Question

Find the dimensions of moment of inertia $I$.

Answer 1

Step 1: Establish moment of inertia in terms of Standard Quantities

The formula for moment of inertia is given as,

$I=\frac{L}{\omega }$

Where, $L$ is the angular momentum, and $\omega$ is the angular velocity.

The formula for angular momentum is,

$L=mvr$

Where, $m$ is the mass, $v$ is the velocity and $r$ is the radius.

The formula for angular velocity is,

$\omega =\frac{2\mathrm{\pi }}{T}$

Where, $T$ is time period.

$\therefore I=\frac{mvrT}{2\mathrm{\pi }}$

Step 2: Evaluate the Dimensional Formula

Substitute the standard units into the above equation to get,

$I=\frac{kg.m{s}^{-1}.m.s}{2\mathrm{\pi }}$

$=kg.m^2$

$=\left[M{L}^2{T}^0{A}^0\right]$

Hence the dimensional formula for inductance is $\left[M{L}^2{T}^0{A}^0\right]$.