Question

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

• A. \( \frac{2}{3} \, \text{s} \)
• B. \( \frac{1}{\sqrt{3}} \, \text{s} \)
• C. \( \frac{2}{\sqrt{3}} \, \text{s} \)
• D. \( \frac{3}{2} \, \text{s} \)

Hint:

When a magnet is cut along its length, the moment of inertia changes, which affects the time period. The stacking of parts with like poles together increases the moment of inertia.

Answer 1

The Correct Option is C

The time period \( T \) of oscillation of a magnet in a vibration magnetometer is given by: \[ T = 2\pi \sqrt{\frac{I}{mgh}} \] where \( I \) is the moment of inertia, \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the distance of the center of mass from the pivot. When the magnet is cut along its length, the moment of inertia will change.
If the magnet is cut into three equal parts and stacked, the effective moment of inertia will be increased, leading to a change in the time period.
The new time period will be: \[ T_{\text{new}} = \frac{2}{\sqrt{3}} \, T \] Substituting the given time period: \[ T_{\text{new}} = \frac{2}{\sqrt{3}} \times 2 = \frac{2}{\sqrt{3}} \, \text{s} \]
Thus, the correct answer is (c).