The length of a magnet is large compared to its width and breadth. The time period of its oscillation in a vibration magnetometer is 150 ms . The magnet is cut along its length into three equal parts

Question

The length of a magnet is large compared to its width and breadth. The time period of its oscillation in a vibration magnetometer is 150 ms . 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 (in ms ) will be

TestHub · Accepted Answer

The time period of oscillations of magnet

$T = 2 π \sqrt{(\frac{I}{M B_{H}})} \dots (i)$

Where $I =$ moment of inertia of magnet

$= \frac{m L^{2}}{12}$ (m, being the mass of magnet)

When the three equal parts of magnet are placed on one another with their like poles together, then

$I^{'} = \frac{1}{12} (\frac{m}{3}) \times {(\frac{L}{3})}^{2} \times 3$

$= \frac{1}{12} \frac{m L^{2}}{9} = \frac{I}{9}$

and $M^{'} =$ pole strength $\times \frac{L}{3} \times 3 = M$

Hence,

$\Rightarrow T^{'} = \frac{1}{3} \times T$

$\Rightarrow T^{'} = \frac{150}{3} = 50 ms$

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