Radians & Angular Displacement (Cambridge (CIE) A Level Physics): Revision Note

Radians & angular displacement

Angles in radians

• 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

Visual definition of radian

When the angle is equal to one radian, the length of the arc (S) is equal to the radius (r) of the circle

• Radians are commonly written in terms of π

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

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

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

$\theta °\times \left(\frac{\mathrm{\pi }}{180}\right)=\theta \mathrm{rad}$

Table of common degrees to radians conversions

Angular displacement

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

• Angular displacement is defined as:

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

• This can be summarised in equation form:

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

• Where:

  • Δθ = 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)

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

Visual representation of angular displacement equation

An angle in radians, subtended at the centre of a circle, is the arc length divided by the radius of the circle