📚 Angles in Radians | 弧度制下的角度
In A-Level Physics, you will frequently encounter angles expressed in radians rather than degrees. Whether you are analysing circular motion, simple harmonic motion, or wave behaviour, the radian measure simplifies equations and reveals deeper connections between linear and rotational quantities. This article explains what radians are, why they are used in physics, how to convert between degrees and radians, and how radian measure underpins key formulas in the CIE syllabus.
在A-Level物理中，你会经常看到角度用弧度而非度数表示。无论你是在分析圆周运动、简谐运动还是波动行为，弧度制都能简化方程，并揭示线量与旋转量之间更深层的联系。本文将解释什么是弧度、为什么物理学中要使用弧度、如何完成度与弧度的换算，以及弧度制如何支撑CIE考纲中的关键公式。
1. Introduction to Angular Measurement | 角度测量简介
Angles can be measured in several units, such as degrees, revolutions, or radians. In everyday life, we commonly use degrees, where a full circle is 360°. However, in physics, the radian is the standard unit of angular measurement because it is a dimensionless quantity defined by the ratio of two lengths. This makes it particularly useful when relating angular motion to linear motion.
角度可以用多种单位测量，如度、转数或弧度。在日常生活中，我们常用度，一个完整的圆是360°。然而，在物理学中，弧度是角度测量的标准单位，因为它是一个由两个长度之比定义的无量纲量。这在将角运动与线运动联系起来时特别有用。
The concept of a radian may initially seem abstract, but it is simply a way of measuring an angle based on the radius and arc length of a circle. Once you become familiar with radians, many physical relationships become much clearer and more elegant.
弧度的概念起初可能显得抽象，但它只是根据圆的半径和弧长来度量角度的一种方式。一旦你熟悉了弧度，许多物理关系就会变得更加清晰和优美。
2. Defining the Radian | 弧度的定义
One radian is defined as the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle. Mathematically, if an arc of length s is cut off by an angle θ at the centre of a circle of radius r, then:
1弧度定义为：当圆弧的长度等于该圆的半径时，该圆弧所对的圆心角的大小。数学上，如果一段长为s的弧在半径为r的圆的圆心处对应一个角度θ，那么：
θ (in radians) = s / r
This definition implies that in a full circle, the total arc length is the circumference, 2πr. Thus the angle for a full revolution in radians is:
这个定义意味着，对于整个圆，总弧长是周长2πr。因此一整圈所对应的弧度角为：
θfull = 2πr / r = 2π rad
Therefore, 360° equals 2π radians. This fundamental equivalence is the basis for all degree–radian conversions and is essential for solving physics problems involving rotation.
因此，360°等于2π弧度。这个基本等式是所有度与弧度换算的基础，也是解决涉及转动的物理问题的关键。
3. Converting Between Degrees and Radians | 度与弧度的换算
multiply by 180/π. These conversion factors arise directly from the identity 180° = π rad.
要将角度从度转换为弧度，乘以π/180。反之，要从弧度转换为度，乘以180/π。这些换算因子直接来源于恒等式180° = π弧度。
For example, a common angle in physics is 30°, which in radians is:
例如，物理学中常见的角度30°，用弧度表示为：
30° × (π / 180) = π/6 rad ≈ 0.5236 rad
Similarly, an angle of 1 radian in degrees is about 57.3°. Becoming fluent in these conversions is important for quickly interpreting angular data and for checking the consistency of your equations.
类似地，1弧度约等于57.3°。熟练掌握这些换算对于快速解读角度数据以及检查方程的一致性非常重要。
You should memorise the radian equivalents of commonly used angles such as 90° (π/2), 45° (π/4), and 60° (π/3) so that you can work smoothly through exam questions.
你应该熟记常用角度的弧度值，比如90° (π/2)、45° (π/4) 和60° (π/3)，这样在解答考题时就能流畅书写。
4. Why Radians? The Natural Unit for Angle | 为什么用弧度？角度的自然单位
The radian is considered the 'natural' unit for measuring angles because it emerges from the geometry of a circle without arbitrary numbers. When you use radians, many important formulas lose their proportionality constants and become simpler.
