Learn7408Paper 1 · Paper 1Periodic motionKinematics of circular motion

Periodic motion · Subtopic 1 of 5

Kinematics of Circular Motion

Angles in circular motion are measured in radians: $\theta=\dfrac sr$ , so a full turn is $2\pi$ rad. Angular speed $\omega$ is the rate of change of angle, $\omega=\dfrac{2\pi}T=2\pi f$ . A point at radius $r$ moves with linear speed $v=r\omega$ . Every point on a rigid rotating object has the same $\omega$ , but points further out move faster.

Cambridge 9231 Paper 3Edexcel YMA01 WME03Edexcel 9FM0 OptionAQA 7367 Paper 3Cambridge 9702 Paper 4Edexcel YPH11 WPH14≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Define the radian and convert between degrees, revolutions and radians.
Use $\omega=\dfrac{\Delta\theta}{\Delta t}=\dfrac{2\pi}T=2\pi f$ .
Use $v=r\omega$ .
Explain why uniform circular motion has constant speed but changing velocity.

Pause and think

On a spinning roundabout, a child near the edge and a child near the centre both go round once every $4$ s. Are they moving at the same speed?

Section 2 of 14

Core idea

Start hereSame spin, different speeds

On a merry-go-round, a child at the edge and one near the middle go round in the same time, yet the outer child whizzes along much faster. Both share the same angular speed; their linear speeds depend on how far out they are.

The big idea

$\theta=\frac sr$ (radians). $\omega=\frac{\Delta\theta}{\Delta t}=\frac{2\pi}T=2\pi f$ . $v=\omega r$ .

1. Radians and angular speed

v=\omega r

$\theta=\frac sr\Rightarrow s=r\theta$
Definition of the radian.
$\frac st=r\frac\theta t$
Divide by time.
$v=r\omega$
Linear speed and angular speed.

A point moving round a circle at constant speed

θ (rad) = 0.20

The velocity is always along the tangent, at right angles to the radius. Its size stays the same but its direction keeps changing.

\omega=\frac{2\pi}T=2\pi f\qquad v=\omega r

$\omega$: angular speed (rad s⁻¹).
$T$: period.
$f$: revolutions per second.

Worked exampleA wheel at 10 rev s⁻¹

Radius 0.30 m.

Goal: $\omega$ ; rim speed; speed half way out.
Strategy: $\omega=2\pi f$ ; $v=\omega r$ .

$\omega=2\pi\times10=62.8\text{ rad s}^{-1}$
Angular speed.
$v=62.8\times0.30=18.8\text{ m s}^{-1}$
Rim.
$62.8\times0.15=9.4\text{ m s}^{-1}$
Same $\omega$ , half the radius.

Answer

62.8

rad s

^{-1}

;

18.8

and

9.4

m s

^{-1}

✓ CheckCircumference

1.885

m × 10 per second = 18.8 ✓.

Worked exampleEarth's spin

$T=86\,400$ s, $R=6.37\times10^6$ m.

Goal: Equator speed.
Strategy: $\omega$ , then $v$ .

$\omega=\frac{2\pi}{86400}=7.27\times10^{-5}$
rad s $^{-1}$ .
$v=7.27\times10^{-5}\times6.37\times10^6=463\text{ m s}^{-1}$
$\omega r$ .

Answer

463

m s

^{-1}

✓ CheckFaster than sound, yet we don't feel it: constant speed ✓.

Common misconceptionrpm and degrees Convert rpm with

\times\frac{2\pi}{60}

; always radians.

Pause and think

45 rpm in rad s⁻¹?

Summary

$\theta=s/r$ .
$\omega=2\pi/T$ , $v=\omega r$ .

Angular speed: Angle swept per second.
Radian: Angle with arc equal to radius.

HistoryWhy 360 degrees? The Babylonians divided the circle into

360

parts, probably because their counting was based on

60

and the year is roughly

360

days. The radian is far more natural for physics: the term was first printed in

1873

, by James Thomson, brother of Lord Kelvin.

Fun factYou're spinning fast right now. Earth turns once a day,

\omega=7.3\times10^{-5}

rad s⁻¹. At the equator (

r=6.4\times10^6

m), that's

v=r\omega\approx465

m s⁻¹, faster than a passenger jet. In Pakistan, at about

30°

N, it's still roughly

400

m s⁻¹.

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