Time periods of vibration of two bar magnets in sum and difference positions are 4 s and 6 s respectively . The ratio of their magnetic moments `(M_1)/(M_2)` is

Options

A. (a)`6:4`
B. (b)`36 : 16`
C. (c)`2.6:1`
D. (d)`1.5:1`

Answer

To solve the problem, we need to find the ratio of the magnetic moments \( \frac{M_1}{M_2} \) of two bar magnets based on their time periods of vibration in sum and difference positions.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Time period in sum position \( T_1 = 4 \, \text{s} \)
- Time period in difference position \( T_2 = 6 \, \text{s} \)

2. **Use the Formula for Magnetic Moments**:
The ratio of the magnetic moments can be calculated using the formula:
\[ \frac{M_1}{M_2} = \frac{T_2^2 + T_1^2}{T_2^2 - T_1^2} \]

3. **Calculate \( T_1^2 \) and \( T_2^2 \)**:
- \( T_1^2 = 4^2 = 16 \)
- \( T_2^2 = 6^2 = 36 \)

4. **Substitute Values into the Formula**:
\[ \frac{M_1}{M_2} = \frac{36 + 16}{36 - 16} \]

5. **Perform the Addition and Subtraction**:
- \( 36 + 16 = 52 \)
- \( 36 - 16 = 20 \)

6. **Calculate the Ratio**:
\[ \frac{M_1}{M_2} = \frac{52}{20} \]

7. **Simplify the Ratio**:
- Dividing both the numerator and denominator by 4 gives:
\[ \frac{M_1}{M_2} = \frac{13}{5} \quad \text{or} \quad 2.6 : 1 \]

### Final Answer:
The ratio of their magnetic moments \( \frac{M_1}{M_2} \) is \( 2.6 : 1 \).