1. Given data

• Three identical solid spheres
• Mass of each sphere $= m$
• Diameter of each sphere $= d$ so radius of each sphere is $r = \frac d2$
• Their centres form an equilateral triangle of side $d$.

We need the ratio $\frac{I_O}{I_A}$ where:

• $I_O$ = moment of inertia of the system about an axis through the centroid of the triangle and perpendicular to its plane.
• $I_A$ = moment of inertia of the system about an axis through the centre of any one sphere and perpendicular to the plane.

2. Moment of inertia of one solid sphere about its own centre

For a solid sphere, $I_{\text{cm}} = \frac{2}{5}mr^2 = \frac{2}{5}m\left(\frac d2\right)^2 = \frac{md^2}{10}$

3. Distance of each sphere centre from centroid of the equilateral triangle

For an equilateral triangle of side $d$, distance from centroid to each vertex is $R = \frac{d}{\sqrt{3}}$

So for each sphere, using parallel axis theorem about the centroidal axis, $I_{\text{one about }O} = I_{\text{cm}} + mR^2$ $= \frac{md^2}{10} + m\left(\frac{d}{\sqrt{3}}\right)^2$ $= \frac{md^2}{10} + \frac{md^2}{3}$

For three spheres, $I_O = 3\left(\frac{md^2}{10} + \frac{md^2}{3}\right)$ $= 3md^2\left(\frac{1}{10} + \frac{1}{3}\right)$ $= 3md^2\left(\frac{13}{30}\right)$ $= \frac{13}{10}md^2$

4. Moment of inertia about axis through centre of one sphere

Let the axis pass through the centre of sphere $A$ and be perpendicular to the plane.

• For sphere $A$ itself: $I_A^{(1)} = \frac{md^2}{10}$

• The other two sphere centres are each at distance $d$ from this axis, since side of triangle is $d$.

For each of those spheres, $I_A^{(2)} = I_{\text{cm}} + md^2 = \frac{md^2}{10} + md^2 = \frac{11}{10}md^2$

Thus total $I_A = \frac{md^2}{10} + 2\left(\frac{11}{10}md^2\right)$ $= \frac{md^2}{10} + \frac{22}{10}md^2$ $= \frac{23}{10}md^2$

5. Required ratio

$\frac{I_O}{I_A} = \frac{\frac{13}{10}md^2}{\frac{23}{10}md^2} = \frac{13}{23}$

6. Option check

• A: $\frac{13}{23}$ ✅
• B: $\frac{23}{13}$ ❌
• C: $\frac{15}{13}$ ❌
• D: $\frac{13}{15}$ ❌

So the correct option is A.