Rotational kinematics

Key ideas

Rotational motion uses angular versions of the linear quantities, all in radians:
- angular displacement $\theta$ (rad)
- angular velocity $\omega = \dfrac{\Delta\theta}{\Delta t}$ (rad s⁻¹)
- angular acceleration $\alpha = \dfrac{\Delta\omega}{\Delta t}$ (rad s⁻²)
For constant angular acceleration, the four rotational equations mirror the linear equations of motion exactly:

Rotational	Linear analogue
$\omega_f = \omega_i + \alpha t$	$v_f = v_i + at$
$\theta = \dfrac{\omega_i + \omega_f}{2}\,t$	$d = \dfrac{v_i+v_f}{2}t$
$\omega_f^2 = \omega_i^2 + 2\alpha\theta$	$v_f^2 = v_i^2 + 2ad$
$\theta = \omega_i t + \tfrac{1}{2}\alpha t^2$	$d = v_i t + \tfrac{1}{2}at^2$

Linking rotation to a point on the rim at radius $r$ $r$ :
- distance travelled: $d = r\theta$
- speed: $v = r\omega$
- tangential acceleration: $a = r\alpha$
$\omega = 2\pi f$ , where $f$ is the rotation frequency (Hz) — a full turn is $2\pi$ radians.

Worked ExampleA spinning-up wheel

A wheel starts from rest and reaches $30$ rad s⁻¹ in $5.0$ s at constant angular acceleration. Find the angular acceleration and the number of radians turned.

Step 1 — Angular acceleration (the equation with no $\theta$ )

$\alpha = \frac{\omega_f - \omega_i}{t} = \frac{30 - 0}{5.0} = 6.0 \text{ rad s}^{-2}$

Step 2 — Angle turned (the equation with no $\alpha$ )

$\theta = \frac{\omega_i + \omega_f}{2}\,t = \frac{0 + 30}{2}(5.0) = 75 \text{ rad}$

Tips

Work in radians, not degrees or revolutions. Convert revolutions to radians first ( $1$ rev $= 2\pi$ rad); mixing units here is the most common slip.
Use $v = r\omega$ and $a = r\alpha$ to connect the spinning object to a point on its edge — many questions give a rim speed and ask for an angular quantity, or the reverse.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A disc spins at a constant $12$ rad s⁻¹.

Show that a point on the rim, $0.25$ m from the axis, moves at $3.0$ m s⁻¹.

Merit

A turntable slows from $8.0$ rad s⁻¹ to rest in $4.0$ s at constant angular acceleration.

Calculate the angle it turns through while stopping.

Excellence

A wheel of radius $0.30$ m accelerates uniformly from rest to $20$ rad s⁻¹ in $2.5$ s.

Determine the tangential and centripetal accelerations of a point on the rim at the final instant, and explain why they point in different directions.

Rotational kinematics

Covers AS91524

Key ideas

Rotational motion uses angular versions of the linear quantities, all in radians:
- angular displacement $\theta$ (rad)
- angular velocity $\omega = \dfrac{\Delta\theta}{\Delta t}$ (rad s⁻¹)
- angular acceleration $\alpha = \dfrac{\Delta\omega}{\Delta t}$ (rad s⁻²)
For constant angular acceleration, the four rotational equations mirror the linear equations of motion exactly:

Rotational	Linear analogue
$\omega_f = \omega_i + \alpha t$	$v_f = v_i + at$
$\theta = \dfrac{\omega_i + \omega_f}{2}\,t$	$d = \dfrac{v_i+v_f}{2}t$
$\omega_f^2 = \omega_i^2 + 2\alpha\theta$	$v_f^2 = v_i^2 + 2ad$
$\theta = \omega_i t + \tfrac{1}{2}\alpha t^2$	$d = v_i t + \tfrac{1}{2}at^2$

Linking rotation to a point on the rim at radius $r$ $r$ :
- distance travelled: $d = r\theta$
- speed: $v = r\omega$
- tangential acceleration: $a = r\alpha$
$\omega = 2\pi f$ , where $f$ is the rotation frequency (Hz) — a full turn is $2\pi$ radians.

Worked ExampleA spinning-up wheel

A wheel starts from rest and reaches $30$ rad s⁻¹ in $5.0$ s at constant angular acceleration. Find the angular acceleration and the number of radians turned.

Step 1 — Angular acceleration (the equation with no $\theta$ )

$\alpha = \frac{\omega_f - \omega_i}{t} = \frac{30 - 0}{5.0} = 6.0 \text{ rad s}^{-2}$

Step 2 — Angle turned (the equation with no $\alpha$ )

$\theta = \frac{\omega_i + \omega_f}{2}\,t = \frac{0 + 30}{2}(5.0) = 75 \text{ rad}$

Tips

Work in radians, not degrees or revolutions. Convert revolutions to radians first ( $1$ rev $= 2\pi$ rad); mixing units here is the most common slip.
Use $v = r\omega$ and $a = r\alpha$ to connect the spinning object to a point on its edge — many questions give a rim speed and ask for an angular quantity, or the reverse.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A disc spins at a constant $12$ rad s⁻¹.

Show that a point on the rim, $0.25$ m from the axis, moves at $3.0$ m s⁻¹.

Merit

A turntable slows from $8.0$ rad s⁻¹ to rest in $4.0$ s at constant angular acceleration.

Calculate the angle it turns through while stopping.

Excellence

A wheel of radius $0.30$ m accelerates uniformly from rest to $20$ rad s⁻¹ in $2.5$ s.

Determine the tangential and centripetal accelerations of a point on the rim at the final instant, and explain why they point in different directions.

Rotational kinematics

Key ideas

Step 1 — Angular acceleration (the equation with no θ\thetaθ)

Step 2 — Angle turned (the equation with no α\alphaα)

Tips

Test yourself

Practice by grade

Rotational kinematics

Key ideas

Step 1 — Angular acceleration (the equation with no θ\thetaθ)

Step 2 — Angle turned (the equation with no α\alphaα)

Tips

Test yourself

Practice by grade

Step 1 — Angular acceleration (the equation with no $\theta$ )

Step 2 — Angle turned (the equation with no $\alpha$ )

Step 1 — Angular acceleration (the equation with no $\theta$ )

Step 2 — Angle turned (the equation with no $\alpha$ )