Angular velocity | 角速度

📚 Angular velocity | 角速度

Angular velocity is one of the most important ideas in A-Level Physics when you study circular motion. It tells you how fast an object rotates or moves around a circle in terms of the angle swept out per unit time. This article explains the definition, units, vector nature, and the key equations that link angular velocity to linear speed, period, frequency, and centripetal acceleration.

角速度是 A-Level 物理中学习圆周运动时最重要的概念之一。它描述的是物体转动或沿圆周运动的快慢，用单位时间内扫过的角度来表示。本文讲解角速度的定义、单位、矢量性质，以及角速度与线速度、周期、频率和向心加速度之间的关键方程。

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1. What is Angular Velocity? | 什么是角速度？

When a particle moves along a circular path, its position can be described by an angle θ measured from a fixed reference line. Angular velocity is a measure of how quickly this angle changes with time. It is usually given the symbol ω, the Greek letter omega.

当质点沿圆周路径运动时，它的位置可以用相对于一条固定参考线测量的角度 θ 来描述。角速度就是衡量这个角度随时间变化快慢的物理量，通常用希腊字母 ω 表示。

Angular velocity applies to any rotating system, including a wheel, a DVD, a planet orbiting the Sun, or a charged particle moving in a magnetic field. In A-Level Physics, it is fundamental to understanding circular motion and rotation.

角速度适用于任何转动系统，包括车轮、DVD、围绕太阳运行的行星，或在磁场中运动的带电粒子。在 A-Level 物理中，它是理解圆周运动和转动的核心基础。

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2. Radian Measure and Angular Displacement | 弧度制与角位移

Before defining angular velocity precisely, it is essential to measure angles in radians rather than degrees. One radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.

在准确定义角速度之前，必须使用弧度而不是度来测量角度。1 弧度的定义是：圆弧长度等于半径时，圆心所对应的角度。

θ = s / r

Here s is the arc length and r is the radius. Since both s and r are lengths, the radian is a dimensionless unit. The full circle has arc length 2πr, so the angle in radians is 2πr / r = 2π rad.

这里 s 是弧长，r 是半径。由于 s 和 r 都是长度，弧度是一个无量纲单位。整个圆周的弧长为 2πr，因此用弧度表示的角度为 2πr / r = 2π rad。

This means 360° = 2π rad, 180° = π rad, and 90° = π/2 rad. Using radians makes many circular motion formulas much simpler.

这意味着 360° = 2π rad，180° = π rad，90° = π/2 rad。使用弧度制可以使许多圆周运动公式大大简化。

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3. Defining Angular Velocity | 角速度的定义

For an object moving in a circle, the average angular velocity is the angular displacement divided by the time taken. If the angular displacement is Δθ and the time interval is Δt, then the average angular velocity is:

对于做圆周运动的物体，平均角速度等于角位移除以所用时间。如果角位移为 Δθ，时间间隔为 Δt，则平均角速度为：

ω_avg = Δθ / Δt

The instantaneous angular velocity is the limit of this ratio as Δt becomes very small. For uniform circular motion, the angular velocity is constant, so the average and instantaneous values are the same.

瞬时角速度是当 Δt 趋近于零时该比值的极限。对于匀速圆周运动，角速度恒定，因此平均角速度和瞬时角速度相同。

In many exam questions, you will simply see angular velocity written as ω = θ / t when the motion is uniform. It tells you how many radians are swept out each second.

在许多考试题中，当运动为匀速时，你会直接看到角速度写成 ω = θ / t。它告诉你每秒钟扫过多少弧度。

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4. Units and Dimensions | 单位与量纲

Angular displacement is measured in radians, which are dimensionless. Since angular velocity is angular displacement divided by time, its SI unit is radians per second, written as rad s⁻¹.

角位移以弧度为单位，而弧度是无量纲的。角速度是角位移除以时间，因此其国际单位制单位是弧度每秒，写作 rad s⁻¹。

Because radians are dimensionless, the base dimensions of angular velocity are simply reciprocal time, written as [ω] = T⁻¹. However, you should always include rad s⁻¹ in answers to make the physical meaning clear.

