define the radian and express angular displacement in radians

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Cambridge A-Level Physics 9702 – Kinematics of Uniform Circular Motion

Kinematics of Uniform Circular Motion

Objective

Define the radian and express angular displacement in radians.

1. Definition of the Radian

The radian is the SI unit for plane angles. 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.

Suggested diagram: A circle with radius $r$, an arc of length $s = r$, and the central angle labelled $\theta$ (one radian).

2. Relationship Between Arc Length, Radius and Angle

When the angle $\theta$ is measured in radians, the arc length $s$ is given by

$$s = r\theta$$

where

  • $s$ – arc length (m)
  • $r$ – radius of the circle (m)
  • $\theta$ – angular displacement (rad)

3. Converting Between Radians and Degrees

The full circumference corresponds to an angle of $2\pi$ rad, which is also $360^\circ$. Hence

$$1\;\text{rad} = \frac{180^\circ}{\pi}\approx 57.3^\circ$$ $$1^\circ = \frac{\pi}{180}\;\text{rad}\approx 0.01745\;\text{rad}$$

4. Expressing Angular Displacement in Radians

For any motion along a circular path, the angular displacement $\Delta\theta$ can be obtained from the travelled arc length $\Delta s$:

$$\Delta\theta = \frac{\Delta s}{r}$$

This expression is valid only when $\Delta\theta$ is expressed in radians.

5. Example Calculation

  1. Given a wheel of radius $r = 0.30\;\text{m}$ rotates through an arc length of $s = 0.75\;\text{m}$. Find the angular displacement in radians and degrees.

Solution:

$$\Delta\theta = \frac{s}{r} = \frac{0.75}{0.30} = 2.5\;\text{rad}$$

Convert to degrees:

$$\Delta\theta = 2.5\;\text{rad}\times\frac{180^\circ}{\pi}\approx 2.5\times57.3^\circ\approx 143^\circ$$

6. Common Angles

Angle (degrees) Angle (radians)
0° 0
30° \(\frac{\pi}{6}\)
45° \(\frac{\pi}{4}\)
60° \(\frac{\pi}{3}\)
90° \(\frac{\pi}{2}\)
180° \(\pi\)
270° \(\frac{3\pi}{2}\)
360° \(2\pi\)

7. Summary

  • A radian is defined by an arc length equal to the radius.
  • When angles are measured in radians, the simple relation $s = r\theta$ holds.
  • Angular displacement in radians is obtained from $\Delta\theta = \Delta s / r$.
  • Conversion: $2\pi\;\text{rad}=360^\circ$.