## Radian

For the measurement of angles, you might be used to using degrees (e.g. $360^{\circ}$). Radian is a more convenient unit for the measurement of angles.

One **radian** is the angle subtended at the centre of a circle by an arc of length equal to the radius of the circle.

Radian is given by:

$$\text{Angle (in radians)} = \frac{\text{arc length}}{\text{radius}}\tag{1}$$

For a complete circle, circumference is given by $2 \pi r$, where $r$ is the radius. Using Equation 1, the “angle” of a whole circle can be given by:

$$\begin{aligned} \text{Angle (in radians)} &= \frac{\text{arc length}}{\text{radius}} \\ &= \frac{2 \pi r}{r} \\ &= 2 \pi \end{aligned}$$

Hence,

$$\begin{aligned} 360^{\circ} &= 2 \pi \\ 1^{\circ} &= \frac{2 \pi}{360} \text{ rad} \\ 57.3^{\circ} &= 1 \text{ rad}\end{aligned}$$

## Angular Displacement

Angular displacement,* *$\theta$ is the angle subtended at centre of a circle by an arc of equal length to the radius.

**Alternative Definition**: Angular displacement is the change in angle (measured in radians) of a body as it rotates around a circle.

- Unit is radian

$$\theta = \frac{s}{r}$$

, where s: arc length, r: radius of circle

## Angular Velocity

Angular velocity, $\omega$ is the rate of change of angular displacement with respect to time.

- S.I. unit is $\text{rad s}^{-1}$

$$\omega = \frac{d \theta}{dt}$$

## Comparison with Translation Motion

You can see some equivalence with the concepts that you have learnt in translational motion (normal Kinematics)

Translational Motion | Circular Motion |
---|---|

Displacement | Angular Displacement |

Speed | Angular Speed |