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. (a)`5:3`
B. (b)`sqrt(3): sqrt(5)`
C. (c)`sqrt(5):sqrt(3)`
D. (d)`1:1`

Explanation

To solve the problem of finding the ratio of the radii of a hollow sphere and a solid sphere with identical masses and moments of inertia about their respective diameters, we can follow these steps:

### Step 1: Write the Moment of Inertia Formulas
The moment of inertia \( I \) for a hollow sphere (thin spherical shell) and a solid sphere about their diameters is given by:
- For the hollow sphere:
\[ I_1 = \frac{2}{3} m r_1^2 \]
- For the solid sphere:
\[ I_2 = \frac{2}{5} m r_2^2 \]

### Step 2: Set the Moments of Inertia Equal
Since it is given that the moments of inertia are identical, we set \( I_1 \) equal to \( I_2 \):
\[ \frac{2}{3} m r_1^2 = \frac{2}{5} m r_2^2 \]

### Step 3: Cancel Common Terms
We can cancel the mass \( m \) from both sides of the equation:
\[ \frac{2}{3} r_1^2 = \frac{2}{5} r_2^2 \]

### Step 4: Simplify the Equation
Next, we can simplify the equation by multiplying both sides by \( 15 \) (the least common multiple of 3 and 5) to eliminate the fractions:
\[ 15 \cdot \frac{2}{3} r_1^2 = 15 \cdot \frac{2}{5} r_2^2 \]
This simplifies to:
\[ 10 r_1^2 = 6 r_2^2 \]

### Step 5: Rearrange to Find the Ratio of Radii
Now, we can rearrange this equation to find the ratio of the squares of the radii:
\[ \frac{r_1^2}{r_2^2} = \frac{6}{10} = \frac{3}{5} \]

### Step 6: Take the Square Root
Taking the square root of both sides gives us the ratio of the radii:
\[ \frac{r_1}{r_2} = \sqrt{\frac{3}{5}} = \frac{\sqrt{3}}{\sqrt{5}} \]

### Conclusion
Thus, the ratio of the radii \( r_1 \) to \( r_2 \) is:
\[ r_1 : r_2 = \sqrt{3} : \sqrt{5} \]