The length of a magnet is large compared to its width and breadth. The time period of its oscillation in a vibration magnetometer is ` 2 s`. The magnet is cut along its length into three equal parts and these parts are then placed on each other with their like poles together . The time period of this combination will be

Options

A. 2s
B. `(2)/(3)s`
C. `2sqrt3s`
D. `2//sqrt3s`

Answer

To solve the problem, we need to analyze the effect of cutting a long magnet into three equal parts and stacking them together on the time period of oscillation in a vibration magnetometer.

### Step-by-Step Solution:

1. **Understanding the Time Period Formula**:
The time period \( T \) of a magnet in a vibration magnetometer is given by the formula:
\[ T = 2\pi \sqrt{\frac{I}{m \cdot h}} \]
where \( I \) is the moment of inertia, \( m \) is the mass of the magnet, and \( h \) is the distance from the pivot to the center of mass.

2. **Initial Moment of Inertia**:
For a long magnet, the moment of inertia \( I \) about one end is given by:
\[ I = \frac{1}{3} m L^2 \]
where \( L \) is the length of the magnet.

3. **Cutting the Magnet**:
When the magnet is cut into three equal parts, each part has a length of \( \frac{L}{3} \) and mass \( \frac{m}{3} \).

4. **Moment of Inertia of Each Part**:
The moment of inertia for each of the three parts about one end is:
\[ I' = \frac{1}{3} \left(\frac{m}{3}\right) \left(\frac{L}{3}\right)^2 = \frac{1}{3} \cdot \frac{m}{3} \cdot \frac{L^2}{9} = \frac{m L^2}{81} \]

5. **Total Moment of Inertia for Stacked Parts**:
Since the three parts are stacked with like poles together, the total moment of inertia \( I_{total} \) becomes:
\[ I_{total} = 3 \cdot I' = 3 \cdot \frac{m L^2}{81} = \frac{m L^2}{27} \]

6. **Mass of the New Combination**:
The total mass \( m' \) of the stacked magnet remains the same as the original magnet:
\[ m' = m \]

7. **New Time Period Calculation**:
Now we can calculate the new time period \( T' \):
\[ T' = 2\pi \sqrt{\frac{I_{total}}{m' \cdot h}} = 2\pi \sqrt{\frac{\frac{m L^2}{27}}{m \cdot h}} = 2\pi \sqrt{\frac{L^2}{27h}} = \frac{2\pi L}{\sqrt{27h}} \]
Since the original time period \( T \) is given as \( 2 \) seconds, we can relate the new time period to the original:
\[ T' = \frac{1}{\sqrt{3}} T \]

8. **Final Calculation**:
Substituting \( T = 2 \) seconds:
\[ T' = \frac{2}{\sqrt{3}} \approx 1.155 \text{ seconds} \]

### Conclusion:
The new time period \( T' \) after cutting and stacking the magnet is approximately \( 1.155 \) seconds.