Angular Velocity: The Rotational Speed of Circular Motion | 角速度：圆周运动的旋转快慢

📚 Angular Velocity: The Rotational Speed of Circular Motion | 角速度：圆周运动的旋转快慢

When an object moves along a circular path, its position changes both in terms of distance travelled along the arc and in terms of the angle swept out at the centre of the circle. While linear speed tells us how fast the object covers distance, angular velocity tells us how rapidly the object rotates — that is, how quickly the angle changes with time.

当物体沿圆周路径运动时，其位置既沿弧长方向改变，也在圆心处扫过角度。线速度描述的是物体通过距离的快慢，而角速度描述的是物体旋转的快慢——即角度随时间变化的速率。

1. From Arc Length to Angle: The Radian | 从弧长到角度：弧度

To describe rotational motion, we first need a natural unit for angles. The radian is defined as the angle subtended at the centre of a circle when the arc length equals the radius. One full revolution corresponds to 2π radians, because the circumference of a circle is 2πr.

要描述转动，我们首先需要一个天然的角单位。弧度的定义是：当弧长等于半径时，圆心处所对应的角。一整圈对应 2π 弧度，因为圆的周长是 2πr。

θ = s / r

where s is the arc length and r is the radius. This relationship is fundamental: it connects the linear distance along the circle with the angle swept out.

其中 s 为弧长，r 为半径。这一关系至关重要：它将圆周上的弧长与扫过的角度联系了起来。

360° = 2π rad
180° = π rad
90° = π/2 rad

2. Defining Angular Velocity | 角速度的定义

Angular velocity ω is defined as the rate of change of angular displacement. In symbols:

角速度 ω 定义为角位移随时间的变化率。用符号表示：

ω = Δθ / Δt

where Δθ is the angular displacement in radians and Δt is the time interval in seconds. The SI unit of angular velocity is radians per second (rad s⁻¹).

其中 Δθ 是以弧度为单位的角度变化量，Δt 是以秒为单位的时间间隔。角速度的 SI 单位是弧度每秒（rad s⁻¹）。

For uniform circular motion, the angular velocity is constant — every second, the object sweeps out the same angle. This is the rotational analogue of constant linear velocity in a straight line.

对于匀速率圆周运动，角速度恒定——每秒钟扫过的角度相等。这是匀速直线运动中恒定速度的转动对应量。

3. Angular Velocity, Period and Frequency | 角速度、周期与频率

A particle performing circular motion completes one full revolution in a time called the period T. In that time, the angular displacement is exactly 2π radians. Therefore:

做圆周运动的质点完成一整圈所需的时间称为周期 T。在这段时间内，角位移正好是 2π 弧度。因此：

ω = 2π / T

The frequency f of revolution is the number of complete revolutions per second, f = 1/T. Hence we may also write:

转动频率 f 是每秒完成的整圈数，f = 1/T。因此我们也可以写：

ω = 2πf

Note carefully: frequency f is measured in hertz (Hz) or revolutions per second, whereas angular velocity ω is measured in radians per second. They differ by a factor of 2π.

请注意：频率 f 的单位是赫兹（Hz）或每秒转数，而角速度 ω 的单位是弧度每秒。二者相差一个 2π 因子。

4. Connecting Linear Speed and Angular Velocity | 线速度与角速度的联系

Suppose a particle moves through an angle Δθ in time Δt. The arc length travelled is Δs = rΔθ. Dividing both sides by Δt gives:

假设质点在 Δt 时间内转过角度 Δθ。它经过的弧长为 Δs = rΔθ。两边同时除以 Δt 得：

v = rω

where v is the linear speed and r is the radius of the circular path. This is a vital formula for solving problems: if you know the angular velocity and the radius, you can immediately find the linear speed of any point on the rotating body.

