Three solid spheres each of mass $m$ and diameter $d$ are stuck together such that the lines connecting the centres form an equilateral triangle of side of length $d$. The ratio $\dfrac{I_0}{I_A}$ of moment of inertia $I_0$ of the system about an axis passing through the centroid and $I_A$ about the centre of any of the spheres, both perpendicular to the plane of the triangle, is:
Answer: (A) $\dfrac{13}{23}$
Each sphere has radius $d/2$, so about its own centre $I = \dfrac{2}{5}m\left(\dfrac{d}{2}\right)^2 = \dfrac{md^2}{10}$.
About the centroid: each centre is at $\dfrac{d}{\sqrt{3}}$, so by the parallel axis theorem
$$I_0 = 3\left(\frac{md^2}{10} + \frac{md^2}{3}\right) = \frac{13}{10}md^2$$
About the centre of sphere A: the other two centres are at distance $d$:
$$I_A = \frac{md^2}{10} + 2\left(\frac{md^2}{10} + md^2\right) = \frac{23}{10}md^2$$
$$\frac{I_0}{I_A} = \frac{13}{23}$$
Solution by Sreeraj P, M.Sc Physics