Radians (AQA A Level Physics): Revision Note

Radians

• In circular motion, it is more convenient to measure angular displacement in units of radians rather than units of degrees

• The angular displacement θ of a body in circular motion is defined as:

The change in angle, in radians, of a body as it rotates around a circle

• The angular displacement is the ratio of:

$∆\theta =\frac{\mathrm{distance}\mathrm{travelled}\mathrm{around}\mathrm{the}\mathrm{circle}}{\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{circle}}$

• Note: both distances must be measured in the same units, e.g. metres

• A radian (rad) is defined as:

The angle subtended at the centre of a circle by an arc equal in length to the radius of the circle

• Angular displacement can be calculated using the equation:

$∆\theta =\frac{s}{r}$

• Where:

  • $∆\theta$ = angular displacement, or angle of rotation (radians)

  • $s$ = length of the arc, or the distance travelled around the circle (m)

  • $r$ = radius of the circle (m)

• Radians are commonly written in terms of $\pi$

• The angle in radians for a complete circle ($360°$) is equal to:

$\frac{\mathrm{circumference}\mathrm{of}\mathrm{circle}}{\mathrm{radius}}=\frac{2\mathrm{\pi }r}{r}=2\mathrm{\pi }$

Radian conversions

• If an angle of $360°=2\mathrm{\pi }$ radians, then 1 radian in degrees is equal to:

$\frac{360}{2\mathrm{\pi }}=\frac{180}{\mathrm{\pi }}\approx 57.3°$

• Therefore, use the following equation to convert from radians to degrees:

$\theta \mathrm{rads}\times \frac{180}{\mathrm{\pi }}=\theta °$

• Use the following equation to convert from degrees to radians:

$\theta °\times \frac{\mathrm{\pi }}{180}=\theta \mathrm{rads}$