弧度被认为是度量角度的“自然”单位，因为它来源于圆的几何性质，不涉及任意数字。当你使用弧度时，许多重要公式会失去比例常数，从而变得更简单。
For instance, the arc length formula s = rθ holds only when θ ecome neat only when angles are in radians: d(sin θ)/dθ = cos θ. This simplicity extends to physics, where radian measure appears in definitions of angular velocity, centripetal acceleration, and wave equations.
例如，弧长公式s = rθ只有在θ以弧度为单位时才成立。如果使用度数，就需要额外的换算因子。在微积分中，只有当角度采用弧度制时，三角函数的导数才变得简洁：d(sin θ)/dθ = cos θ。这种简洁性延伸到了物理学中，弧度制出现在角速度、向心加速度和波动方程的定义里。
Using radians also makes it easier to compare angular and linear dynamics, because the radian is a ratio of two lengths and thus dimensionless. This property helps in dimensional analysis and confirms that equations like v = ωr are dimensionally consistent.
使用弧度还能更容易地比较角动力学与线动力学，因为弧度是两个长度的比值，因而是无量纲的。这一特性有助于量纲分析，并确认v = ωr这类方程在量纲上是一致的。
5. Arc Length and Angular Displacement | 弧长与角位移
One of the most immediate applications of radian measure in physics is the relationship between arc length, angular displacement, and radius. When an object moves along a circular path, the distance travelled along the circumference, s, is given by:
弧度制在物理学中最直接的应用之一就是弧长、角位移和半径之间的关系。当一个物体沿圆形路径运动时，沿圆周运动的距离s由下式给出：
s = rθ
Here θ must be in radians. This formula is extremely useful when calculating the displacement of a point on a rotating wheel, or when linking the angular motion of a pendulum bob with its linear path.
这里的θ必须以弧度为单位。这个公式在计算旋转轮上某一点的位移，或者将单摆摆锤的角运动与其线性路径联系起来时，非常有用。
Angular displacement itself is measured in radians and can be positive or negative depending on the direction of rotation. Conventionally, anti-clockwise rotation is taken as positive. For small angles, the arc length is approximately equal to the chord length, which is why radian measure is central to the small-angle approximations used in simple harmonic motion.
角位移本身以弧度为单位，根据旋转方向可正可负。通常规定逆时针转动为正。对于小角度，弧长约等于弦长，这就是为什么弧度制是简谐运动中小角度近似法的核心。
6. Angular Velocity in Radians per Second | 角速度（弧度每秒）
Angular velocity, typically denoted by the Greek letter ω (omega), is defined as the rate of change of angular displacement. In symbols:
角速度，通常用希腊字母ω（omega）表示，定义为角位移的变化率。用符号表示为：
ω = Δθ / Δt
If the angular displacement is measured in radians and time in seconds, the unit of angular velocity is radian per second, abbreviated as rad s⁻¹. This is the standard unit in the CIE A-Level Physics specification.
如果角位移以弧度为单位，时间以秒为单位，那么角速度的单位就是弧度每秒，缩写为rad s⁻¹。这是CIE A-Level物理规范中的标准单位。
Knowing the angular velocity allows you to find the instantaneous linear speed of a point at a distance r from the centre of rotation:
知道了角速度，你就能求出距转动中心r处一点的瞬时线速度：
v = ωr
This formula is valid only when ω is expressed in rad s⁻¹. If you were using degrees per second, the relationship would require a conversion factor and would not be as elegant. It is the use of radians that makes v = ωr a clean and powerful connection between angular and linear worlds.
这个公式仅在ω以rad s⁻¹为单位时成立。如果你使用度每秒，这个关系就需要一个换算因子，也就不会如此简洁。正是弧度制的使用，使得v = ωr成为角世界与线世界之间清晰而强大的联系。
7. Radians in Simple Harmonic Motion (SHM) | 简谐运动中的弧度
Simple harmonic motion (SHM) is a form of oscillation in which the acceleration is directly proportional to the negative of the displacement. The standard mathematical description uses sine and cosine functions whose arguments are angles in radians.
简谐运动（SHM）是一种振动形式，其加速度与位移的负值成正比。标准的数学描述使用正弦和余弦函数，这些函数的自变量就是以弧度表示的角度。
In SHM, displacement is often written as:
在简谐运动中，位移通常写作：
x = A sin(ωt) or x = A cos(ωt)
Here (ωt) is an angular quantity measured in radians. The term ω is the angular frequency, with units rad s⁻¹. Because radians are dimensionless, the argument of the trigonometric function is a pure number, which is required for mathematical consistency.
