Question

The moment of inertia of a sphere of Mass $M$ and radius $R$ about it's diameter is $I$. Four such spheres are arranged as shown in the figure. Find the moment of inertia of the system about axis $X{X}^{\prime }$.

Answer 1

Given, Moment of inertia(MOI) of sphere about its diameter$=\frac{2}{5}M{R}^2=I$

Using Parallel axis theorem, $I=I_{cm}+M{x}^2$ where $I_{cm}$ is MOI about centre of mass $x$ is distance between centre of mass and axis of rotation.

$\Rightarrow \text{MOI of system}=2\times \frac{2}{5}M{R}^2+2\times (\frac{2}{5}M{R}^2+M{R}^2)=\frac{18}{5}M{R}^2=9I$