Angular displacement and angular velocityEdexcel International A Level Physics: Revision notes

The angular displacement $\theta$ of a body moving on a circle is the angle through which the radius to the body has turned. One radian is the angle subtended at the centre by an arc equal in length to the radius, so for an arc of length $s$ on a circle of radius $r$:

$\theta = \dfrac{s}{r}$ (in radians)

A full circle has circumference $2\pi r$, so one revolution is $2\pi$ rad.

Since $2\pi\ \text{rad} = 360^\circ$, $\pi\ \text{rad} = 180^\circ$:

• degrees to radians: multiply by $\dfrac{\pi}{180}$
• radians to degrees: multiply by $\dfrac{180}{\pi}$

For example $72^\circ = 72 \times \dfrac{\pi}{180} = 1.26$ rad, and $\dfrac{\pi}{2}$ rad $= 90^\circ$.

The angular velocity $\omega$ is the rate of change of angular displacement: $\omega = \dfrac{\Delta\theta}{\Delta t}$, measured in rad s⁻¹.

For one full revolution, $\theta = 2\pi$ and $\Delta t = T$, the period, so $\omega = \dfrac{2\pi}{T}$. With frequency $f = 1/T$, $\omega = 2\pi f$.

To convert revolutions per minute (rpm) to rad s⁻¹, multiply by $2\pi/60$.

A point at radius $r$ travels an arc $s = r\theta$ in a time $t$, so its speed is

$v = \omega r$

All points on a rotating rigid body have the same $\omega$ and period, but points further from the axis have greater $v$. The direction of $v$ is along the tangent to the circle.

A hard disk of radius 4.7 cm spins at 7200 rpm.

• $\omega = 7200 \times 2\pi \div 60 = 754$ rad s⁻¹
• $T = 2\pi / \omega = 8.3\times10^{-3}$ s
• $v = \omega r = 754 \times 0.047 = 35$ m s⁻¹
• Angle turned in 2.0 ms: $754 \times 0.0020 = 1.51$ rad $= 86^\circ$