The time period of a freely suspended magnet is 4 seconds. If it is broken in length into two equal parts and one part is suspended in the same way, then its time period will be

Options

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

Answer

To solve the problem, we need to analyze how the time period of a freely suspended magnet changes when it is cut into two equal parts.

### Step-by-Step Solution:

1. **Understanding the Given Data**:
- The initial time period \( T \) of the magnet is given as 4 seconds.

2. **Cutting the Magnet**:
- When the magnet is cut into two equal parts, the length of each part becomes half of the original length. If we denote the original length as \( L \), then the new length \( L' = \frac{L}{2} \).

3. **Magnetic Dipole Moment**:
- The magnetic dipole moment \( \mu \) of a magnet is given by the product of its pole strength \( p \) and its length \( L \):
\[ \mu = p \cdot L \]
- After cutting the magnet, the new magnetic dipole moment \( \mu' \) for one part becomes:
\[ \mu' = p \cdot \frac{L}{2} = \frac{\mu}{2} \]

4. **Moment of Inertia**:
- The moment of inertia \( I \) for a magnet fixed at its center is given by:
\[ I = \frac{mL^2}{12} \]
- When the magnet is cut into two equal parts, the mass of each part becomes \( \frac{m}{2} \) and the new length is \( \frac{L}{2} \). Therefore, the new moment of inertia \( I' \) becomes:
\[ I' = \frac{\frac{m}{2} \left(\frac{L}{2}\right)^2}{12} = \frac{m}{2} \cdot \frac{L^2}{4} \cdot \frac{1}{12} = \frac{mL^2}{96} = \frac{I}{8} \]

5. **Time Period Formula**:
- The time period \( T \) of a magnet suspended in a magnetic field is given by:
\[ T = 2\pi \sqrt{\frac{I}{\mu B}} \]
- For the new part, the time period \( T' \) becomes:
\[ T' = 2\pi \sqrt{\frac{I'}{\mu' B}} = 2\pi \sqrt{\frac{\frac{I}{8}}{\frac{\mu}{2} B}} = 2\pi \sqrt{\frac{I}{8} \cdot \frac{2}{\mu B}} = 2\pi \sqrt{\frac{I}{\mu B}} \cdot \frac{1}{\sqrt{4}} = \frac{T}{2} \]

6. **Calculating the New Time Period**:
- Since the original time period \( T \) is 4 seconds, the new time period \( T' \) will be:
\[ T' = \frac{4}{2} = 2 \text{ seconds} \]

### Final Answer:
The new time period of the suspended magnet after cutting it into two equal parts is **2 seconds**.

---