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 follow these steps:

### Step 1: Understand the Time Period Formula
The time period \( T \) of a magnet in a vibration magnetometer is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{M_{\text{net}}} \]
where \( I \) is the moment of inertia and \( M_{\text{net}} \) is the net magnetic moment acting on the magnet.

### Step 2: Define the Time Periods for Each Case
Let:
- \( T_1 \) be the time period when the unlike poles are together.
- \( T_2 \) be the time period when the like poles are together.

For the case of unlike poles together:
\[ M_{\text{net}} = M_2 - M_1 \]
Thus, the time period \( T_1 \) can be expressed as:
\[ T_1 = 2\pi \sqrt{\frac{I}{M_2 - M_1}} \]

For the case of like poles together:
\[ M_{\text{net}} = M_2 + M_1 \]
Thus, the time period \( T_2 \) can be expressed as:
\[ T_2 = 2\pi \sqrt{\frac{I}{M_2 + M_1}} \]

### Step 3: Set Up the Ratio of Time Periods
According to the problem, the ratio of the time periods is given as:
\[ \frac{T_1}{T_2} = \frac{2}{1} \]

### Step 4: Substitute the Time Period Expressions
Substituting the expressions for \( T_1 \) and \( T_2 \):
\[ \frac{2\pi \sqrt{\frac{I}{M_2 - M_1}}}{2\pi \sqrt{\frac{I}{M_2 + M_1}}} = \frac{2}{1} \]
This simplifies to:
\[ \frac{\sqrt{\frac{I}{M_2 - M_1}}}{\sqrt{\frac{I}{M_2 + M_1}}} = 2 \]

### Step 5: Square Both Sides
Squaring both sides gives:
\[ \frac{I}{M_2 - M_1} \cdot \frac{M_2 + M_1}{I} = 4 \]
This simplifies to:
\[ \frac{M_2 + M_1}{M_2 - M_1} = 4 \]

### Step 6: Cross-Multiply and Rearrange
Cross-multiplying gives:
\[ M_2 + M_1 = 4(M_2 - M_1) \]
Expanding this results in:
\[ M_2 + M_1 = 4M_2 - 4M_1 \]
Rearranging terms yields:
\[ M_1 + 4M_1 = 4M_2 - M_2 \]
Thus:
\[ 5M_1 = 3M_2 \]

### Step 7: Find the Ratio of Magnetic Moments
Dividing both sides by \( M_2 \) gives:
\[ \frac{M_1}{M_2} = \frac{3}{5} \]
Thus, the ratio \( M_1 : M_2 \) is:
\[ M_1 : M_2 = 3 : 5 \]

### Conclusion
Since it is given that \( M_2 > M_1 \), the final answer is:
\[ M_1 : M_2 = 3 : 5 \]