由于弧度无量纲，角速度的基本量纲就是时间的倒数，写作 [ω] = T⁻¹。不过，答题时仍应使用 rad s⁻¹，以明确物理意义。

Sometimes angular speed is given in revolutions per minute, or rpm. To convert to rad s⁻¹, multiply by 2π and divide by 60:

有时角速度以每分钟转数（rpm）给出。要换算成 rad s⁻¹，应乘以 2π 并除以 60：

ω (rad s⁻¹) = rpm × 2π / 60

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5. Period and Frequency | 周期与频率

For uniform circular motion, one complete revolution corresponds to an angular displacement of 2π radians. The time taken for one full revolution is called the period, T. Therefore the angular velocity is:

对于匀速圆周运动，完整转动一圈对应的角位移为 2π 弧度。完成一整圈所需的时间称为周期 T。因此角速度为：

ω = 2π / T

Frequency f is the number of complete revolutions per second, so f = 1 / T. This gives another very common form:

频率 f 是每秒完整转动的圈数，因此 f = 1 / T。由此得到另一个常见公式：

ω = 2πf

These equations are useful when a question gives rotation rate in revolutions per second, revolutions per minute, or period. Always make sure the frequency is in hertz before applying ω = 2πf.

当题目给出的转速单位是每秒转数、每分钟转数或周期时，这些方程非常有用。在应用 ω = 2πf 之前，务必确保频率单位是赫兹。

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6. Relationship between Linear and Angular Velocity | 线速度与角速度的关系

A particle moving in a circle of radius r sweeps out an arc length s = rθ. Differentiating this with respect to time gives the link between linear speed v and angular speed ω:

一个在半径为 r 的圆周上运动的质点，扫过的弧长为 s = rθ。对时间求导就可以得到线速度 v 与角速度 ω 的关系：

v = rω

Here v is the tangential speed along the circle, r is the radius, and ω must be in rad s⁻¹. This relation is only valid when angles are measured in radians.

这里 v 是沿圆周的切向速度，r 是半径，ω 必须以 rad s⁻¹ 为单位。此关系只有在角度使用弧度时才成立。

The following table summarises the comparison between linear and angular quantities:

下表总结了线量与角量的对比：

Linear quantity | 线量	Angular quantity | 角量	Relationship | 关系
Displacement s | 位移	Angle θ | 角度	s = rθ
Velocity v | 速度	Angular velocity ω | 角速度	v = rω
Acceleration a | 加速度	Angular acceleration α | 角加速度	a = rα

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7. Angular Velocity as a Vector | 作为矢量的角速度

Although angular speed is a scalar, angular velocity is a vector. It has both magnitude and direction. Its magnitude is ω, and its direction is determined by the right-hand rule.

虽然角速率是标量，但角速度是矢量。它既有大小也有方向。其大小为 ω，方向由右手定则确定。

If the fingers of your right hand curl in the direction of rotation, your thumb points along the axis of rotation. That thumb direction is the direction of the angular velocity vector.

如果你右手的手指沿旋转方向弯曲，拇指就指向转轴方向。该拇指方向就是角速度矢量的方向。

For anticlockwise rotation viewed from above, the angular velocity vector points upward. For clockwise rotation, it points downward. The direction is perpendicular to the plane of rotation.

从上方观察逆时针旋转时，角速度矢量指向上方；顺时针旋转时，角速度矢量指向下方。角速度方向垂直于旋转平面。

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8. Centripetal Acceleration and Angular Velocity | 向心加速度与角速度

An object moving in a circle at constant speed is still accelerating because its direction changes continuously. This acceleration is called centripetal acceleration and it points towards the centre of the circle.

物体以恒定速率做圆周运动时仍然在加速，因为其速度方向不断改变。这种加速度称为向心加速度，方向指向圆心。

The magnitude of centripetal acceleration can be expressed in two equivalent forms:

向心加速度的大小可以表示为两种等价形式：

a = v² / r = rω²

Using v = rω, you can quickly switch between these expressions. If a question gives the angular velocity, use a = rω²; if it gives the linear speed, use a = v² / r.

利用 v = rω，你可以快速在这两个表达式之间转换。如果题目给出角速度，使用 a = rω²；如果给出线速度，则使用 a = v² / r。

This centripetal acceleration is caused by a net inward force, often called centripetal force, given by F = mv² / r or F = mrω². Angular velocity therefore connects circular motion to Newton’s second law.

这种向心加速度由净指向圆心的力引起，通常称为向心力，表示为 F = mv² / r 或 F = mrω²。因此，角速度将圆周运动与牛顿第二定律联系了起来。

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9. Non-uniform Circular Motion and Angular Acceleration | 非匀速圆周运动与角加速度

When the angular velocity of a rotating object changes with time, the motion is non-uniform. The rate of change of angular velocity is called angular acceleration, denoted by α.

