A body is said to be in circular motion if it is travelling in a closed circular path of radius r. The same body is said to be in uniform circular motion if it has uniform speed in each and every point of its travel. Following are the few terms used in Circular Motion:
Angular Displacement
The central angle traced out by the radius vector of the body in uniform circular motion from the point of reference is called angular displacement and is denoted by $\theta$ (theta).
It is measured in radian (rad) in the SI unit. Its dimensional formula is $[M^0L^0T^0]$
Remember
Radian has no dimensional formula.
Angular Velocity
The rate of change of angular displacement is defined as angular displacement and is denoted by $\omega$ (omega).
Mathematically,
$$\text{Angular Velocity} = \frac{ \text{Change in angular displacement} }{ \text{Time taken}}$$
$$\omega = \frac{ \theta_2 \ - \ \theta_1}{t_2 \ - \ t_1}$$
where $\omega$ is angular velocity, $\theta_2$ is final angular displacement and $\theta_1$ is initial angular displacement and $t_2$ and $t_1$ are final and initial time taken respectively.
Using derivatives,
$$\omega = \frac{d \theta}{d t}$$
It is measured in radians per second (rad/s) in the SI unit. Its dimensional formula is $[M^0L^0T^{-1}]$.
Angular Acceleration
The rate of change of angular velocity is defined as angular acceleration and is denoted by $\alpha$ (alpha).
Mathematically,
$$\text{Angular Acceleration} = \frac{ \text{Change in angular velocity} }{ \text{Time taken}}$$
$$\alpha = \frac{ \omega \ - \ \omega_o }{t \ - \ t_o}$$
where alpha is the angular acceleration. $\omega$ is the final velocity and $\omega_o$ is the initial velocity. Similarly, $t$ and $t_o$ are the final and initial time respectively.
The SI unit of angular acceleration is rad / s2. Its dimensional formula is $[M^0L^0T^{-2}]$.
Relation between Angular and Linear Motion
| Physical Quantity | Angular Motion | Linear Motion | Relation |
|---|---|---|---|
| Displacement | s | $\theta$ | $s \ = \ r \theta$ |
| Velocity | v | $\omega$ | $v \ = \ r \omega$ |
| Acceleration | a | $\alpha$ | $a \ = \ r \alpha$ |
Relation between Angular and Linear Motion
Solved Examples
EXAMPLE 1: If the angular displacement of a body is 6 radians and the time taken for the displacement is 2 seconds. What is its angular velocity?
Solution:
Here,
Angular displacement ($\theta$) = 6 rad
Time taken ($t$ = 2 s
We know,
Angular velocity ($\omega$) = $\frac{\theta}{t}$
$= \frac{6}{2}$
$= 3 \text{rad/s}$
Hence, the required angular velocity of the body is 3 radians per second.
EXAMPLE 2: What is the angular velocity of a particle moving in a circular path whose $\theta = 3t^2 + 5t + 6$ radian and $t = 5$ seconds?
Solution:
Here,
Angular displacement ($\theta$) = $3t^2 + 5t + 6$ rad
Time taken ($t$ = 5 s
We know,
Angular velocity ($\omega$) = $\frac{d \theta}{d t}$
$= \frac{d}{dt} (3t^2 + 5t + 6)$
$= 3 \text{x} 2t + 5 x \text{1} + 6 \text{x} 0$
$= 6t + 5$
$= 6 \text{x} 5 + 5$
$= 30 + 5$
$= 35$ radian per second.
Hence, the required angular velocity of the body is 35 radians per second.