Angular Displacement & Velocity


Nice animation showing 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:

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


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

Angular Displacement

Angular displacement

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 MotionCircular Motion
DisplacementAngular Displacement
SpeedAngular Speed

Back To Circular Motion (A Level)

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