这里(ωt)是一个以弧度为单位的角量。ω是角频率，单位为rad s⁻¹。由于弧度是无量纲的，三角函数的自变量是一个纯数，这是数学一致性所要求的。
A crucial application of radian measure in SHM arises with the small-angle approximation. For a simple pendulum, the restoring force is proportional to sin θ. When θ his approximation simplifies the motion to true SHM and yields the familiar period formula T = 2π√L/g. Without radians, this approximation would not hold in such a straightforward manner.
弧度制在简谐运动中的一个关键应用来自小角度近似。对于单摆，回复力正比于sin θ。当θ很小且以弧度为单位时，sin θ ≈ θ。这个近似将运动简化为真正的简谐运动，并得出熟悉的周期公式T = 2π√L/g。如果没有弧度，这种近似就不会如此直接成立。
8. Centripetal Acceleration and Radian Measure | 向心加速度与弧度制
Centripetal acceleration describes the acceleration directed towards the centre of a circle for an object moving at constant speed along a circular path. Its magnitude is given by two equivalent expressions:
向心加速度描述的是沿圆形路径匀速运动的物体指向圆心的加速度。其大小由两个等价的表达式给出：
ac = v² / r = ω² r
Again, these formulas rely on radian measure for ω. Deriving ac = ω²r from the definition of centripetal acceleration uses the fact that the change in velocity vector direction is related to the angular displacement in radians. If ω were in degrees per second, the geometry would produce an incorrect factor.
同样，这些公式依赖于ω的弧度度量。从向心加速度的定义推导ac = ω²r时，就利用了速度矢量方向的变化与以弧度表示的角位移相关这一事实。如果ω以度每秒为单位，几何图形就会产生错误的因子。
Understanding that centripetal acceleration formulas are built on radian angular velocity helps you avoid common mistakes. For example, when a question gives rotational speed in revolutions per minute (rpm), you must first convert to rad s⁻¹ before using ac = ω²r.
理解向心加速度公式建立在弧度角速度的基础上，有助于你避免常见错误。例如，当题目给出旋转速度以每分钟转数（rpm）为单位时，你必须先将其转换为rad s⁻¹，然后再使用ac = ω²r。
9. Angular Acceleration | 角加速度
When the angular velocity of a rotating object changes, we speak of angular acceleration, usually denoted by α (alpha). Angular acceleration is defined as the rate of change of angular velocity:
当旋转物体的角速度发生变化时，我们称之为角加速度，通常用α（alpha）表示。角加速度定义为角速度的变化率：
α = Δω / Δt
The unit of angular acceleration is rad s⁻². The link to the linear tangential acceleration at of a point on the rotating body is:
角加速度的单位是rad s⁻²。它与旋转体上一点的线切向加速度at的联系为：
at = rα
Once more, this simple proportionality exists only when α is expressed in rad s⁻². Equations for uniformly accelerated rotational motion mirror those for linear motion under constant acceleration, but with angular quantities substituted: ωf = ωi + αt, θ = ωit + ½αt², and ωf² = ωi² + 2αθ.
又一次地，这种简单的比例关系仅在α以rad s⁻²为单位时才存在。匀加速转动运动的方程与匀加速直线运动的方程相似，只是将角量代入：ωf = ωi + αt、θ = ωit + ½αt²以及ωf² = ωi² + 2αθ。
When solving problems, always check that your angular quantities are in radians, radians per second, and radians per second squared. Many errors arise from mixing degrees and radians in the same calculation.
在解题时，始终要检查角量是否以弧度、弧度每秒和弧度每二次方秒为单位。许多错误都源于在同一计算中混用了度与弧度。
10. Quick Reference: Common Angles in Degrees and Radians | 常用角度及弧度对照表
The following table summarises the most frequently encountered angles and their radian equivalents. Use it for quick revision and to reinforce your conversion skills.
