Question

$SI$ unit of the angular velocity is
(A) $m/s$
(B) $rad$
(C) $rad/s$
(D) $m/rad$

Answer 1

The term angular velocity is used only when the motion is angular motion, that is the motion takes place either circular or any angular way. It denotes the speed of the angular motion that the displacement varies with the time taken for the rotation.

Useful formula:
The formula of the angular velocity is given as

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

Where $\omega$ is the angular velocity and $\theta$ is the angular displacement.

Complete step by step solution:
Let us consider the formula of the angular velocity,

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

The angular velocity is the derived quantity that is formed from the fundamental quantities like angular displacement and the time taken. The standard system of unit of the angular displacement is radian and the standard system of unit of the time is seconds. Hence substituting the unit in the formula, we get

$\omega = \dfrac{{d\theta }}{{dt}}\left[ {\dfrac{{rad}}{s}} \right]$

Hence the $SI$ unit of the angular velocity is $rad{s^{ - 1}}$ .

Thus the option (C) is correct.

Additional information: The linear velocity is similar to that of the angular velocity but the motion takes place in a straight line without any angular changes. It is the rate of the change of the linear distance with respect to that of the time. The example of this is the velocity of the car that is moving on the road.

Note: Let us consider an example of the roulette wheel, it rotates. Since the rotation takes place, the angular motion takes place. The time taken for the one complete rotation is $20\,s$ , the total perimeter of the outer circle is $40\,rad$ . The angular displacement is the perimeter of the wheel and hence the angular velocity is obtained as $\dfrac{{40}}{{20}} = 2\,rad{s^{ - 1}}$ .