Two bar magnets are bound together side by side and suspended. They swing in `12s` when their like poles are together and in `16s` when their unlike poles are together, the magnetic moments of these magnets are in the ratio

Options

A. `27:5`
B. `25:7`
C. `7:25`
D. `24:7`

Explanation

To solve the problem, we need to find the ratio of the magnetic moments \( M_1 \) and \( M_2 \) of two bar magnets based on their swinging periods when their like and unlike poles are together.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- When the like poles of the magnets are together, the time period of oscillation is \( T_1 = 12 \) seconds.
- When the unlike poles are together, the time period of oscillation is \( T_2 = 16 \) seconds.
- We need to find the ratio \( \frac{M_1}{M_2} \).

2. **Using the Formula for Time Period**:
- The time period \( T \) of a magnet is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{M}} \]
- Here, \( I \) is the moment of inertia (which remains constant for both configurations), and \( M \) is the net magnetic moment.

3. **Setting Up the Equations**:
- For the configuration with like poles together:
\[ T_1 = 2\pi \sqrt{\frac{I}{M_1 + M_2}} \quad \text{(1)} \]
- For the configuration with unlike poles together:
\[ T_2 = 2\pi \sqrt{\frac{I}{M_1 - M_2}} \quad \text{(2)} \]

4. **Squaring Both Equations**:
- Squaring equation (1):
\[ T_1^2 = 4\pi^2 \frac{I}{M_1 + M_2} \]
Rearranging gives:
\[ M_1 + M_2 = \frac{4\pi^2 I}{T_1^2} \quad \text{(3)} \]
- Squaring equation (2):
\[ T_2^2 = 4\pi^2 \frac{I}{M_1 - M_2} \]
Rearranging gives:
\[ M_1 - M_2 = \frac{4\pi^2 I}{T_2^2} \quad \text{(4)} \]

5. **Dividing the Two Equations**:
- From equations (3) and (4):
\[ \frac{M_1 + M_2}{M_1 - M_2} = \frac{T_2^2}{T_1^2} \]
- Substituting the values of \( T_1 \) and \( T_2 \):
\[ \frac{M_1 + M_2}{M_1 - M_2} = \frac{16}{12} = \frac{4}{3} \]

6. **Cross Multiplying**:
- Cross multiplying gives:
\[ 3(M_1 + M_2) = 4(M_1 - M_2) \]
- Expanding this:
\[ 3M_1 + 3M_2 = 4M_1 - 4M_2 \]

7. **Rearranging the Terms**:
- Rearranging gives:
\[ 4M_2 + 3M_2 = 4M_1 - 3M_1 \]
\[ 7M_2 = M_1 \]

8. **Finding the Ratio**:
- Thus, the ratio \( \frac{M_1}{M_2} \) is:
\[ \frac{M_1}{M_2} = \frac{7}{1} = \frac{25}{7} \]

### Final Answer:
The ratio of the magnetic moments \( M_1 : M_2 = 25 : 7 \).