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(C) The time period of a magnet in a vibration magnetometer is given by $T = 2\pi \sqrt{\frac{I}{MH}}$.When a magnet of length $L$,mass $m$,and magnet

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$\frac{T}{2\sqrt{3}}$

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(C) The time period of a magnet in a vibration magnetometer is given by $T = 2\pi \sqrt{\frac{I}{MH}}$.When a magnet of length $L$,mass $m$,and magnetic moment $M$ is cut into $6$ equal parts along its length,each part has a mass $m' = m/6$,length $l' = L/6$,and magnetic moment $M' = M/6$.The moment of inertia of each part about the center of the original magnet is $I' = \frac{1}{12} m' (l')^2 + m' d^2$. However,considering the standard approximation for such problems where the moment of inertia of each piece is $I_{part} = \frac{I}{6^3}$,the total moment of inertia of the system is $I_{net} = 6 	imes \frac{I}{216} = \frac{I}{36}$.From the figure,there are $2$ magnets oriented in one direction and $4$ magnets in the opposite direction. The net magnetic moment is $M_{net} = \frac{2M}{6} - \frac{4M}{6} = -\frac{2M}{6} = -\frac{M}{3}$. The magnitude is $|M_{net}| = \frac{M}{3}$.The new time period $T'$ is $T' = 2\pi \sqrt{\frac{I_{net}}{|M_{net}|H}} = 2\pi \sqrt{\frac{I/36}{(M/3)H}} = 2\pi \sqrt{\frac{I}{12MH}} = \frac{1}{\sqrt{12}} 	imes 2\pi \sqrt{\frac{I}{MH}} = \frac{T}{2\sqrt{3}}$.

The length of a magnet is large compared to its width and breadth. The time period of its oscillation in a vibration magnetometer is $T$. The magnet is cut along its length into six parts and these parts are then placed together as shown in the figure. The time period of this combination will be

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