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. `6:4`
B. `36 : 16`
C. `2.6:1`
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. The time periods are given as follows:

- Time period in sum position \( T_1 = 4 \, \text{s} \)
- Time period in difference position \( T_2 = 6 \, \text{s} \)

### Step-by-Step Solution:

1. **Identify the Formula**:
The ratio of the magnetic moments \( \frac{M_1}{M_2} \) can be calculated using the formula:
\[ \frac{M_1}{M_2} = \frac{T_2^2 + T_1^2}{T_2^2 - T_1^2} \]

2. **Substitute the Values**:
Substitute \( T_1 \) and \( T_2 \) into the formula:
\[ \frac{M_1}{M_2} = \frac{6^2 + 4^2}{6^2 - 4^2} \]

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

4. **Plug in the Squared Values**:
Now substitute these squared values back into the equation:
\[ \frac{M_1}{M_2} = \frac{36 + 16}{36 - 16} \]

5. **Simplify the Numerator and Denominator**:
- Numerator: \( 36 + 16 = 52 \)
- Denominator: \( 36 - 16 = 20 \)

6. **Calculate the Ratio**:
Now we can calculate the ratio:
\[ \frac{M_1}{M_2} = \frac{52}{20} \]

7. **Simplify the Fraction**:
Simplifying \( \frac{52}{20} \):
\[ \frac{M_1}{M_2} = 2.6 \]

### Final Answer:
Thus, the ratio of their magnetic moments \( \frac{M_1}{M_2} \) is:
\[ \frac{M_1}{M_2} = 2.6 : 1 \]