A LevelPhysicsAQARevision NotesEngineering PhysicsRotational DynamicsRotational Motion

Rotational Motion (AQA A Level Physics): Revision Note

Exam code: 7408

Written by: Ashika

Reviewed by: Tim

Updated on 14 October 2026

Guided study available on this topic

Understand this topic with deeper explanations and understanding checks.

Angular displacement, velocity & acceleration

A rigid rotating body can be described using the following properties:
- angular displacement
- angular velocity
- angular acceleration

These properties can be inferred from the properties of objects moving in a straight line combined with the geometry of circles and arcs

Angular displacement

Angular displacement is defined as:

The change in angle through which a rigid body has rotated relative to a fixed point

Angular displacement is measured in radians

Angular displacement to linear displacement

The linear displacement $s$ at any point along a segment that is in rotation can be calculated using:

$s = r θ$

Where:
- $s$ = length of the arc, or the linear distance travelled along a circular path (m)
- $r$ = radius of a circular path, or distance from the axis of rotation (m)
- $θ$ = angular displacement, or change in angle (radians)

A bar rotates clockwise about an axis at the centre, with the angle θ subtended by two radius lines, radius r and arc length s labelled. The equation states θ (rad) = s/r. — **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 velocity $ω$ of a rigid rotating body is defined as:

The rate of change in angular displacement with respect to time

Angular velocity is measured in rad s^–1
This can be expressed as an equation:

$ω = \frac{∆ θ}{∆ t}$

Where:
- $ω$ = angular velocity (rad s^–1)
- $∆ θ$ = angular displacement (rad)
- $∆ t$ = change in time (s)

Angular velocity to linear velocity

The linear speed $v$ is related to the angular speed $ω$ by the equation:

$v = r ω$

Where:
- $v$ = linear speed (m s^–1)
- $r$ = distance from the axis of rotation (m)
- $ω$ = angular velocity (rad s^–1)

Taking the angular displacement of a complete cycle as 2π, angular velocity $ω$ can also be expressed as:

$ω = \frac{v}{r} = 2 π f = \frac{2 π}{T}$

Rearranging gives the expression for linear speed:

$v = 2 π f r = \frac{2 π r}{T}$

Where:
- $f$ = frequency of the rotation (Hz)
- $T$ = time period of the rotation (s)

Angular acceleration

Angular acceleration $α$ is defined as:

The rate of change of angular velocity with time

Angular acceleration is measured in rad s⁻²
This can be expressed as an equation:

$α = \frac{∆ ω}{∆ t}$

Where:
- $α$ = angular acceleration (rad s⁻²)
- $∆ ω$ = change in angular velocity, or $∆ ω = ω_{f} - ω_{i}$ (rad s⁻¹)
- $∆ t$ = change in time (s)

Angular acceleration to linear acceleration

Using the definition of angular velocity $ω$ with the equation for angular acceleration $α$ gives:

$∆ ω = \frac{∆ v}{r}$

$α = \frac{∆ ω}{∆ t} = \frac{∆ v}{r ∆ t} = \frac{a}{r}$

Rearranging gives the expression for linear acceleration:

$a = r α$

Where:
- $a$ = linear acceleration (m s⁻²)
- $r$ = distance from the axis of rotation (m)
- $∆ v$ = change in linear velocity, or $∆ v = v - u$ (m s⁻¹)

Examiner Tips and Tricks

While there are many similarities between the angular quantities used in this topic and the angular quantities used in the circular motion topic, make sure you are clear on the distinctions between the two, for example, angular acceleration and centripetal acceleration are not the same thing.

Graphs of rotational motion

Graphs of rotational motion can be interpreted in the same way as linear motion graphs

Graphs of angular displacement, velocity and acceleration against time. Gradient of angular displacement time graph gives angular velocity omega, and gradient of angular velocity time graph gives angular acceleration alpha. Areas under the omega- and alpha-time graphs give angular displacement theta and angular velocity omega respectively. — **Graphs of angular displacement, angular velocity and angular acceleration**

Angular displacement, $θ$ is equal to:
- the area under the angular velocity-time graph

Angular velocity, $ω$ is equal to:
- the gradient of the angular displacement-time graph
- the area under the angular acceleration-time graph

Angular acceleration, $α$ is equal to:
- the gradient of the angular velocity-time graph
The following table gives a summary of linear and angular variables:

Variable	Linear	Angular
Displacement	$s = r θ$	$θ = \frac{s}{r}$
Velocity	$v = r ω$	$ω = \frac{v}{r}$
Acceleration	$a = r α$	$α = \frac{a}{r}$

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