A magnent when suspended freely in earth's magnetic field, oscillates with a time period T. It is replaced by a magnet of magnetic moment four times that of the original one. What will be the new time period?

Answer

To solve the problem, we need to understand how the time period of oscillation of a magnet in a magnetic field is affected by its magnetic moment.

### Step-by-Step Solution:

1. **Understand the Time Period Formula**:
The time period \( T \) of a magnet oscillating in a magnetic field is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{MB}} \]
where:
- \( I \) is the moment of inertia of the magnet,
- \( M \) is the magnetic moment of the magnet,
- \( B \) is the magnetic field strength (which remains constant).

2. **Identify the Initial Conditions**:
Let the initial magnetic moment of the first magnet be \( M \). Thus, the time period for the first magnet is:
\[ T_1 = 2\pi \sqrt{\frac{I}{MB}} \]

3. **Determine the New Magnetic Moment**:
The new magnet has a magnetic moment that is four times that of the original magnet. Therefore, the new magnetic moment \( M_2 \) is:
\[ M_2 = 4M \]

4. **Calculate the New Time Period**:
Using the same formula for the new magnet, we can express the new time period \( T_2 \):
\[ T_2 = 2\pi \sqrt{\frac{I}{M_2B}} = 2\pi \sqrt{\frac{I}{4MB}} \]
Simplifying this, we get:
\[ T_2 = 2\pi \sqrt{\frac{I}{4MB}} = 2\pi \sqrt{\frac{1}{4}} \sqrt{\frac{I}{MB}} = \frac{1}{2} \cdot 2\pi \sqrt{\frac{I}{MB}} = \frac{T_1}{2} \]

5. **Conclusion**:
Since \( T_1 = T \), we find that:
\[ T_2 = \frac{T}{2} \]

Thus, the new time period \( T_2 \) is half of the original time period \( T \).

### Final Answer:
The new time period \( T_2 \) is \( \frac{T}{2} \).

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