Two bar-magnets having their moment of inertia in the ratio `2: 3` oscillate in a horizontal plane with time periods `(5)/(2)` s and `(9)/(2)` s respectively . The ratio of their magnetic moments is

Options

A. `27 : 8`
B. `54: 25`
C. `25: 7`
D. `32 : 9`

Answer

To solve the problem, we need to find the ratio of the magnetic moments of two bar magnets given their moments of inertia and time periods of oscillation. Let's denote the two bar magnets as Magnet 1 and Magnet 2.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Moment of inertia ratio: \( I_1 : I_2 = 2 : 3 \)
- Time period of Magnet 1: \( T_1 = \frac{5}{2} \) s
- Time period of Magnet 2: \( T_2 = \frac{9}{2} \) s

2. **Write the Formula for Time Period:**
The time period \( T \) of a bar magnet oscillating in a magnetic field is given by:
\[ T = 2\pi \sqrt{\frac{I}{MB}} \]
where \( I \) is the moment of inertia, \( M \) is the magnetic moment, and \( B \) is the magnetic field strength.

3. **Set Up the Ratio of Time Periods:**
Since both magnets are in the same magnetic field \( B \), we can write the ratio of their time periods:
\[ \frac{T_1}{T_2} = \sqrt{\frac{I_1}{M_1}} \Bigg/ \sqrt{\frac{I_2}{M_2}} \]
This simplifies to:
\[ \frac{T_1}{T_2} = \sqrt{\frac{I_1}{I_2}} \cdot \sqrt{\frac{M_2}{M_1}} \]

4. **Square Both Sides:**
Squaring both sides gives:
\[ \left(\frac{T_1}{T_2}\right)^2 = \frac{I_1}{I_2} \cdot \frac{M_2}{M_1} \]

5. **Rearranging for Magnetic Moment Ratio:**
Rearranging the equation for the ratio of magnetic moments \( \frac{M_1}{M_2} \):
\[ \frac{M_1}{M_2} = \frac{I_1}{I_2} \cdot \left(\frac{T_2}{T_1}\right)^2 \]

6. **Substituting Known Values:**
- From the moment of inertia ratio \( \frac{I_1}{I_2} = \frac{2}{3} \)
- Calculate \( \frac{T_2}{T_1} = \frac{\frac{9}{2}}{\frac{5}{2}} = \frac{9}{5} \)

7. **Calculating the Square of the Time Period Ratio:**
\[ \left(\frac{T_2}{T_1}\right)^2 = \left(\frac{9}{5}\right)^2 = \frac{81}{25} \]

8. **Final Calculation:**
Now substitute these values back into the equation:
\[ \frac{M_1}{M_2} = \frac{2}{3} \cdot \frac{81}{25} = \frac{162}{75} \]

9. **Simplifying the Ratio:**
Simplifying \( \frac{162}{75} \):
\[ \frac{M_1}{M_2} = \frac{54}{25} \]

### Conclusion:
The ratio of their magnetic moments is:
\[ \frac{M_1}{M_2} = \frac{54}{25} \]