We have two spheres, one of which is hollow and the other solid. They have identical masses and moment of intertia about their respective diameters. The ratio of their radius is given by.

Options

A. `5:7`
B. `3:5`
C. `sqrt(3):sqrt(5)`
D. `3:7`

Answer

To solve the problem, we need to analyze the moment of inertia for both the hollow and solid spheres and establish a relationship between their radii given that they have identical masses and moments of inertia.

### Step-by-Step Solution:

1. **Identify the Moment of Inertia Formulas**:
- For a hollow sphere (thin spherical shell), the moment of inertia about its diameter is given by:
\[ I_{\text{hollow}} = \frac{2}{3} m R^2 \]
- For a solid sphere, the moment of inertia about its diameter is given by:
\[ I_{\text{solid}} = \frac{2}{5} m r^2 \]

2. **Set Up the Equation**:
Since the problem states that the moments of inertia are equal for both spheres, we can set the two equations equal to each other:
\[ \frac{2}{3} m R^2 = \frac{2}{5} m r^2 \]

3. **Cancel Common Terms**:
We can cancel the mass \( m \) from both sides since they are identical:
\[ \frac{2}{3} R^2 = \frac{2}{5} r^2 \]

4. **Eliminate the Coefficients**:
Next, we can eliminate the factor of 2 from both sides:
\[ \frac{1}{3} R^2 = \frac{1}{5} r^2 \]

5. **Cross-Multiply**:
To eliminate the fractions, we cross-multiply:
\[ 5 R^2 = 3 r^2 \]

6. **Rearrange for the Ratio**:
Now, we can express the ratio of the radii:
\[ \frac{R^2}{r^2} = \frac{3}{5} \]

7. **Take the Square Root**:
Finally, we take the square root of both sides to find the ratio of the radii:
\[ \frac{R}{r} = \sqrt{\frac{3}{5}} = \frac{\sqrt{3}}{\sqrt{5}} \]

### Conclusion:
Thus, the ratio of the radius of the hollow sphere to the radius of the solid sphere is:
\[ \frac{R}{r} = \frac{\sqrt{3}}{\sqrt{5}} \]