11.1.1 Rotational Motion

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:
- θ = angular displacement, or change in angle (radians)
- 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)

1-4-3-angular-displacement-rigid-body-1

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)
- Δθ = angular displacement (rad)
- Δt = 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)

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)

Exam Tip

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

1-4-3-angular-graphs-of-motion

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

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	$a = r α$	$α = \frac{a}{r}$

AQA A Level Physics

Revision Notes