Rotational Motion

Engineering Physics | AQA A-Level Physics

Key Definition

Angular displacement ($\theta$): The change in angle through which a rigid body has rotated relative to a fixed point. Measured in radians.

Describing rotational motion

A rigid rotating body can be described using three key properties: angular displacement, angular velocity and angular acceleration.
These properties are the rotational equivalents of the linear quantities you already know from mechanics. The key part is that they can all be inferred from the geometry of circles and arcs.

Angular displacement and arc length

The linear displacement $s$ at any point along a rotating segment is linked to the angular displacement by: $$s = r\theta$$
Where: $\theta$ = angular displacement (rad), $s$ = arc length (m), $r$ = radius or distance from the axis of rotation (m).
Rearranging gives the definition of a radian: $$\theta = \frac{s}{r}$$
Fundamentally, an angle in radians, subtended at the centre of a circle, is the arc length divided by the radius of the circle.

Angular velocity

The angular velocityThe rate of change of angular displacement with respect to time. Measured in rad s^-1. $\omega$ of a rigid rotating body is defined as the rate of change in angular displacement with respect to time: $$\omega = \frac{\Delta\theta}{\Delta t}$$
Angular velocity is measured in rad s$^{-1}$.
The linear speed $v$ is related to the angular speed $\omega$ by: $$v = r\omega$$
Since one complete cycle corresponds to $2\pi$ radians, angular velocity can also be expressed as: $$\omega = \frac{v}{r} = 2\pi f = \frac{2\pi}{T}$$
Where $f$ is the frequency of rotation (Hz) and $T$ is the time period (s).

Angular acceleration

The angular accelerationThe rate of change of angular velocity with time. Measured in rad s^-2. $\alpha$ is defined as the rate of change of angular velocity with time: $$\alpha = \frac{\Delta\omega}{\Delta t}$$
Angular acceleration is measured in rad s$^{-2}$.
Using the relationship $v = r\omega$, the angular acceleration is related to the linear acceleration by: $$a = r\alpha$$

Graphs of rotational motion

Graphs of rotational motion are interpreted in the same way as linear motion graphs:
Angular displacement $\theta$ equals the area under the angular velocity-time graph.
Angular velocity $\omega$ equals the gradient of the angular displacement-time graph, and the area under the angular acceleration-time graph.
Angular acceleration $\alpha$ equals the gradient of the angular velocity-time graph.

Common Mistake

Students often confuse angular acceleration with centripetal acceleration. Crucially, angular acceleration ($\alpha$) describes how quickly an object speeds up or slows down its rotation. Centripetal acceleration ($a = \frac{v^2}{r}$) is the inward acceleration that keeps an object moving in a circle. They are not the same thing.