1. Recall the moments of inertia about a diameter

For a solid sphere of mass $M$ and radius $R$: $I_A=\frac{2}{5}MR^2$

For a hollow sphere (spherical shell) of mass $M$ and radius $R$: $I_B=\frac{2}{3}MR^2$

Since the problem states that both spheres have the same mass and the same outer radius, we can directly compare these two expressions.

2. Compare $I_A$ and $I_B$

$I_A=\frac{2}{5}MR^2, \qquad I_B=\frac{2}{3}MR^2$

Because $\frac{2}{5}<\frac{2}{3}$ we get $I_A<I_B$

3. Check the options

• A: $I_A<I_B$ ✅ Correct
• B: $I_A>I_B$ ❌
• C: $I_A=I_B$ ❌
• D: $\dfrac{I_A}{I_B}=\dfrac{d_A}{d_B}$ ❌

Let us verify option D explicitly: $\frac{I_A}{I_B}=\frac{\frac{2}{5}MR^2}{\frac{2}{3}MR^2}=\frac{3}{5}$

Now densities are not equal because masses are same but the solid sphere and hollow sphere have different material volumes. Also there is no general reason that $\frac{I_A}{I_B}=\frac{d_A}{d_B}$ So D is false.

4. Final answer

The correct option is: $\boxed{A}$

5. Comparison with stored answer

Stored correct answer: A

This matches our derived answer.