Two bar magnets of the same length and breadth but having magnetic moments `M` and `2M` are joined with like poles together and suspended by a string. The time of oscillation of this assembly in a magnetic field of strength `B` is `3` sec. What will be the period of oscillation, if the polarity of one of the magnets is changed and the combination is again made to oscillate in the same field ?

Options

A. `sqrt3s`
B. `3sqrt3s`
C. `3s`
D. 6s

Explanation

To solve the problem, we need to analyze the time period of oscillation of two bar magnets under different configurations.

### Step-by-step Solution:

1. **Understanding the Configuration**:
- We have two bar magnets with magnetic moments \( M \) and \( 2M \).
- Initially, they are joined with like poles together (both north or both south).

2. **Time Period Formula for Like Poles**:
- The time period \( T_1 \) for the oscillation of the assembly with like poles together can be expressed as:
\[ T_1 = 2\pi \sqrt{\frac{I_1 + I_2}{(M + 2M) \cdot h}} \]
- Here, \( I_1 \) and \( I_2 \) are the moments of inertia of the two magnets, \( h \) is the distance from the pivot to the center of mass, and \( M + 2M = 3M \) is the effective magnetic moment.

3. **Given Information**:
- We know that \( T_1 = 3 \) seconds.

4. **Changing the Configuration**:
- Now, we change the polarity of one of the magnets, making it an unlike pole configuration.
- The effective magnetic moment in this case becomes \( 2M - M = M \).

5. **Time Period Formula for Unlike Poles**:
- The time period \( T_2 \) for the oscillation of the assembly with unlike poles can be expressed as:
\[ T_2 = 2\pi \sqrt{\frac{I_1 + I_2}{(2M - M) \cdot h}} = 2\pi \sqrt{\frac{I_1 + I_2}{M \cdot h}} \]

6. **Relating the Two Time Periods**:
- We can relate \( T_2 \) to \( T_1 \):
\[ \frac{T_2}{T_1} = \sqrt{\frac{I_1 + I_2}{I_1 + I_2}} \cdot \sqrt{\frac{3M \cdot h}{M \cdot h}} = \sqrt{3} \]
- Thus, we have:
\[ T_2 = T_1 \cdot \sqrt{3} \]

7. **Calculating \( T_2 \)**:
- Substituting \( T_1 = 3 \) seconds:
\[ T_2 = 3 \cdot \sqrt{3} = 3\sqrt{3} \text{ seconds} \]

### Final Answer:
The period of oscillation when the polarity of one of the magnets is changed is \( 3\sqrt{3} \) seconds.