Physics 9702 · Paper 4

Radians & Angular Displacement

A2 physics measures angles in a new unit built from the circle itself. Every formula in this chapter assumes it.

The radian

\theta = \dfrac{s}{r}

Symbols

$\theta$ = angular displacement (rad (a ratio of lengths, so no dimensions))
$s$ = arc length travelled (m)
$r$ = radius of the circle (m)

One radian: the angle at the centre when the arc has the same length as the radius. The arc drawn here really spans 1 rad (57.3°).

One radian is the angle at the centre of a circle when the arc length equals the radius. This exact sentence is a banked one-marker: it was asked in both 2024 and 2025 (9702/42/M/J/25 Q1(a)).
A full circle has arc $2\pi r$ , so $2\pi \text{ rad} = 360^\circ$ and $1 \text{ rad} \approx 57.3^\circ$ .
Convert with the $2\pi \leftrightarrow 360^\circ$ pair: degrees × $\pi/180$ gives radians; radians × $180/\pi$ gives degrees.

Worked example

Smallest case: converting an angle

Write 60° in radians.

$\theta = 60 \times \dfrac{\pi}{180} = \dfrac{\pi}{3} = 1.05 \text{ rad}$ .

Answer

\theta = \pi/3 \approx 1.05 \text{ rad}

Worked example

Exam version: an angle from an arc

A chain moves 0.24 m around a cog wheel of radius 0.15 m. Find the angle the cog turns through. (9702/42/M/J/25 Q1(b)(iv))

$\theta = \dfrac{s}{r} = \dfrac{0.24}{0.15} = 1.6 \text{ rad}$ .

Answer

\theta = 1.6 \text{ rad}

Common mistake

Set your calculator to radian mode for this chapter, and check which unit a question uses before quoting an angle. An answer of 1.6 means radians here: writing 1.6° would be a different (and wrong) angle.

NextAngular Speed & v = rω

Radians & Angular Displacement

A2 physics measures angles in a new unit built from the circle itself. Every formula in this chapter assumes it.

The radian

\theta = \dfrac{s}{r}

Symbols

$\theta$ = angular displacement (rad (a ratio of lengths, so no dimensions))
$s$ = arc length travelled (m)
$r$ = radius of the circle (m)

One radian: the angle at the centre when the arc has the same length as the radius. The arc drawn here really spans 1 rad (57.3°).

One radian is the angle at the centre of a circle when the arc length equals the radius. This exact sentence is a banked one-marker: it was asked in both 2024 and 2025 (9702/42/M/J/25 Q1(a)).
A full circle has arc $2\pi r$ , so $2\pi \text{ rad} = 360^\circ$ and $1 \text{ rad} \approx 57.3^\circ$ .
Convert with the $2\pi \leftrightarrow 360^\circ$ pair: degrees × $\pi/180$ gives radians; radians × $180/\pi$ gives degrees.

Worked example

Smallest case: converting an angle

Write 60° in radians.

$\theta = 60 \times \dfrac{\pi}{180} = \dfrac{\pi}{3} = 1.05 \text{ rad}$ .

Answer

\theta = \pi/3 \approx 1.05 \text{ rad}

Worked example

Exam version: an angle from an arc

A chain moves 0.24 m around a cog wheel of radius 0.15 m. Find the angle the cog turns through. (9702/42/M/J/25 Q1(b)(iv))

$\theta = \dfrac{s}{r} = \dfrac{0.24}{0.15} = 1.6 \text{ rad}$ .

Answer

\theta = 1.6 \text{ rad}

Common mistake