understand and use the concept of angular speed

Kinematics of Uniform Circular Motion

When an object moves in a circle at constant speed, its motion is called uniform circular motion (UCM). Although the speed is constant, the direction of the velocity continuously changes, giving the object a centripetal (centre‑seeking) acceleration.

Angular Quantities

In circular motion it is convenient to describe the motion using angular quantities rather than linear ones.

• Angular displacement ($\theta$) – the angle swept out by the radius vector, measured in radians (rad).
• Angular speed ($\omega$) – the rate of change of angular displacement, $$\omega = \frac{\Delta\theta}{\Delta t}\;,$$ with SI unit rad s\(^{-1}\). For uniform circular motion $\omega$ is constant.
• Angular acceleration ($\alpha$) – the rate of change of angular speed, $$\alpha = \frac{\Delta\omega}{\Delta t}\;,$$ measured in rad s\(^{-2}\). In uniform circular motion $\alpha = 0$.

Relation Between Angular and Linear Quantities

The linear (tangential) speed $v$ of a point at a distance $r$ from the centre of the circle is directly proportional to the angular speed:

$$v = \omega r$$

Similarly, the linear (centripetal) acceleration $a_c$ can be expressed in two equivalent forms:

$$a_c = \frac{v^{2}}{r} = \omega^{2} r$$

Key Equations Summary

Quantity	Symbol	Formula	Units
Angular displacement	$\theta$	$\theta = s/r$	rad
Angular speed	$\omega$	$\omega = \Delta\theta/\Delta t$	rad s\(^{-1}\)
Linear (tangential) speed	$v$	$v = \omega r$	m s\(^{-1}\)
Centripetal acceleration	$a_c$	$a_c = \omega^{2} r = v^{2}/r$	m s\(^{-2}\)
Period of revolution	$T$	$T = 2\pi/\omega$	s
Frequency of revolution	$f$	$f = 1/T = \omega/2\pi$	Hz

Using Angular Speed in Problems

Typical steps when solving A‑Level questions involving angular speed:

1. Identify the radius $r$ of the circular path and any given linear quantities (e.g., speed $v$ or period $T$).
2. Convert any linear data to angular form using $v = \omega r$ or $T = 2\pi/\omega$.
3. Apply the appropriate formula from the table above to find the required quantity (e.g., $\omega$, $v$, $a_c$).
4. Check units and, if necessary, convert between revolutions per minute (rpm), hertz (Hz), or rad s\(^{-1}\).
5. State the final answer with the correct number of significant figures.

Example Problem

Question: A car travels round a circular track of radius $30\,$m at a constant speed of $15\,$m s\(^{-1}\). Determine the angular speed $\omega$ and the centripetal acceleration $a_c$ of the car.

Solution:

1. Calculate angular speed using $v = \omega r$: $$\omega = \frac{v}{r} = \frac{15\;\text{m s}^{-1}}{30\;\text{m}} = 0.50\;\text{rad s}^{-1}.$$
2. Find centripetal acceleration: $$a_c = \omega^{2} r = (0.50\;\text{rad s}^{-1})^{2}\times 30\;\text{m}=7.5\;\text{m s}^{-2}.$$
3. Answer: $\omega = 0.50\;\text{rad s}^{-1}$, $a_c = 7.5\;\text{m s}^{-2}$.

Common Misconceptions

• Confusing angular speed $\omega$ with angular displacement $\theta$. $\omega$ is a rate (rad s\(^{-1}\)), whereas $\theta$ is an angle (rad).
• Assuming that a constant linear speed implies zero acceleration. In circular motion the direction changes, giving a non‑zero centripetal acceleration.
• Neglecting the factor $2\pi$ when converting between period $T$, frequency $f$, and angular speed $\omega$.

Summary

Understanding angular speed is essential for analysing uniform circular motion. The core relationships $v = \omega r$ and $a_c = \omega^{2} r$ link angular and linear descriptions, allowing you to move seamlessly between them in A‑Level physics problems.