Question

A short bar magnet in a vibrating magnetometer makes $16$ oscillations in $4$ seconds. Another short magnet with same length and width having moment of inertia $1.5$ times the first one is placed over the first magnet and oscillated. Neglecting the induced magnetization, the time period of the combination is:

• A. $2\sqrt{10}s$
• B. $20\sqrt{10}s$
• C. $\frac{2}{\sqrt{10}}s$
• D. $\frac{2.5}{\sqrt{10}}s$

Answer 1

Answer: (C)

$\frac{2}{\sqrt{10}}s$

The magnetic torque is proportional to the magnetic moment of the magnet and angle with the magnetic field. Hence,

$\tau =I{\omega }^2\theta =BM\theta$

$\omega \propto \sqrt{\frac{M}{I}}$

$T\propto \sqrt{\frac{I}{M}}$

Since the second magnet is kept over the other surface.

$I_2=(1.5+1)I_1=2.5I\&M_2=2M_1=2M$

$\frac{T}{T_0}=\frac{T}{0.25}=\sqrt{\frac{2.5I}{2M}}\sqrt{\frac{M}{I}}$

$T=\frac{1}{8}\sqrt{5}s$

$T=\frac{2}{\sqrt{10}}s$