Two bar magnets of the same mass, length and breadth but magnetic moment M amd 2M respectively, when placed in same position, time period is 3 sec. What will be the time period when they are placed in different positio?

Options

A. `sqrt(3) sec`
B. `3sqrt(3) sec`
C. `3 sec`
D. `6 sec`

Explanation

To solve the problem step by step, we will analyze the situation involving the two bar magnets with different magnetic moments and derive the time period when they are placed in different positions.

### Step 1: Understand the Given Information
We have two bar magnets:
- Magnet 1 has a magnetic moment \( M_1 = M \)
- Magnet 2 has a magnetic moment \( M_2 = 2M \)
- When placed in the same position, the time period \( T_s = 3 \) seconds.

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

### Step 3: Time Period for Same Position
When both magnets are placed in the same position, their effective magnetic moment is the sum of their individual magnetic moments:
\[ M_s = M_1 + M_2 = M + 2M = 3M \]
Thus, the time period when they are together is:
\[ T_s = 2\pi \sqrt{\frac{I}{3M \cdot B}} \]
Given that \( T_s = 3 \) seconds, we can write:
\[ 3 = 2\pi \sqrt{\frac{I}{3M \cdot B}} \]

### Step 4: Time Period for Different Positions
When the magnets are placed in different positions, the effective magnetic moment is the difference of their magnetic moments:
\[ M_d = M_2 - M_1 = 2M - M = M \]
Thus, the time period when they are apart is:
\[ T_d = 2\pi \sqrt{\frac{I}{M \cdot B}} \]

### Step 5: Relate the Time Periods
We can relate the time periods \( T_s \) and \( T_d \) using their effective magnetic moments:
\[ \frac{T_s^2}{T_d^2} = \frac{M_d}{M_s} = \frac{M}{3M} = \frac{1}{3} \]
This implies:
\[ T_d^2 = 3T_s^2 \]

### Step 6: Calculate \( T_d \)
Substituting \( T_s = 3 \) seconds into the equation:
\[ T_d^2 = 3 \cdot (3)^2 = 3 \cdot 9 = 27 \]
Taking the square root gives:
\[ T_d = \sqrt{27} = 3\sqrt{3} \text{ seconds} \]

### Final Answer
The time period when the magnets are placed in different positions is:
\[ T_d = 3\sqrt{3} \text{ seconds} \]

---