Question

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 $I_0/I_A$ of moment of inertia $I_0$ of the system about an axis passing the centroid and about center of any of the spheres $I_A$ and perpendicular to the plane of the triangle is :

• A. $\frac{15}{13}$
• B. $\frac{13}{15}$
• C. $\frac{13}{23}$
• D. $\frac{23}{13}$

Answer 1

The Correct Option is C

To solve the problem of finding the ratio $I_0/I_A$, we need to calculate the moment of inertia of the system about two different axes: one passing through the centroid of the equilateral triangle formed by the centers of the spheres ($I_0$), and another passing through the center of any one of the spheres and perpendicular to the plane of the triangle ($I_A$).

1. Calculate the moment of inertia $I_0$ of the system about the centroid of the triangle:
   • The centroid of an equilateral triangle formed by spheres is at equal distance from all vertices. The distance of the centroid from any of the vertices (sphere centers) is $\frac{d}{\sqrt{3}}$. This is due to geometric properties of an equilateral triangle.
   • The moment of inertia $I_0$ about the centroid can be calculated using the parallel axis theorem:
   • The formula for the moment of inertia of a single sphere about its center is $I_{\text{sphere}} = \frac{2}{5}mr^2 = \frac{2}{5}m\left(\frac{d}{2}\right)^2 = \frac{md^2}{10}$.
   • Using the parallel axis theorem for one sphere with the distance $\frac{d}{\sqrt{3}}$ from the centroid, the moment of inertia is $I_{\text{parallel}} = I_{\text{sphere}} + m\left(\frac{d}{\sqrt{3}}\right)^2$.
   • Therefore, $I_{\text{parallel}} = \frac{md^2}{10} + m\frac{d^2}{3} = \frac{md^2}{10} + \frac{md^2}{3} = \frac{13md^2}{30}$.
   • Since there are three spheres arranged symmetrically: $I_0 = 3 \cdot \frac{13md^2}{30} = \frac{13md^2}{10}$.
2. Calculate the moment of inertia $I_A$ of the system about the center of one sphere:
   • The moment of inertia $I_A$ for one sphere about its own center is $I_{\text{sphere}} = \frac{md^2}{10}$.
   • For the other two spheres, each is at a distance $d$ from this center. Using the parallel axis theorem for each of these two spheres, the moment of inertia is $I_{\text{parallel, other}} = \frac{md^2}{10} + md^2 = \frac{11md^2}{10}$ for each of these spheres.
   • Summing all contributions, $I_A = \frac{md^2}{10} + 2 \cdot \frac{11md^2}{10} = \frac{23md^2}{10}$.
3. Calculate the ratio $I_0/I_A$: $$\frac{I_0}{I_A} = \frac{\frac{13md^2}{10}}{\frac{23md^2}{10}} = \frac{13}{23}.$$

Therefore, the ratio of the moments of inertia is $\frac{13}{23}$, which is the correct answer.