Question

The ratio of radius of gyration of a solid sphere of mass M and radius R about its own axis to the radius of gyration of the thin hollow sphere of same mass and radius about its axis is:

• A. 5:2
• B. 3:5
• C. 5:3
• D. 2:5

Answer 1

The Correct Option is B

To solve this problem, we need to find the ratio of the radius of gyration of a solid sphere to that of a thin hollow sphere, both having the same mass \(M\) and radius \(R\). 

1. Radius of gyration: The radius of gyration \(K\) of an object is defined as the distance from the axis of rotation at which the entire mass of the body can be assumed to be concentrated to provide a moment of inertia \(I\). Mathematically, \(K = \sqrt{\frac{I}{M}}\).

2. Solid sphere: For a solid sphere, the moment of inertia about its axis is \(I_s = \frac{2}{5}MR^2\). Thus, the radius of gyration \(K_s\) is:

\(K_s = \sqrt{\frac{\frac{2}{5}MR^2}{M}} = \sqrt{\frac{2}{5}R^2} = \frac{R}{\sqrt{5}}\).

3. Thin hollow sphere: For a thin hollow sphere, the moment of inertia about its axis is \(I_h = \frac{2}{3}MR^2\). Thus, the radius of gyration \(K_h\) is:

\(K_h = \sqrt{\frac{\frac{2}{3}MR^2}{M}} = \sqrt{\frac{2}{3}R^2} = \frac{R}{\sqrt{3}}\).

4. Ratio: The ratio of the radii of gyration of the solid sphere to the hollow sphere is given by:

\(\frac{K_s}{K_h} = \frac{\frac{R}{\sqrt{5}}}{\frac{R}{\sqrt{3}}} = \frac{\sqrt{3}}{\sqrt{5}}\).

Simplifying, we multiply both numerator and denominator by \(\sqrt{15}\) to get:

\(\frac{\sqrt{3}}{\sqrt{5}} \times \frac{\sqrt{15}}{\sqrt{15}} = \frac{3}{5}\).

The ratio is therefore 3:5, which matches the given correct option.

Answer 2

The correct option is (B): 3:5
The radius of gyration: K=\(\sqrt {\frac{I}{m}}\)
\(\frac{K_{solid\,sphere}}{K_{hollow\,sphere}}=\frac{\sqrt{(2mR^2/5m)}}{\sqrt{(2mR^2/3m)}}\)
=3:5