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 analyze the relationship between the time period of oscillation of a magnet and its moment of inertia. The formula for the time period \( T \) of a magnet oscillating in a magnetic field is given by:

\[ T = 2\pi \sqrt{\frac{I}{mB}} \]

Where:
- \( T \) is the time period of oscillation,
- \( I \) is the moment of inertia,
- \( m \) is the magnetic moment,
- \( B \) is the magnetic field strength.

### Step-by-Step Solution:

1. **Identify the Given Ratios**:
We know that the ratio of the time periods of oscillation of two magnets is given as \( T_1 : T_2 = 2 : 1 \).

2. **Write the Time Periods**:
Let \( T_1 = 2T \) and \( T_2 = T \) for some time period \( T \).

3. **Use the Time Period Formula**:
According to the formula, we can express the time periods in terms of their respective moments of inertia:
\[ T_1 = 2\pi \sqrt{\frac{I_1}{mB}} \quad \text{and} \quad T_2 = 2\pi \sqrt{\frac{I_2}{mB}} \]

4. **Set Up the Ratio**:
Since \( T_1 : T_2 = 2 : 1 \), we can write:
\[ \frac{T_1}{T_2} = \frac{2}{1} \]
Substituting the expressions for \( T_1 \) and \( T_2 \):
\[ \frac{2\pi \sqrt{\frac{I_1}{mB}}}{2\pi \sqrt{\frac{I_2}{mB}}} = \frac{2}{1} \]

5. **Simplify the Equation**:
The \( 2\pi \) and \( mB \) terms cancel out:
\[ \frac{\sqrt{I_1}}{\sqrt{I_2}} = 2 \]

6. **Square Both Sides**:
Squaring both sides gives:
\[ \frac{I_1}{I_2} = 4 \]

7. **Final Ratio**:
Thus, the ratio of the moments of inertia of the two magnets is:
\[ I_1 : I_2 = 4 : 1 \]

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