Question

Two spheres of equal masses, one of which is a thin spherical shell and the other a solid, have the same moment of inertia about their respective diameters. The ratio of their radii will be

• A. \(5:7\)
• B. \(3:5\)
• C. \(\sqrt{3}:\sqrt{5}\)
• D. \(\sqrt{3}:\sqrt{7}\)

Hint:

Thin spherical shell: \(I = \frac{2}{3}MR^2\); Solid sphere: \(I = \frac{2}{5}MR^2\).

Answer 1

The Correct Option is C

Step 1: Concept:
Moment of inertia of thin spherical shell about diameter: \(I_s = \frac{2}{3}MR_1^2\).
Moment of inertia of solid sphere about diameter: \(I_{solid} = \frac{2}{5}MR_2^2\).
Step 2: Explanation:
Given \(I_s = I_{solid} \Rightarrow \frac{2}{3}MR_1^2 = \frac{2}{5}MR_2^2\).
\(\frac{R_1^2}{R_2^2} = \frac{2/5}{2/3} = \frac{3}{5} \Rightarrow \frac{R_1}{R_2} = \sqrt{\frac{3}{5}}\).
Thus, \(R_1 : R_2 = \sqrt{3} : \sqrt{5}\).
Step 3: Conclusion:
Thus, ratio of radii = \(\sqrt{3}:\sqrt{5}\).