物体经常沿着弯曲路径运动。为了精确描述这种运动，我们需要角位移、角速度、周期和向心加速度等角量。本文逐步解释这些概念，重点放在圆周运动的运动学描述上：即在不考虑力的情况下，用来描述运动的术语和方程。
1. Why Study Circular Motion? | 为什么要研究圆周运动？
Circular motion appears in countless real-world situations. A satellite in orbit, a cyclist rounding a bend and a proton in a particle accelerator all follow circular or nearly circular paths. In the CIE A-Level syllabus, circular motion forms the bridge between linear kinematics and dynamics. Mastering the kinematic description first makes it much easier to understand centripetal force later.
圆周运动出现在无数真实情境中。在轨卫星、转弯的自行车骑行者、粒子加速器中的质子，都在沿圆形或近似圆形的路径运动。在 CIE A-Level 考试大纲中，圆周运动是连接直线运动学与动力学的桥梁。先掌握运动学描述，后面理解向心力就会容易得多。
Kinematics of circular motion focuses on how the position, angle, speed and acceleration of an object change with time. It does not ask why the motion happens; that belongs to dynamics. This article keeps strictly to kinematic descriptions, using radians as the natural unit of angle.
圆周运动的运动学关注物体的位置、角度、速度和加速度如何随时间变化。它不探讨运动发生的原因，那属于动力学范畴。本文严格限于运动学描述，并使用弧度作为角度的自然单位。
2. Radians: The Natural Unit of Angle | 弧度：角度的自然单位
In circular motion, angles are usually measured in radians rather than degrees. The radian is defined from geometry: the angle θ in radians is the ratio of the arc length s travelled along the circle to the radius r of that circle.
在圆周运动中，角度通常用弧度而不是度来度量。弧度由几何定义：弧度制下的角度 θ 等于物体沿圆弧走过的弧长 s 与该圆半径 r 之比。
θ = s / r
Rearranging gives the arc length formula s = rθ. One complete revolution corresponds to the full circumference 2πr, so θ = 2πr / r = 2π rad. Therefore 360° = 2π rad, or π rad = 180°. A useful approximation is 1 rad ≈ 57.3°.
整理后得到弧长公式 s = rθ。一整圈对应整个圆周 2πr，所以 θ = 2πr / r = 2π rad。因此 360° = 2π rad，或者说 π rad = 180°。常用近似为 1 rad ≈ 57.3°。
Using radians keeps equations simple because trigonometric and angular-speed formulas are derived with radians. In A-Level exams, always convert degrees to radians before calculating angular speed or arc length.
使用弧度可以使公式保持简洁，因为三角函数和角速度公式都是在弧度制下推导出来的。在 A-Level 考试中，计算角速度或弧长之前，一定要先把角度从度转换为弧度。
3. Angular Displacement and Angular Velocity | 角位移与角速度
Angular displacement θ describes the angle swept out by the radius line as an object moves around a circle. It is measured in radians. If an object moves 2.5 complete circles, its angular displacement is 5π rad, not 2.5π rad, because each circle contributes 2π rad.
角位移 θ 描述物体绕圆运动时半径线扫过的角度，单位是弧度。如果物体转了 2.5 整圈，其角位移是 5π rad，而不是 2.5π rad，因为每一圈贡献 2π rad。
Angular velocity ω is the rate of change of angular displacement. For constant angular speed, it is defined as
角速度 ω 是角位移随时间的变化率。在角速度恒定时，它定义为
ω = Δθ / Δt
The SI unit of angular velocity is radian per second, written rad s⁻¹. For a very small time interval, this definition gives the instantaneous angular velocity. Although ω is often treated as a scalar in A-Level kinematics, it can also be represented as a vector pointing along the axis of rotation.
