A thin rectangular magnet suspnded freely has a period of oscillation of 4s . If it is broken into one of the pieces is suspened similarly . The period of its oscillation will be

Options

A. 4s
B. 2s
C. `0.5s`
D. `0.25s`

Explanation

To solve the problem, we need to determine the new period of oscillation of a piece of the broken magnet. We will use the formulas related to the period of oscillation of a magnet and the changes in its magnetic moment and moment of inertia after it is broken.

### Step-by-Step Solution:

1. **Understand the Period of Oscillation**:
The period of oscillation \( T \) of a thin rectangular magnet is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{mB}} \]
where:
- \( I \) is the moment of inertia,
- \( m \) is the magnetic moment,
- \( B \) is the magnetic field.

2. **Given Values**:
We know that the initial period \( T \) is 4 seconds. Therefore:
\[ T = 4 \text{ s} \]

3. **Breaking the Magnet**:
When the magnet is broken into two equal parts, the new magnetic moment \( m' \) of one piece becomes:
\[ m' = \frac{m}{2} \]
where \( m \) is the original magnetic moment.

4. **Moment of Inertia**:
The moment of inertia \( I \) of a thin rectangular magnet is given by:
\[ I = \frac{1}{12} mL^2 \]
After breaking the magnet, the moment of inertia \( I' \) of one piece becomes:
\[ I' = \frac{1}{12} \left(\frac{m}{2}\right) \left(\frac{L}{2}\right)^2 = \frac{1}{12} \cdot \frac{m}{2} \cdot \frac{L^2}{4} = \frac{mL^2}{96} \]

5. **New Moment of Inertia**:
We can express the new moment of inertia in terms of the original moment of inertia \( I \):
\[ I' = \frac{I}{8} \]

6. **Calculate New Period of Oscillation**:
The new period of oscillation \( T' \) can be calculated using the new values of \( I' \) and \( m' \):
\[ T' = 2\pi \sqrt{\frac{I'}{m'B}} \]
Substituting \( I' = \frac{I}{8} \) and \( m' = \frac{m}{2} \):
\[ T' = 2\pi \sqrt{\frac{\frac{I}{8}}{\frac{m}{2}B}} = 2\pi \sqrt{\frac{2I}{8mB}} = 2\pi \sqrt{\frac{I}{4mB}} = \frac{1}{2} T \]

7. **Final Calculation**:
Now substituting the value of \( T \):
\[ T' = \frac{1}{2} \cdot 4 \text{ s} = 2 \text{ s} \]

### Conclusion:
The period of oscillation of the piece of the magnet after it is broken will be **2 seconds**.

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