Mathematics of Circular Motion

Angular Displacement

The angle which an object moves around a circle is called the angular displacement. This angle is usually expressed in radians.

\begin{aligned} \pi = 180^\circ \end{aligned}

Hence, there are 2 \pi radians in one complete circle.

The relationship between the angle in radian and the circle is

\begin{aligned} \theta =\frac{s}{r} \end{aligned}

where s is the angular displacement in radians, and r is the radius of the circle. The unit of \omega is \text{rad s}^{-1} .

Angular Velocity

Angular velocity is the rate of change of angular displacement.

Mathematically, angular velocity \omega is

\begin{aligned} \omega = \frac{\Delta \theta}{\Delta t} \end{aligned}

Since an object moves one circumference in one period time, we have

\begin{aligned} \omega &= \frac{2 \pi}{T}\\v &= \frac{2 \pi r}{T}\end{aligned}

Eliminating T ,

\begin{aligned} v=r\omega \end{aligned}

Review

Question 1

An object moves round a circle of 15 cm radius at a constant speed. It completes one revolution in 8.0 s. Calculate its angular velocity and its velocity.

Solution

\begin{aligned} \omega &= \frac{2 \pi}{8.0}\\&= 0.785\text{ rad s}^{-1}\end{aligned} \begin{aligned} v &= r \omega\\&= 15 \times 0.785\\&= 11.8\text{ cm s}^{-1}\end{aligned}