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 $T^{\text{'}}$, the ratio $\frac{T^{\text{'}}}{T}$ is

When magnet is divided into two equal halves, mass is reduced by a factor of 2 and length is also reduced by factor of 2. 

New moment of inertia is 1/8th of the initial moment of inertial. 

Also, magnetic moment=pole strength × length Pole strength is unchanged and the, length is halved. 

So, new magnetic moment is one-half of the initial magnetic moment.

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

$\text{ Now }=2\mathrm{\pi }\sqrt{\frac{\mathrm{I}/8}{\frac{\mathrm{M}}{2}\mathrm{B}}}=\frac{\mathrm{T}}{\sqrt{4}}=\frac{\mathrm{T}}{2}\Rightarrow \frac{{\mathrm{T}}^{\text{'}}}{\mathrm{T}}=\frac{1}{2}$