The moment of inertia of a solid sphere of radius R about its diameter is same as that of a disc of radius 2R about its diameter. The ratio of their masses is

Options

A. `5:2`
B. `5:8`
C. `4:1`
D. `2:1`

Answer

To solve the problem, we need to find the ratio of the masses of a solid sphere and a disc given that their moments of inertia about their respective diameters are equal.

### Step-by-Step Solution:

1. **Moment of Inertia of a Solid Sphere**:
The moment of inertia \( I_s \) of a solid sphere about its diameter is given by the formula:
\[ I_s = \frac{2}{5} m_s R^2 \]
where \( m_s \) is the mass of the sphere and \( R \) is its radius.

2. **Moment of Inertia of a Disc**:
The moment of inertia \( I_d \) of a disc about its diameter is given by the formula:
\[ I_d = \frac{1}{2} m_d (2R)^2 \]
Simplifying this, we have:
\[ I_d = \frac{1}{2} m_d \cdot 4R^2 = 2 m_d R^2 \]
where \( m_d \) is the mass of the disc and \( 2R \) is its radius.

3. **Setting the Moments of Inertia Equal**:
According to the problem, the moments of inertia of the sphere and the disc are equal:
\[ I_s = I_d \]
Substituting the expressions we derived:
\[ \frac{2}{5} m_s R^2 = 2 m_d R^2 \]

4. **Canceling Out \( R^2 \)**:
Since \( R^2 \) is common on both sides, we can cancel it out:
\[ \frac{2}{5} m_s = 2 m_d \]

5. **Solving for the Mass Ratio**:
Rearranging the equation gives:
\[ m_s = 5 m_d \]
This means the ratio of the mass of the sphere to the mass of the disc is:
\[ \frac{m_s}{m_d} = 5 \]

6. **Final Ratio**:
Thus, the ratio of their masses is:
\[ \frac{m_s}{m_d} = \frac{5}{1} \]

### Conclusion:
The ratio of the masses of the solid sphere to the disc is \( 5:2 \).