Physics 9702 · Paper 4

Angular Speed & v = rω

Two speeds describe circular motion: how fast the angle grows, and how fast the object itself moves. One equation links them.

Angular speed ω

\omega = \dfrac{\Delta\theta}{\Delta t} = \dfrac{2\pi}{T} = 2\pi f

Symbols

$\omega$ = angular speed (omega) (rad s⁻¹)
$\Delta\theta$ = angle turned through (rad)
$T$ = period of one revolution (s)
$f$ = revolutions per unit time (Hz)

Angular speed is the angle turned through per unit time. One full revolution is $2\pi$ rad, so a period T gives $\omega = 2\pi/T$ .
Real periods arrive in awkward units: convert 24 hours or one minute to seconds before dividing. The standard first mark in these calculations is exactly this conversion (9702/42/O/N/25 Q1(a)).

Worked example

Smallest case: a clock's second hand

The second hand turns once every 60 s. Find its angular speed.

$\omega = \dfrac{2\pi}{T} = \dfrac{2\pi}{60} = 0.105 \text{ rad s}^{-1}$ .

Answer

\omega = 0.105 \text{ rad s}^{-1}

Linking the two speeds: v = rω

v = r\omega

Symbols

$v$ = speed along the circle (m s⁻¹)
$r$ = radius of the circle (m)
$\omega$ = angular speed (rad s⁻¹)

It comes straight from the radian: in time t the object turns $\theta = \omega t$ and travels arc $s = r\theta = r\omega t$ . Divide by t: $v = r\omega$ .
Every point of a spinning object shares the same ω, but points further out move faster ( $v \propto r$ ).

Worked example

Exam version: a bicycle wheel

A bicycle moves at 17 m s⁻¹. Its rear wheel has radius 0.46 m. Find the wheel's angular speed. (9702/42/M/J/25 Q1(b)(i))

$\omega = \dfrac{v}{r} = \dfrac{17}{0.46} = 37 \text{ rad s}^{-1}$ .

Answer

\omega = 37 \text{ rad s}^{-1}

The two sharing rules the exam tests: parts of the same rigid object share ω (so v grows with r) (9702/42/M/J/24 Q1(d)); wheels linked by a chain or belt share the chain's linear speed v (so the smaller cog has the larger ω) (9702/42/M/J/25 Q1(c)).

Common mistake

Read radius data carefully. Papers give a diameter (halve it), a distance measured in from the rim (subtract it from the rim radius), or lengths in cm (convert). The first credited line in the mark scheme is usually this radius step (9702/42/M/J/24 Q1(c)).

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Angular Speed & v = rω

Two speeds describe circular motion: how fast the angle grows, and how fast the object itself moves. One equation links them.

Angular speed ω

\omega = \dfrac{\Delta\theta}{\Delta t} = \dfrac{2\pi}{T} = 2\pi f

Symbols

$\omega$ = angular speed (omega) (rad s⁻¹)
$\Delta\theta$ = angle turned through (rad)
$T$ = period of one revolution (s)
$f$ = revolutions per unit time (Hz)

Angular speed is the angle turned through per unit time. One full revolution is $2\pi$ rad, so a period T gives $\omega = 2\pi/T$ .
Real periods arrive in awkward units: convert 24 hours or one minute to seconds before dividing. The standard first mark in these calculations is exactly this conversion (9702/42/O/N/25 Q1(a)).

Worked example

Smallest case: a clock's second hand

The second hand turns once every 60 s. Find its angular speed.

$\omega = \dfrac{2\pi}{T} = \dfrac{2\pi}{60} = 0.105 \text{ rad s}^{-1}$ .

Answer

\omega = 0.105 \text{ rad s}^{-1}

Linking the two speeds: v = rω

v = r\omega

Symbols

$v$ = speed along the circle (m s⁻¹)
$r$ = radius of the circle (m)
$\omega$ = angular speed (rad s⁻¹)

It comes straight from the radian: in time t the object turns $\theta = \omega t$ and travels arc $s = r\theta = r\omega t$ . Divide by t: $v = r\omega$ .
Every point of a spinning object shares the same ω, but points further out move faster ( $v \propto r$ ).

Worked example

Exam version: a bicycle wheel

A bicycle moves at 17 m s⁻¹. Its rear wheel has radius 0.46 m. Find the wheel's angular speed. (9702/42/M/J/25 Q1(b)(i))

$\omega = \dfrac{v}{r} = \dfrac{17}{0.46} = 37 \text{ rad s}^{-1}$ .

Answer

\omega = 37 \text{ rad s}^{-1}

The two sharing rules the exam tests: parts of the same rigid object share ω (so v grows with r) (9702/42/M/J/24 Q1(d)); wheels linked by a chain or belt share the chain's linear speed v (so the smaller cog has the larger ω) (9702/42/M/J/25 Q1(c)).

Common mistake