The time period of oscillation of a magnet in a vibration magnetometer is `1.5` seconds. The time period of oscillation of another of another magnet similar in size, shap and mass but having one-fourth magnetic moment than that of first magnet, oscillating at same place will be

Options

A. 0.75 sec
B. Zero
C. 3.0 sec
D. 6.0 sec

Answer

To solve the problem, we need to determine the time period of oscillation of a second magnet with a magnetic moment that is one-fourth that of the first magnet. We will use the relationship between the time period of oscillation and the magnetic moment.

### Step-by-Step Solution:

1. **Understand the Formula**:
The time period \( T \) of a magnet in a vibration magnetometer is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{M \cdot B}} \]
where \( I \) is the moment of inertia, \( M \) is the magnetic moment, and \( B \) is the magnetic field.

2. **Identify Constants**:
In this problem, we are told that the second magnet is similar in size, shape, and mass to the first magnet. Therefore, the moment of inertia \( I \) and the magnetic field \( B \) remain constant for both magnets.

3. **Relate Time Periods**:
Since \( I \) and \( B \) are constant, we can express the time period \( T \) in terms of the magnetic moment \( M \):
\[ T \propto \frac{1}{\sqrt{M}} \]
This means that the time period is inversely proportional to the square root of the magnetic moment.

4. **Set Up the Ratio**:
Let \( M_1 \) be the magnetic moment of the first magnet and \( M_2 \) be the magnetic moment of the second magnet. We know that \( M_2 = \frac{M_1}{4} \). Thus, we can write:
\[ \frac{T_2}{T_1} = \sqrt{\frac{M_1}{M_2}} = \sqrt{\frac{M_1}{\frac{M_1}{4}}} = \sqrt{4} = 2 \]

5. **Substitute Known Values**:
We know from the problem that the time period of the first magnet \( T_1 = 1.5 \) seconds. Therefore:
\[ T_2 = 2 \cdot T_1 = 2 \cdot 1.5 = 3 \text{ seconds} \]

6. **Conclusion**:
The time period of oscillation of the second magnet will be \( 3 \) seconds.

### Final Answer:
The time period of oscillation of the second magnet is \( 3 \) seconds.

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