Two rings have their M.I. in the ratio 2:1. If their diameters are in the ratio of 2:1, then the ratio of their masses will be

Options

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

Explanation

To solve the problem, we need to find the ratio of the masses of two rings given the ratio of their moments of inertia and diameters.

### Step-by-Step Solution:

1. **Understand the Given Ratios**:
- The ratio of the moments of inertia (M.I.) of the two rings is given as:
\[ \frac{I_1}{I_2} = \frac{2}{1} \]
- The ratio of the diameters of the two rings is given as:
\[ \frac{d_1}{d_2} = \frac{2}{1} \]

2. **Relate Diameter to Radius**:
- Since diameter is twice the radius, the ratio of the radii will be the same as the ratio of the diameters:
\[ \frac{r_1}{r_2} = \frac{d_1/2}{d_2/2} = \frac{2/2}{1/2} = \frac{2}{1} \]

3. **Use the Formula for Moment of Inertia**:
- The moment of inertia \(I\) of a ring about its axis is given by:
\[ I = m r^2 \]
- Thus, for the two rings, we can write:
\[ I_1 = m_1 r_1^2 \quad \text{and} \quad I_2 = m_2 r_2^2 \]

4. **Set Up the Ratio of Moments of Inertia**:
- Using the given ratio of moments of inertia:
\[ \frac{I_1}{I_2} = \frac{m_1 r_1^2}{m_2 r_2^2} = \frac{2}{1} \]

5. **Substitute the Radius Ratio**:
- We know that \(\frac{r_1}{r_2} = \frac{2}{1}\), so we can express \(r_1\) in terms of \(r_2\):
\[ r_1 = 2r_2 \]
- Substitute this into the moment of inertia ratio:
\[ \frac{m_1 (2r_2)^2}{m_2 r_2^2} = \frac{2}{1} \]
- Simplifying this gives:
\[ \frac{m_1 \cdot 4r_2^2}{m_2 r_2^2} = \frac{2}{1} \]
- The \(r_2^2\) terms cancel out:
\[ \frac{4m_1}{m_2} = 2 \]

6. **Solve for the Mass Ratio**:
- Rearranging gives:
\[ 4m_1 = 2m_2 \quad \Rightarrow \quad \frac{m_1}{m_2} = \frac{2}{4} = \frac{1}{2} \]

### Conclusion:
The ratio of the masses of the two rings is:
\[ \frac{m_1}{m_2} = \frac{1}{2} \]