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 |
