Question

Write the dimensions of angular velocity.

Answer 1

First of all, we will find out the expression of angular velocity. Then we will write the dimensions of the physical quantities which are involved in the expression for angular velocity. We will manipulate accordingly to obtain the result.

Complete step by step solution:
In the given question, we are given the physical quantity called angular velocity.
We are asked to find the dimension angular velocity.To begin with, let us discuss a bit about dimension.In terms of simple quantities, the dimensional formula is an acronym for the unit of a physical quantity. Mass $\left( M \right)$ , length $\left( L \right)$ , and time $\left( T \right)$ are the fundamental quantities. In terms of $M$ , $L$ and $T$ powers, a dimensional formula is expressed.
We know, the expression for angular velocity, which is given by:
$v = r\omega$ …… (1)
Where,
$v$ indicates the linear velocity of the body moving around a fixed path.
$r$ indicates the radius of the path.
$\omega$ indicates the angular velocity of a body which is moving around a fixed path.
From equation (1), we can write by doing simple rearrangement a shown below:
$\omega = \dfrac{v}{r}$ …… (2)
Since, $v$ indicates the linear velocity of the body moving around a fixed path. Its unit is ${\text{m}}{{\text{s}}^{ - 1}}$.
So, its dimension will be $\left[ {{M^0}{L^1}{T^{ - 1}}} \right]$.
Again, $r$ indicates the radius of the path. Its unit is ${\text{m}}$.
So, its dimension will be $\left[ {{M^0}{L^1}{T^0}} \right]$.
Now we write the dimension of the respective physical quantities in the equation (2) and we get:
$\omega = \dfrac{v}{r} \\\ \Rightarrow \omega = \dfrac{{\left[ {{M^0}{L^1}{T^{ - 1}}} \right]}}{{\left[ {{M^0}{L^1}{T^0}} \right]}} \\\ \Rightarrow \omega = \left[ {{T^{ - 1}}} \right] \\\ \therefore \omega = \left[ {{M^0}{L^0}{T^{ - 1}}} \right]$

Hence, the dimensions of angular velocity are $\left[ {{M^0}{L^0}{T^{ - 1}}} \right]$.

Note: While finding dimensions, always remember that dimensions are those physical quantities which can be measured. It tells us which are the fundamental physical quantities involved in a complex physical quantity. It is important to remember that the dimension of angle is $\left[ {{M^0}{L^0}{T^0}} \right]$ as it does not contain mass, length and time.