Magnets `A` and `B` are geometrically similar but the magnetic moment of `A` is twice that of `B`. If `T_(1)` and `T_(2)` be the time periods of the oscillation when their like poles and unlike poles are kept togather respectively, then `T_(1)/T_(2)` will be

Options

A. `(1)/(3)`
B. `(1)/(2)`
C. `(1)/(sqrt3)`
D. `sqrt3`

Explanation

To solve the problem, we need to find the ratio of the time periods \( T_1 \) and \( T_2 \) when magnets A and B are placed with like poles and unlike poles together, respectively.

### Step-by-Step Solution:

1. **Understanding the Time Periods**:
- The time period \( T_1 \) when like poles are together is given by:
\[ T_1 = 2\pi \sqrt{\frac{I_1 + I_2}{M_1 + M_2} \cdot \frac{1}{B_H}} \]
- The time period \( T_2 \) when unlike poles are together is given by:
\[ T_2 = 2\pi \sqrt{\frac{I_1 + I_2}{M_1 - M_2} \cdot \frac{1}{B_H}} \]

2. **Taking the Ratio \( \frac{T_1}{T_2} \)**:
- We can write the ratio of the time periods as follows:
\[ \frac{T_1}{T_2} = \frac{2\pi \sqrt{\frac{I_1 + I_2}{M_1 + M_2}}}{2\pi \sqrt{\frac{I_1 + I_2}{M_1 - M_2}}} \]
- The \( 2\pi \) and \( \sqrt{I_1 + I_2} \) terms cancel out:
\[ \frac{T_1}{T_2} = \sqrt{\frac{M_1 - M_2}{M_1 + M_2}} \]

3. **Substituting the Magnetic Moments**:
- Given that the magnetic moment of A (\( M_1 \)) is twice that of B (\( M_2 \)), we can express \( M_1 \) and \( M_2 \) as:
\[ M_1 = 2M \quad \text{and} \quad M_2 = M \]
- Now substituting these values into the ratio:
\[ \frac{T_1}{T_2} = \sqrt{\frac{(2M) - M}{(2M) + M}} = \sqrt{\frac{M}{3M}} = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \]

4. **Final Result**:
- Therefore, the ratio \( \frac{T_1}{T_2} \) is:
\[ \frac{T_1}{T_2} = \frac{1}{\sqrt{3}} \]

### Conclusion:
The final answer is \( \frac{T_1}{T_2} = \frac{1}{\sqrt{3}} \).