其中 v 是线速度，r 是圆周运动的半径。这是解题中极为重要的公式：已知角速度和半径，即可立即求得旋转体上任意一点的线速度。

Notice that for a fixed angular velocity, points farther from the centre move faster linearly: the outer edge of a spinning disc has a higher linear speed than a point near the centre.

注意，对于固定的角速度，离圆心越远的点线速度越大：旋转圆盘边缘的线速度高于靠近圆心处的点。

5. Angular Velocity as a Vector | 角速度作为矢量

Although in introductory problems we often treat angular velocity as a signed scalar (positive for anticlockwise, negative for clockwise), it is in fact a vector quantity. Its direction is given by the right-hand grip rule: curl the fingers of your right hand in the direction of rotation, and your thumb points along the axis of rotation in the direction of the angular velocity vector.

虽然在入门问题中我们常将角速度视为带符号的标量（逆时针为正，顺时针为负），但角速度实际上是矢量。其方向由右手定则确定：用右手手指指向旋转方向弯曲，拇指所指即为角速度矢量沿转轴的方向。

This vector nature becomes important when we study angular momentum or precession, where the direction of rotation must be accounted for explicitly.

角速度的矢量性在研究角动量或进动时十分重要，因为此时必须明确考虑旋转方向。

6. Constant Angular Velocity vs Constant Linear Speed | 恒定角速度与恒定线速度

In uniform circular motion, the magnitude of the linear velocity (the speed) remains constant, and the angular velocity remains constant. However, the linear velocity vector changes direction continuously because the object is always turning toward the centre.

在匀速率圆周运动中，线速度的大小（速率）保持恒定，角速度也保持恒定。然而，线速度矢量的方向不断变化，因为物体始终朝向圆心转向。

|v| = constant; |ω| = constant
Direction of v changes continuously
ω is constant in both magnitude and direction (for motion in one plane)

|v| 恒定；|ω| 恒定
v 的方向不断改变
ω 的大小和方向都恒定（对同一平面内的运动而言）

Thus, uniform circular motion is an example of accelerated motion: even though speeds do not change, velocities do — the acceleration is directed toward the centre of the circle.

因此，匀速率圆周运动是加速运动的典型例子：虽然速率不变，但速度方向在改变——加速度指向圆心。

7. Centripetal Acceleration in Terms of Angular Velocity | 用角速度表示向心加速度

The centripetal acceleration required to keep an object moving in a circle can be expressed in two equivalent ways:

使物体保持圆周运动所需的向心加速度可以表示为两种等价形式：

a = v² / r = rω²

The second form is particularly convenient when v is not directly known but ω is given. For example, a particle on a disc rotating at 3 rad s⁻¹ at a distance of 0.4 m from the centre experiences an acceleration of a = 0.4 × 3² = 3.6 m s⁻² directed toward the centre.

当 v 未知而 ω 已知时，第二种形式尤其方便。例如，一个质点在以 3 rad s⁻¹ 旋转的圆盘上，距中心 0.4 m，其向心加速度为 a = 0.4 × 3² = 3.6 m s⁻²，方向指向圆心。

This acceleration must be provided by a centripetal force, such as tension, friction or gravity, depending on the physical situation.

这个加速度必须由向心力提供，例如张力、摩擦力或重力，具体取决于物理情境。

8. Angular Displacement vs Linear Displacement | 角位移与线位移的比较

Angular displacement is not simply the circular analogue of linear displacement in every respect. Linear displacement is a vector that can be added by the parallelogram law directly; angular displacements about different axes require more careful treatment because they do not commute.

角位移并不是线位移在所有方面都简单的转动对应物。线位移是可直接按平行四边形法则相加的矢量；而绕不同轴的角位移需要更细致的处理，因为它们不满足交换律。

For small angular displacements, however, the vector treatment works cleanly, and angular velocity as the time rate of change of angular displacement is well-defined. This subtlety is rarely tested at A-Level but is worth knowing for deeper understanding.

