Two similar magnets of magnets moments `M_(1)` and `M_(2)` are taken and vibration magnetometer their (a) Unlike poles together (b) Like poles together respectively .
The ratio of their time period is `2 : 1` . Then the ratio `M_(1) : M_(2)` is `(M_(2) gt M_(1))`

Options

A. `0.8`
B. `0.6`
C. `0.5`
D. `0.7`

Answer

To solve the problem, we will analyze the time periods of the two configurations of the magnets in a vibration magnetometer.

### Step-by-Step Solution:

1. **Understanding the Time Periods**:
- When the unlike poles are together, the effective magnetic moment \( M_{\text{net}} \) is given by:
\[ M_{\text{net}} = M_2 - M_1 \]
- When the like poles are together, the effective magnetic moment \( M_{\text{net}} \) is:
\[ M_{\text{net}} = M_2 + M_1 \]

2. **Formulating the Time Periods**:
- The time period \( T \) of a vibration magnetometer is given by:
\[ T = 2\pi \sqrt{\frac{I}{M_{\text{net}}}} \]
- Therefore, the time periods for the two configurations are:
- For unlike poles together:
\[ T_1 = 2\pi \sqrt{\frac{I}{M_2 - M_1}} \]
- For like poles together:
\[ T_2 = 2\pi \sqrt{\frac{I}{M_2 + M_1}} \]

3. **Setting Up the Ratio**:
- We are given that the ratio of the time periods is \( \frac{T_1}{T_2} = \frac{2}{1} \).
- Substituting the expressions for \( T_1 \) and \( T_2 \):
\[ \frac{T_1}{T_2} = \frac{2\pi \sqrt{\frac{I}{M_2 - M_1}}}{2\pi \sqrt{\frac{I}{M_2 + M_1}}} = \frac{\sqrt{M_2 + M_1}}{\sqrt{M_2 - M_1}} = \frac{2}{1} \]

4. **Squaring Both Sides**:
- Squaring the equation gives:
\[ \frac{M_2 + M_1}{M_2 - M_1} = 4 \]

5. **Cross-Multiplying**:
- Cross-multiplying leads to:
\[ M_2 + M_1 = 4(M_2 - M_1) \]

6. **Expanding and Rearranging**:
- Expanding the right side:
\[ M_2 + M_1 = 4M_2 - 4M_1 \]
- Rearranging gives:
\[ M_1 + 4M_1 = 4M_2 - M_2 \]
\[ 5M_1 = 3M_2 \]

7. **Finding the Ratio**:
- Dividing both sides by \( M_2 \):
\[ \frac{M_1}{M_2} = \frac{3}{5} \]
- Thus, the ratio \( M_1 : M_2 \) is \( 3 : 5 \).

### Conclusion:
The ratio \( M_1 : M_2 \) is \( 3 : 5 \).