A spherical shell has mass one fourth of mass of a solid sphere and both have same moment of inertia about their respective diameters. The ratio of their radii will be

Options

A. `1:2`
B. `sqrt12: sqrt5`
C. `sqrt3:sqrt20`
D. `sqrt5: sqrt20`

Explanation

To solve the problem, we need to find the ratio of the radii of a hollow sphere and a solid sphere given that the hollow sphere has a mass that is one fourth of the mass of the solid sphere, and both have the same moment of inertia about their respective diameters.

### Step-by-Step Solution:

1. **Define the Masses:**
Let the mass of the solid sphere (SS) be \( M \). Then, the mass of the hollow sphere (HS) is given as:
\[ m_{HS} = \frac{M}{4} \]

2. **Moment of Inertia Formulas:**
The moment of inertia \( I \) about the diameter for a solid sphere is given by:
\[ I_{SS} = \frac{2}{5} M R_{SS}^2 \]
where \( R_{SS} \) is the radius of the solid sphere.

The moment of inertia for a hollow sphere about its diameter is:
\[ I_{HS} = \frac{2}{3} m_{HS} R_{HS}^2 \]
where \( R_{HS} \) is the radius of the hollow sphere.

3. **Set the Moments of Inertia Equal:**
According to the problem, the moments of inertia are equal:
\[ I_{SS} = I_{HS} \]
Substituting the formulas we have:
\[ \frac{2}{5} M R_{SS}^2 = \frac{2}{3} \left(\frac{M}{4}\right) R_{HS}^2 \]

4. **Simplify the Equation:**
Cancel \( M \) from both sides (assuming \( M \neq 0 \)):
\[ \frac{2}{5} R_{SS}^2 = \frac{2}{3} \cdot \frac{1}{4} R_{HS}^2 \]
This simplifies to:
\[ \frac{2}{5} R_{SS}^2 = \frac{1}{6} R_{HS}^2 \]

5. **Cross-Multiply to Solve for the Radii:**
Cross-multiplying gives:
\[ 2 \cdot 6 R_{SS}^2 = 5 R_{HS}^2 \]
Simplifying this, we get:
\[ 12 R_{SS}^2 = 5 R_{HS}^2 \]

6. **Find the Ratio of the Radii:**
Rearranging gives:
\[ \frac{R_{HS}^2}{R_{SS}^2} = \frac{12}{5} \]
Taking the square root of both sides:
\[ \frac{R_{HS}}{R_{SS}} = \sqrt{\frac{12}{5}} = \frac{\sqrt{12}}{\sqrt{5}} \]

7. **Final Ratio:**
Thus, the ratio of the radii of the hollow sphere to the solid sphere is:
\[ R_{HS} : R_{SS} = \sqrt{12} : \sqrt{5} \]