Interpret the axis

The four identical solid spheres are placed at the four corners of a square of side $b$.

The axis is along one side of the square and lies in the plane of the square.

So:

• Two spheres have their centres on the axis.
• The other two spheres are at perpendicular distance $b$ from the axis.

Moment of inertia of one solid sphere about an axis through its centre

For a solid sphere, $I_{\text{cm}}=\frac{2}{5}ma^2$

Spheres whose centres lie on the axis

There are 2 such spheres.

Since the axis passes through their centres, $I_1=I_2=\frac{2}{5}ma^2$

Total contribution: $I_{\text{on axis}}=2\times \frac{2}{5}ma^2=\frac{4}{5}ma^2$

Spheres whose centres are at distance $b$ from the axis

There are 2 such spheres.

By the parallel axis theorem, $I=I_{\text{cm}}+md^2=\frac{2}{5}ma^2+mb^2$ where $d=b$.

So for 2 spheres, $I_{\text{off axis}}=2\left(\frac{2}{5}ma^2+mb^2\right)=\frac{4}{5}ma^2+2mb^2$

Total moment of inertia

$I_{\text{total}}=I_{\text{on axis}}+I_{\text{off axis}}$ $I_{\text{total}}=\frac{4}{5}ma^2+\left(\frac{4}{5}ma^2+2mb^2\right)$ $I_{\text{total}}=\frac{8}{5}ma^2+2mb^2$

Match with the options

$\boxed{I=\frac{8}{5}ma^2+2mb^2}$

This corresponds to Option D.