Question

The dimensional formula for angular velocity is:
A) $M^{-1}{L}^{-1}{T}^0$
B) $M^0{L}^{-1}{T}^{-1}$
C) $M^{-1}{L}^{-1}{T}^0$
D) $M^0{L}^0{T}^{-1}$

Answer 1

Hint: In order to find the correct answer of the given question, we need to know about angular velocity. Also, we need to know the dimensions of the quantities of the angular velocity. Then only we can conclude with the correct solution of the given question.

Complete solution:
First of all let us find out about angular velocity.
Angular velocity: It is defined as the rate of velocity at which an object or a particle is rotating around a point. We can define angular velocity in a uniform circular motion as a vector quantity which is the ratio of angular displacement and change in time. Mathematically, angular velocity can be represented as,
$\omega ={\displaystyle \frac{\mathrm{\Delta }\theta }{\mathrm{\Delta }t}}$
Where, $\mathrm{\Delta }\theta$ is the change in angular rotation
$\mathrm{\Delta }t$= change in time
$\omega$= angular velocity
Now, in order to find the dimension of angular velocity, first of all we need to find the dimensions of angular rotation and time.
Already we know that an angle is a dimensionless quantity. Therefore, the dimension of angular rotation $\theta$ is $M^0{L}^0{T}^0$
Now, the dimension of time is $M^0{L}^0{T}^1$
Now, we need to put the dimensions of angular rotation and time in the equation of angular velocity.
Therefore, $\omega ={\displaystyle \frac{M^0{L}^0{T}^0}{M^0{L}^0{T}^1}}$
$\therefore \omega =M^0{L}^0{T}^{-1}$

Hence, option (D), i.e. $M^0{L}^0{T}^{-1}$ is the correct choice of the given question.

Note: We should know that the unit of the angular rotation $\theta$ is radian but it is a dimensionless quantity. Also, we represent the dimensions of any quantity in terms of $MLT$. Therefore, the dimension of angular rotation is $M^0{L}^0{T}^0$.