Angles in Radians | 弧度制角度

📚 Angles in Radians | 弧度制角度

In A-Level Physics, angular measurements are not usually given in degrees. Instead, the radian is the standard SI unit for angle because it makes rotational equations simpler and connects linear and angular quantities directly. Understanding radians is essential for circular motion, simple harmonic motion, and wave phase calculations.

在 A-Level 物理中，角度测量通常不使用度数。弧度是角度的标准国际单位，因为它使旋转运动方程更简洁，并直接联系线量与角量。理解弧度对于圆周运动、简谐运动以及波的相位计算都至关重要。

1. Why Radians Matter in Physics | 为什么物理中弧度很重要

In A-Level Physics, rotational and circular motion equations are written in their simplest form when angles are measured in radians.

在 A-Level 物理中，当角度以弧度为单位时，旋转和圆周运动的方程形式最为简单。

For example, the arc length formula s = rθ is true by definition only if θ is in radians. If degrees were used, an extra conversion factor of π/180 would be needed.

例如，弧长公式 s = rθ 只有在 θ 以弧度为单位时才作为定义成立。如果使用度数，则需要额外的换算系数 π/180。

The radian is also a dimensionless unit, so it can be omitted in dimensional analysis without changing the nature of the quantity.

弧度还是无量纲单位，因此在量纲分析中可以省略，而不会改变物理量的性质。

2. Definition of the Radian | 弧度的定义

One radian is defined as the angle subtended at the centre of a circle by an arc length equal to the radius of the circle.

一弧度定义为圆心角所对的弧长等于半径时的角度。

θ = s / r

In this equation, θ is the angle in radians, s is the arc length, and r is the radius. When s = r, the angle is exactly 1 rad.

在这个方程中，θ 是以弧度为单位的角度，s 是弧长，r 是半径。当 s = r 时，角度恰好为 1 rad。

A complete circle has circumference 2πr, so a full revolution corresponds to θ = 2πr / r = 2π rad.

一个完整圆的周长为 2πr，因此一整圈对应的角度为 θ = 2πr / r = 2π rad。

3. Converting Between Degrees and Radians | 度与弧度的换算

Because 360° is equivalent to 2π rad, the conversion between degrees and radians is based on π rad = 180°.

因为 360° 等于 2π rad，所以度与弧度的换算基于 π rad = 180°。

angle in radians = angle in degrees × π / 180

angle in degrees = angle in radians × 180 / π

For example, 90° = 90 × π/180 = π/2 rad, and 1 rad ≈ 57.3°.

例如，90° = 90 × π/180 = π/2 rad，而 1 rad ≈ 57.3°。

Degrees / 度数	Radians / 弧度
0°	0
30°	π/6
45°	π/4
60°	π/3
90°	π/2
180°	π
360°	2π

4. Arc Length Formula | 弧长公式

When θ is in radians, the arc length s along a circular path is given by s = rθ.

当 θ 以弧度为单位时，沿圆路径的弧长 s 由 s = rθ 给出。

s = rθ

This equation is used to find the distance travelled by a point on a rotating object or by an object moving along a circular arc.

该方程用于求旋转物体上一点或沿圆弧运动的物体所经过的路程。

For example, if a wheel of radius 0.50 m rotates through an angle of π/3 rad, the arc length travelled by a point on the rim is s = 0.50 × π/3 ≈ 0.52 m.

例如，如果一个半径为 0.50 m 的轮子转过 π/3 rad 的角度，轮缘上一点通过的弧长为 s = 0.50 × π/3 ≈ 0.52 m。

If the angle is given in degrees, convert it to radians before using this formula.

如果角度以度数给出，在使用该公式之前必须先换算为弧度。

5. Angular Displacement | 角位移

Angular displacement Δθ is the angle through which an object rotates in a given time interval. It is measured in radians.

角位移 Δθ 是物体在给定时间间隔内转过的角度，以弧度为单位。

For uniform circular motion, the angular displacement is often expressed as θ = ωt, where ω is the angular velocity.

对于匀速圆周运动，角位移通常表示为 θ = ωt，其中 ω 是角速度。

θ = ωt

Angular displacement can be positive or negative depending on the chosen rotational direction.

角位移可以是正值或负值，取决于所选的旋转方向。

Because angular displacement is measured in radians, it is directly proportional to the linear distance travelled along the circumference.

由于角位移以弧度为单位，它与沿圆周运动的线距离成正比。

6. Angular Velocity | 角速度

Angular velocity ω is the rate of change of angular displacement: ω = Δθ / Δt. Its SI unit is rad s⁻¹.

角速度 ω 是角位移的变化率：ω = Δθ / Δt。其国际单位为 rad s⁻¹。

ω = Δθ / Δt

For one complete revolution taking period T, ω = 2π / T since the angle is 2π rad.

对于周期为 T 的一整圈旋转，因为角度为 2π rad，所以 ω = 2π / T。

ω = 2π / T = 2πf

Angular velocity is also related to frequency f by ω = 2πf.

角速度还与频率 f 相关：ω = 2πf。

The linear speed v of a point at radius r is v = rω. This relationship only holds when ω is in radians per second.

半径为 r 处一点的线速度 v 为 v = rω。这个关系只有在 ω 以弧度每秒为单位时才成立。

7. Angular Acceleration | 角加速度

Angular acceleration α is the rate of change of angular velocity: α = Δω / Δt. Its SI unit is rad s⁻².

角加速度 α 是角速度的变化率：α = Δω / Δt。其国际单位为 rad s⁻²。

α = Δω / Δt

The tangential acceleration aₜ of a point at radius r is aₜ = rα, provided α is in rad s⁻².

半径为 r 处一点的切向加速度为 aₜ = rα，前提是 α 以 rad s⁻² 为单位。

aₜ = rα

In uniform circular motion, α = 0 and the angular velocity is constant.

在匀速圆周运动中，α = 0，角速度保持不变。

8. Radians in Circular Motion Equations | 圆周运动方程中的弧度

The equations of rotational kinematics mirror linear kinematics, but angles must be in radians.

旋转运动学方程与直线运动学方程形式相似，

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Angles in Radians | 弧度制角度