One solid sphere A and another hollow sphere B are of same mass and same outer radii. Their moment of inertia about their diameters are respectively $$I_A$$ and $$I_B$$ such that

Let both spheres have the same mass $$M$$ and the same outer radius $$R$$.

1. For Solid Sphere A:

The mass of a solid sphere is uniformly distributed from the center out to its outer radius $$R$$. Its moment of inertia about a diameter is given by the formula:

$$I_A = \frac{2}{5}MR^2 = 0.4MR^2$$

2. For Hollow Sphere B:

The entire mass of a thin hollow sphere (or spherical shell) is concentrated completely at its outer shell surface, at a distance $$R$$ from the center. Its moment of inertia about a diameter is given by the formula:

$$I_B = \frac{2}{3}MR^2 \approx 0.67MR^2$$

Conceptual Comparison:

Moment of inertia is a measure of how far the mass of an object is distributed relative to the axis of rotation.

In the solid sphere, a significant portion of the mass is located closer to the axis of rotation (near the core).

 In the hollow sphere, all of the mass is pushed as far away from the axis as possible (at the outer radius $$R$$).

Since mass located further from the axis contributes more to the moment of inertia, the hollow sphere will have a greater moment of inertia than the solid sphere of the same mass and radius.

Comparing the numerical coefficients:

$$0.4MR^2 < 0.67MR^2$$

$$I_A < I_B$$  Option C