然而，对于微小角位移，矢量的处理方式仍然成立，角速度作为角位移随时间的变化率有明确的定义。这一微妙之处在 A-Level 中很少考查，但对深入理解很有价值。

9. Measuring Angular Velocity in the Laboratory | 实验室中测量角速度

In practice, angular velocity can be measured using several techniques:

在实践中，角速度可以用多种技术来测量：

Method \| 方法	Principle \| 原理
Optical encoder	Counts pulses per revolution to find f, then ω = 2πf
Strobe light	Match flashing frequency to appear stationary; ω = 2πf_strobe
Timing a marker	Measure time T for one revolution; ω = 2π/T

Each method involves measuring either a period or a frequency, and then converting to angular velocity using the factor 2π.

每种方法都是测量周期或频率，然后利用因子 2π 转换为角速度。

10. Rotational Kinetic Energy and Angular Velocity | 转动动能与角速度

An object rotating with angular velocity ω possesses rotational kinetic energy. For a point mass m at distance r from the axis, the linear speed is v = rω, giving:

以角速度 ω 旋转的物体具有转动动能。对于距离转轴 r 处的质点 m，其线速度为 v = rω，因此：

Eₖ = ½mv² = ½m(rω)² = ½mr²ω²

For a rigid body, we sum over all particles to obtain Eₖ = ½Iω², where I = Σmr² is the moment of inertia. Note the parallels: mass m translates to moment of inertia I, and linear velocity v translates to angular velocity ω.

对于刚体，将所有质点的贡献相加得到 Eₖ = ½Iω²，其中 I = Σmr² 是转动惯量。注意对应关系：质量 m 对应转动惯量 I，线速度 v 对应角速度 ω。

This formula is frequently examined when discussing energy conservation in rolling or rotating systems.

这一公式在讨论滚动或旋转系统中的能量守恒时经常被考查。

11. Typical CIE Exam Questions and Common Pitfalls | CIE 典型考题与常见错误

Examiners often set questions that test whether candidates can distinguish between angular velocity, frequency, and linear speed. A common trap is writing v = ω when the radius has been omitted, or mixing up rad s⁻¹ and Hz.

考官常设置考查学生能否区分角速度、频率和线速度的题目。一个常见的陷阱是漏掉半径直接写 v = ω，或混淆 rad s⁻¹ 与 Hz。

Always check units: rad s⁻¹ for ω, Hz for f, m s⁻¹ for v.

始终检查单位：ω 为 rad s⁻¹，f 为 Hz，v 为 m s⁻¹。

Another common error is using degrees instead of radians when computing ω. When the question provides data in revolutions per minute (rpm), first convert to revolutions per second, then to radians per second:

另一个常见错误是用角度制代替弧度制来计算 ω。当题目给出每分钟转数（rpm）时，先转换为每秒转数，再转换为每秒弧度：

ω = (rpm × 2π) / 60

12. Worked Example | 例题精解

Question: A 0.50 m radius wheel accelerates from rest to an angular velocity of 12 rad s⁻¹ in 4.0 s. Find (a) the angular acceleration (assumed constant), (b) the linear speed of a point on the rim at t = 4.0 s, and (c) the number of revolutions completed in those 4.0 s.

题目：一个半径为 0.50 m 的车轮从静止开始，在 4.0 s 内加速到角速度 12 rad s⁻¹。求（a）角加速度（设恒定），（b）在 t = 4.0 s 时轮缘上一点的线速度，（c）在这 4.0 s 内完成的转数。

Solution:

解答：

(a) α = Δω / Δt = (12 − 0) / 4.0 = 3.0 rad s⁻².

(b) v = rω = 0.50 × 12 = 6.0 m s⁻¹.

This example illustrates how angular quantities (ω, α, θ) combine to determine linear quantities (v, a, s) via the radius.

这个例题展示了角量（ω、α、θ）如何通过半径决定线量（v、a、s）。

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Angular Velocity: The Rotational Speed of Circular Motion | 角速度：圆周运动的旋转快慢