角速度的 SI 单位是弧度每秒，写作 rad s⁻¹。对于非常小的时间间隔，该定义得到瞬时角速度。在 A-Level 运动学中，ω 通常被视为标量，但它也可以表示为沿转轴方向的矢量。
Angular velocity tells us how fast the angle changes, independent of the radius. Two points on the same spinning disc have the same ω, but their linear speeds are different if their radii are different.
角速度告诉我们角度变化的快慢，与半径无关。同一旋转圆盘上的两个点具有相同的 ω，但如果半径不同，它们的线速度也不同。
4. Period and Frequency | 周期与频率
The period T is the time taken for one complete revolution, measured in seconds. The frequency f is the number of complete revolutions per second, measured in hertz (Hz) or s⁻¹. They are reciprocals of each other:
周期 T 是完成一整圈运动所用的时间，单位为秒。频率 f 是每秒钟完成的完整圈数，单位为赫兹 (Hz) 或 s⁻¹。它们互为倒数：
T = 1 / f
Since one complete revolution is an angular displacement of 2π rad, the angular velocity is directly related to period and frequency:
由于一整圈对应的角位移是 2π rad，角速度与周期和频率直接相关：
ω = 2π / T = 2π f
For example, if a wheel completes 5 revolutions per second, then f = 5 Hz, T = 0.2 s, and ω = 2π × 5 ≈ 31.4 rad s⁻¹. These conversions appear frequently in exam problems, so remember them well.
例如，如果一个轮子每秒完成 5 圈，则 f = 5 Hz，T = 0.2 s，ω = 2π × 5 ≈ 31.4 rad s⁻¹。这些换算在考试题目中经常出现，一定要牢记住。
5. Linear Speed and Its Relation to Angular Speed | 线速度与角速度的关系
The linear speed v of an object moving in a circle is the distance travelled along the circumference per unit time. For one full revolution, the distance is 2πr and the time is T, so the average speed is
物体做圆周运动的线速度 v 是单位时间内沿圆周运动过的距离。一整圈的距离为 2πr，时间为 T，因此平均速度为
v = 2πr / T
Using ω = 2π / T, we obtain the key relationship
利用 ω = 2π / T，我们得到关键关系式
v = r ω
The direction of the linear velocity is always tangent to the circular path at the position of the object. This is why the linear velocity is sometimes called the tangential velocity. In uniform circular motion, the magnitude v is constant, but the direction changes continuously.
线速度的方向始终在物体所在位置与圆相切。因此线速度有时也称为切向速度。在匀速圆周运动中，v 的大小恒定，但方向不断变化。
It is important not to confuse v with ω. The angular speed ω is the same for the whole object, while the linear speed v depends on the distance from the axis. If you double the radius while keeping ω constant, the linear speed doubles.
注意不要把 v 和 ω 混淆。角速度 ω 对整个物体相同，而线速度 v 取决于到转轴的距离。如果保持 ω 不变而半径加倍，线速度也加倍。
6. Why Is Circular Motion Accelerated? | 为什么圆周运动是加速运动？
Acceleration is defined as the rate of change of velocity. Velocity is a vector, so a change in direction counts as a change in velocity, even if the speed remains constant. In uniform circular motion, the speed is constant but the direction of motion is continuously turning. Therefore the object is accelerating.
加速度定义为速度的变化率。速度是矢量，因此方向的变化也属于速度变化，即使速度大小保持不变。在匀速圆周运动中，速度大小不变，但运动方向不断转变，因此物体具有加速度。
For uniform circular motion, the acceleration is always directed towards the centre of the circle. This inward-pointing acceleration is called centripetal acceleration, from the Latin word for “seeking the centre”. It is perpendicular to the velocity vector at every instant.
在匀速圆周运动中，加速度始终指向圆心。这个指向内侧的加速度称为向心加速度，拉丁语意为“寻求圆心”。它在每一瞬间都与速度矢量垂直。
This idea surprises many students: acceleration does not always mean speeding up. A satellite moving at constant speed in orbit is still accelerating because its direction is changing.