当旋转物体的角速度随时间变化时，运动就是非匀速的。角速度的变化率称为角加速度，用 α 表示。

α = Δω / Δt

Angular acceleration is measured in rad s⁻². A positive α means the angular velocity is increasing in the chosen positive direction, while a negative α means it is decreasing or increasing in the opposite direction.

角加速度的单位是 rad s⁻²。正的 α 表示角速度沿选定的正方向增大，而负的 α 表示角速度减小或沿反方向增大。

In non-uniform circular motion, the object still has centripetal acceleration from the changing direction, and it also has tangential acceleration from the changing speed. The two components combine to give the total acceleration.

在非匀速圆周运动中，物体仍然因方向变化而产生向心加速度，同时还会因速率变化而产生切向加速度。这两个分量合起来构成总加速度。

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10. Rotational Kinematics Equations | 转动运动学方程

For constant angular acceleration, the angular versions of the SUVAT equations apply. They have exactly the same structure as the linear equations but with θ, ω, and α replacing s, v, and a.

对于恒定的角加速度，可以使用 SUVAT 方程的角量版本。它们与线量方程结构完全相同，只是用 θ、ω 和 α 代替 s、v 和 a。

ω = ω₀ + αt

θ = ω₀t + ½αt²

ω² = ω₀² + 2αθ

In these equations, ω₀ is the initial angular velocity, ω is the final angular velocity, α is the constant angular acceleration, t is time, and θ is the angular displacement in radians.

在这些方程中，ω₀ 是初始角速度，ω 是末角速度，α 是恒定角加速度，t 是时间，θ 是以弧度为单位的角位移。

You can use these equations exactly like linear SUVAT problems. Identify the known quantities, choose the equation without the unknown you do not need, and solve.

你可以像解直线 SUVAT 问题一样使用这些方程。先找出已知量，选择不含你不需要的未知量的方程，然后求解。

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11. Worked Example and Common Pitfalls | 例题与常见误区

Worked example: A disc accelerates uniformly from rest with an angular acceleration of 2.0 rad s⁻² for 5.0 s. Calculate the final angular velocity and the angular displacement.

例题：一个圆盘从静止开始以 2.0 rad s⁻² 的角加速度均匀加速 5.0 秒。计算末角速度和角位移。

Using ω = ω₀ + αt, with ω₀ = 0, α = 2.0 rad s⁻² and t = 5.0 s:

使用 ω = ω₀ + αt，已知 ω₀ = 0，α = 2.0 rad s⁻²，t = 5.0 s：

ω = 0 + (2.0)(5.0) = 10 rad s⁻¹

Using θ = ω₀t + ½αt²:

使用 θ = ω₀t + ½αt²：

θ = 0 + ½(2.0)(5.0)² = 25 rad

Common pitfalls include using degrees instead of radians, forgetting to convert rpm to rad s⁻¹, and mixing linear and angular quantities. Always check that the equation matches the physical situation.

常见误区包括使用度数而不是弧度、忘记将 rpm 转换为 rad s⁻¹，以及混淆线量和角量。务必检查方程是否符合物理情景。

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12. Summary and Exam Tips | 总结与应试技巧

Angular velocity describes how fast an angle changes in circular motion. Its core definition is ω = Δθ / Δt, and for uniform circular motion it links to period and frequency by ω = 2π / T = 2πf.

角速度描述圆周运动中角度变化的快慢。其核心定义是 ω = Δθ / Δt，对于匀速圆周运动，它与周期和频率的关系为 ω = 2π / T = 2πf。

The relationship v = rω connects angular and linear speed, while centripetal acceleration can be written as a = rω². Angular velocity is a vector directed along the rotation axis by the right-hand rule.

关系式 v = rω 将角速率与线速率联系起来，而向心加速度可以写成 a = rω²。角速度是矢量，方向由右手定则确定，沿转轴方向。

In the exam, write down the relevant equation first, convert all angles to radians, and state the units of every answer. These habits reduce careless mistakes and help you earn method marks.

考试时，先写出相关方程，将所有角度转换为弧度，并标明每个答案的单位。这些习惯可以减少粗心错误，并帮助你获得方法分。

Published by TutorHao | Physics Revision Series | aleveler.com

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