Question

A thin rectangular magnet suspended freely has a period of oscillation equal to $T$. Now it is broken into two equal halves (each having half of the original length) and one piece is made to oscillate freely in the same field. If its period of oscillation is $T^{\prime }$, the ratio $\frac{T^{\prime }}{T}$ is

• A. $\frac{1}{2\sqrt{2}}$
• B. $\frac{1}{2}$
• C. $4$
• D. $\frac{1}{4}$

Answer 1

The correct option is B $\frac{1}{2}$

Time period of oscillation of a bar magnet having magnetic moment $M$ and moment of inertia $I$ in a magnetic field $B$ is given by

$T=2\pi \sqrt{\frac{I}{MB}}$


Now, $M=ml$

where, $m=$ magnetic pole strength

When the magnet is broken into two halves, the magnetic pole strength remains the same, but the magnetic length will change.

The new magnetic dipole strength will be

$M^{\prime }=m\left(\frac{l}{2}\right)=\frac{M}{2}$

Moment of inertia of magnet,

$I=\frac{Mass\times (length{)}^2}{12}=\frac{m^{\prime }{l}^2}{12}$

When the magnet is broken into two pieces, the mass is also halved. The new moment of inertia will be

$I^{\prime }=\frac{\left(\frac{m^{\prime }}{2}\right){\left(\frac{l}{2}\right)}^2}{12}=\frac{I}{8}$

As, $B$ is constant,

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

$\Rightarrow \frac{T^{\prime }}{T}=\sqrt{\frac{I^{\prime }}{I}\times \frac{M}{M^{\prime }}}$

$=\sqrt{\frac{\frac{I}{8}\times M}{I\times \frac{M}{2}}}=\frac{1}{2}$

$\therefore \frac{T^{\prime }}{T}=\frac{1}{2}$

Hence, option $\left(B\right)$ is correct.