S5.2 Angular positions, velocities and accelerations
[A] Using vectors to describe rotations
- To describe the rotation of a rigid body, we need to specify
- the axis of rotation
- the sense of the rotation (clockwise/anti-clockwise)
- the magnitude of the rotation.
- We can describe a rotation using a rotation vector:
- the magnitude of the axial vector is given by the rotation angle in radians.
- the direction of the rotation vector is given by the axis of rotation in conjunction with the corkscrew rule
Corkscrew rule: Take a corkscrew and align it with the axis of rotation. Turn it in the sense of the rotation.
The direction it moves defines the direction of the rotation vector. - Under a rotation every point on the body moves in a circle centred on the rotation axis.
[B] Angular position
- To define the angular position of a rigid body, we need to
specify the rotation axis, and an origin.
We choose the origin as follows:
- we draw an imaginary line in the rigid body, which will rotate with the body
- we draw a line outside the body, which will remain fixed.
- the angular position of the body can then be defined as the angle, θ, the line in the body makes with the line fixed outside the body.
- The definitions of angular displacement, velocity and acceleration can now be completed in a straightforward manner.
Commentary
Reveal

Commentary
- We do not reset θ to zero with each complete rotation of the reference line. If the reference line completes two revolutions, then the angular position is θ=4π rad.
- Avoid the popular misconception that something is moving along the rotation vector. Instead, something is moving around the direction of the vector, i.e. the vector defines the axis of rotation. (Nonetheless, the vector does define the motion through its magnitude and sense).
[C] Angular displacement
Key Point 5.2
Angular Displacement: If a rigid body rotates about a fixed axis changing the angular position from θ1 to θ2, the body undergoes an angular displacement
Note: angular displacement is measured in radians.
[D] Angular velocity
Key Point 5.3
Average angular velocity: If a rigid body rotates around a fixed axis producing an angular displacement Δθ in time Δt, the average angular velocity about the axis is
![\[ \omega_{av} = \frac{\Delta \theta}{\Delta t} \]](mastermathpng-0.png)
Key Point 5.4
Instantaneous angular velocity is defined as the average angular velocity over the next infinitesimal time interval
![\[ \omega = \lim_{\Delta t \rightarrow 0} \frac{\Delta \theta}{\Delta t} = \frac{d \theta}{d t} \]](mastermathpng-1.png)
Note: the units of angular velocity are rad s-1.
- Note: strictly speaking, what we have defined above are angular speeds, not velocities, since we have not specified the direction of rotation.
- We can define an angular velocity vector,
. The axis of rotation determines the direction of the angular velocity vector; the magnitude of the vector is the angular velocity ω as defined above and the sense of rotation is specified by the corkscrew rule.
[E] Angular acceleration
Key Point 5.5
Average angular acceleration: If a rigid body accelerates about a fixed axis producing a change in angular velocity Δω in time Δt the average angular acceleration is
![\[ \alpha_{av} = \frac{\Delta \omega}{\Delta t} \]](mastermathpng-3.png)
Key Point 5.6
Instantaneous angular acceleration is defined as the average angular acceleration over the next infinitesimal time interval
![\[ \alpha = \lim_{\Delta t \rightarrow 0} \frac{\Delta \omega}{\Delta t} = \frac{d \omega}{d t} \]](mastermathpng-4.png)
Note: the units of angular acceleration are rad s-2.
- We can define an angular acceleration vector,
. The rotation axis specifies the direction of the angular acceleration vector; the magnitude is the angular acceleration α as defined above; the sense is given by the corkscrew rule.

Worked examples
A wheel accelerates from rest with constant angular acceleration to an angular velocity of 100 rad s-1 in 20 s. Find the angular acceleration.
Solution
The angular velocity changes by 100 rad s-1 in 20s. Thus
Worked Example
During a time interval t the flywheel of a generator turns through the angle
Solution
Angular velocity:
![\[ \omega=\frac{d\theta}{dt}=\xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{a+3bt^2-4ct^3} \]](mastermathpng-6.png)
Angular acceleration:
![\[ \alpha=\frac{d\omega}{dt}=\xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{6bt-12ct^2} \]](mastermathpng-7.png)
Learning Resources
![]() | HRW Chapter 10.1-10.3 |
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