The ratio of time periods of oscillaiton of two magnets in the same field is 2:1 . If magnetic moment of both the magnets is equals , the ratio of their moment of inertias will be .

Options

A. `4:1`
B. `1:4`
C. `sqrt(2) : 1`
D. `1: sqrt(2)`

Explanation

To solve the problem, we need to determine the ratio of the moments of inertia of two magnets based on the given ratio of their time periods of oscillation.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- The ratio of the time periods of oscillation of two magnets is given as \( T_1 : T_2 = 2 : 1 \).
- The magnetic moments of both magnets are equal, i.e., \( M_1 = M_2 = M \).

2. **Using the Formula for Time Period:**
- The time period \( T \) of a vibrating magnet can be expressed using the formula:
\[ T = 2\pi \sqrt{\frac{I}{M \cdot B}} \]
where \( I \) is the moment of inertia, \( M \) is the magnetic moment, and \( B \) is the magnetic field strength.

3. **Setting Up the Ratios:**
- Since the magnetic moments \( M_1 \) and \( M_2 \) are equal and the magnetic field \( B \) is the same for both magnets, we can simplify the relationship:
\[ T_1 \propto \sqrt{I_1} \quad \text{and} \quad T_2 \propto \sqrt{I_2} \]
- Therefore, we can write:
\[ \frac{T_1}{T_2} = \frac{\sqrt{I_1}}{\sqrt{I_2}} \]

4. **Substituting the Given Ratio:**
- From the problem, we know:
\[ \frac{T_1}{T_2} = \frac{2}{1} \]
- Thus, we can equate:
\[ \frac{2}{1} = \frac{\sqrt{I_1}}{\sqrt{I_2}} \]

5. **Squaring Both Sides:**
- To eliminate the square root, we square both sides:
\[ \left(\frac{2}{1}\right)^2 = \frac{I_1}{I_2} \]
- This simplifies to:
\[ \frac{4}{1} = \frac{I_1}{I_2} \]

6. **Conclusion:**
- Therefore, the ratio of the moments of inertia of the two magnets is:
\[ I_1 : I_2 = 4 : 1 \]

### Final Answer:
The ratio of their moments of inertia is \( 4 : 1 \).

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