这个概念让许多学生惊讶：加速度并不总是意味着加速。在轨道上匀速运行的卫星仍然在加速，因为它的方向在不断改变。
7. Centripetal Acceleration: Formula and Derivation | 向心加速度：公式与推导
Consider an object moving at constant speed v along a circle of radius r. In a small time interval Δt, the object sweeps out a small angle Δθ = ω Δt. The velocity vector also rotates through the same angle Δθ while keeping the same magnitude v.
考虑一个物体以恒定速度 v 沿半径为 r 的圆运动。在很小的时间间隔 Δt 内，物体扫过一个小角度 Δθ = ω Δt。速度矢量也旋转了相同的角度 Δθ，但大小保持为 v。
For very small Δθ, the magnitude of the change in velocity Δv is approximately v Δθ, because the chord between the two velocity vectors is nearly equal to the arc length. The centripetal acceleration is therefore
当 Δθ 很小时，速度变化量的大小 Δv 近似等于 v Δθ，因为两个速度矢量之间的弦长近似等于弧长。因此向心加速度为
a = Δv / Δt = v Δθ / Δt = v ω
Since ω = v / r, we can write the three equivalent forms:
由于 ω = v / r，我们可以写出三种等价形式：
a = v² / r = r ω² = v ω
The direction of this acceleration is radially inward. The units are m s⁻². Use the form that is most convenient for the quantities given in the question.
该加速度的方向是径向向内的。单位是 m s⁻²。解题时根据题目给出的量，选择最方便使用的公式形式。
8. Uniform Circular Motion: Kinematic Summary | 匀速圆周运动的运动学总结
In uniform circular motion, the angular speed, linear speed, period and frequency are all constant. The velocity direction and acceleration direction change continuously, but their magnitudes remain fixed. The acceleration always points to the centre, while the velocity is tangent to the circle.
在匀速圆周运动中，角速度、线速度、周期和频率都恒定不变。速度方向和加速度方向不断变化，但大小保持不变。加速度始终指向圆心，而速度沿圆的切线方向。
| Quantity | Equation | Key idea |
| Angular speed | ω = 2π / T = 2π f | Same for all points on a rigid rotating body |
| Linear speed | v = r ω = 2πr f | Tangent to the circle; depends on radius |
| Centripetal acceleration | a = v² / r = r ω² | Directed towards the centre of the circle |
In an exam, it is helpful to draw a diagram showing the radius, velocity vector and acceleration vector at the instant of interest. This prevents sign errors and helps you see the geometry clearly.
在考试中，画一个示意图，标出特定时刻的半径、速度矢量和加速度矢量会很有帮助。这样可以避免符号错误，并帮助你更清楚地看清几何关系。
9. Non-Uniform Circular Motion: Tangential and Radial Acceleration | 非匀速圆周运动：切向与径向加速度
If the speed of an object in circular motion is changing, the motion is non-uniform. The velocity vector changes in both magnitude and direction. The total acceleration can then be resolved into two perpendicular components: radial and tangential.
如果物体做圆周运动时速度大小也在变化，这种运动就是非匀速圆周运动。速度矢量的大小和方向都在变化。此时总加速度可以分解为两个互相垂直的分量：径向分量和切向分量。
The radial component a_r is the centripetal acceleration, always given by a_r = v² / r, where v is the instantaneous speed. It is responsible for changing the direction of the velocity. The tangential component a_t = Δv / Δt is responsible for changing the speed along the direction of motion.
径向分量 a_r 就是向心加速度，始终由 a_r = v² / r 给出，其中 v 是瞬时速度。它的作用是改变速度方向。切向分量 a_t = Δv / Δt 的作用是改变沿运动方向的速度大小。
The magnitude of the total acceleration is found using Pythagoras:
总加速度的大小用勾股定理计算：
a = √(a_r² + a_t²)
For example, a pendulum bob swinging in a circular arc has both components. At the bottom of the swing, speed is maximum, so the tangential acceleration is zero, and the radial acceleration is large. Near the extreme positions, speed is small, so the radial acceleration is small, and the tangential acceleration is large.