下表总结了一些最常遇到的角度及其弧度等价值。用它来快速复习并强化你的换算技能。
| Degrees (°) | Radians (rad) |
|---|---|
| 0° | 0 |
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 120° | 2π/3 |
| 135° | 3π/4 |
| 180° | π |
| 270° | 3π/2 |
| 360° | 2π |
Notice that all entries in the radian column are expressed as multiples or fractions of π. This is standard practice in physics problems and you should aim to leave your answers in terms of π unless a decimal is specifically requested.
请注意，弧度一栏中的所有条目都以π的倍数或分数表示。这是物理题目的标准做法，除非特别要求小数，否则你应尽量将答案保留为含π的形式。
11. Worked Examples | 计算示例
Example 1: A car wheel of radius 0.30 m turns through an angle of 150°. Calculate the distance travelled by a point on the rim.
示例1：一个半径为0.30 m的汽车车轮转过150°角。计算轮缘上一点运动所经过的距离。
First, convert degrees to radians: 150° = 150 × (π/180) = 5π/6 rad. Then use s = rθ: s = 0.30 × (5π/6) = 0.25π m ≈ 0.785 m.
首先，将度转换为弧度：150° = 150 × (π/180) = 5π/6 rad。然后使用s = rθ：s = 0.30 × (5π/6) = 0.25π m ≈ 0.785 m。
Example 2: A spinning disc increases its angular velocity from 10 rad s⁻¹ to 30 rad s⁻¹ in 5.0 s. Find the angular acceleration and the total angular displacement during this time.
示例2：一个旋转圆盘在5.0 s内角速度从10 rad s⁻¹增加到30 rad s⁻¹。求角加速度以及这段时间内的总角位移。
Angular acceleration: α = Δω/Δt = (30 - 10)/5.0 = 4.0 rad s⁻². For angular displacement, use θ = ωit + ½αt²: θ = 10×5.0 + ½×4.0×(5.0)² = 50 + 50 = 100 rad.
角加速度：α = Δω/Δt = (30 - 10)/5.0 = 4.0 rad s⁻²。角位移使用θ = ωit + ½αt²：θ = 10×5.0 + ½×4.0×(5.0)² = 50 + 50 = 100 rad。
Example 3: ntre. If it sweeps out an angle of 1.2 × 10⁻³ rad in one second, what is its orbital speed?
示例3：一颗通信卫星在距地心4.22 × 10⁷ m的轨道上运行。如果它在一秒内扫过1.2 × 10⁻³ rad的角度，其轨道线速度是多少？
Since the angle is already in radians, use v = ωr but first find ω = θ/t = 1.2 × 10⁻³ rad / 1 s = 1.2 × 10⁻³ rad s⁻¹. Then v = (1.2 × 10⁻³)(4.22 × 10⁷) ≈ 5.06 × 10⁴ m s⁻¹.
由于角度已经是弧度，使用v = ωr，但首先求出ω = θ/t = 1.2 × 10⁻³ rad / 1 s = 1.2 × 10⁻³ rad s⁻¹。然后v = (1.2 × 10⁻³)(4.22 × 10⁷) ≈ 5.06 × 10⁴ m s⁻¹。
12. Conclusion | 总结
Radian measure is not an arbitrary alternative to degrees; it is a fundamental tool that unifies circular and linear physics. From arc length and angular velocity to the dynamics of simple harmonic motion and centripetal acceleration, radians make the mathematics consistent and physically meaningful. As you progress through the CIE A-Level Physics course, you will find that comfort with radians is essential for mastering topics ranging from rotational mechanics to wave phenomena.
弧度制并不是度数的随意替代，它是一种将圆周运动与直线运动统一起来的基本工具。从弧长和角速度，到简谐运动与向心加速度的动力学，弧度使得数学表达具有一致性并富有物理意义。在CIE A-Level物理课程的学习过程中，你会发现熟练掌握弧度对于掌握从转动力学到波动现象的各种主题是至关重要的。
Make a habit of practising conversions, using radian-based formulas confidently, and checking dimensional consistency. With time, thinking in radians will become second nature, giving you an edge in both understanding and examinations.
养成练习换算的习惯，自信地使用基于弧度的公式，并检查量纲的一致性。随着时间的推移，用弧度思考将成为你的第二天性，让你在理解和考试中都占据优势。
Published by TutorHao | Physics Revision Series | aleveler.com
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