Question

Four spheres of diameter $2a$ and mass $M$ are placed with their centres on the four corners of a square of side $b$. Then moment of inertia of the system about an axis about one of the sides of the square is

• A. ${Ma}^2+{2Mb}^2$
• B. ${Ma}^2$
• C. ${Ma}^2+{4Mb}^2$
• D. $\frac{8}{5}{Ma}^2+{2Mb}^2$

Answer 1

The correct option is D $\frac{8}{5}{Ma}^2+{2Mb}^2$

Moment of Inertia (MOI) of sphere about centre of mass $=\frac{2}{5}M{a}^2$

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

$\Rightarrow$ MOI of system about one side of the square $=2\left(\frac{2}{5}M{a}^2\right)+2(\frac{2}{5}M{a}^2+M{b}^2)=\frac{8}{5}M{a}^2+2M{b}^2$