例如，沿圆弧摆动的单摆摆球同时具有这两个分量。在最低点，速度最大，切向加速度为零，径向加速度很大。在接近极端位置时，速度很小，径向加速度小，而切向加速度较大。
10. Common Exam Pitfalls | 常见考试易错点
Many marks are lost in circular motion questions through small but repeated mistakes. Here are the most common ones to avoid.
在圆周运动题目中，许多分数由于细小而反复的错误被扣掉。以下是需要避免的最常见错误。
-
Confusing angular speed ω with linear speed v. Remember ω is measured in rad s⁻¹, while v is measured in m s⁻¹.
混淆角速度 ω 与线速度 v。记住 ω 的单位是 rad s⁻¹，而 v 的单位是 m s⁻¹。
-
Using frequency f directly in v = rω. Always multiply by 2π first, because ω = 2πf.
在 v = rω 中直接使用频率 f。一定要先乘以 2π，因为 ω = 2πf。
-
Forgetting that the centripetal acceleration points towards the centre, not away from it. The so-called “centrifugal acceleration” is not a real acceleration in an inertial frame.
忘记向心加速度指向圆心而不是背离圆心。所谓的“离心加速度”在惯性系中并不是真实的加速度。
-
Using the diameter instead of the radius. Always check whether the question gives r or d, and convert if necessary.
使用直径而不是半径。始终检查题目给出的是 r 还是 d，必要时进行换算。
-
Using degrees in angular calculations. Convert all angles to radians before applying formulas such as s = rθ or ω = Δθ/Δt.
在角度计算中使用度。在应用 s = rθ 或 ω = Δθ/Δt 等公式之前，将所有角度转换为弧度。
11. Worked Example | 例题
Let us apply the kinematic equations to a simple problem. A particle moves in a circular path of radius 0.50 m with a constant frequency of 2.0 Hz. Calculate its angular speed, linear speed and centripetal acceleration.
让我们将运动学方程应用到一道简单题目中。一个粒子在半径为 0.50 m 的圆形路径上运动，恒定频率为 2.0 Hz。计算它的角速度、线速度和向心加速度。
First, find the angular speed using ω = 2πf.
首先，用 ω = 2πf 求角速度。
ω = 2π × 2.0 = 4π ≈ 12.6 rad s⁻¹
Next, find the linear speed using v = rω.
接下来，用 v = rω 求线速度。
v = 0.50 × 4π = 2π ≈ 6.28 m s⁻¹
Finally, find the centripetal acceleration using a = rω².
最后，用 a = rω² 求向心加速度。
a = 0.50 × (4π)² = 8π² ≈ 79 m s⁻²
Notice that the linear speed could also have been found from v = 2πrf, and the acceleration from a = v²/r. All equivalent formulas give the same result. Always include units in your final answer.
注意线速度也可以用 v = 2πrf 求得，加速度也可以用 a = v²/r 求得。所有等价公式都会给出相同的结果。最终答案中一定要包含单位。
12. Key Takeaways | 核心要点总结
The kinematics of circular motion can be summarised in a few essential ideas that connect every equation together.
圆周运动的运动学可以用几个关键思想来总结，这些思想将每个方程联系在一起。
-
Radians are the natural unit for angular displacement and angular speed: 2π rad = 360°.
弧度是角位移和角速度的自然单位：2π rad = 360°。
-
Angular speed ω = Δθ/Δt = 2π/T = 2πf, with unit rad s⁻¹.
角速度 ω = Δθ/Δt = 2π/T = 2πf，单位为 rad s⁻¹。
-
Linear speed and angular speed are connected by v = rω, with velocity tangent
Published by TutorHao | A-Level Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply