If the earth were to suddenly contract to ${\frac{1}{n}}^{\text{th }}$ of its present radius without any change in its mass, then the duration of the new day will be nearly

Let mass of earth is M and radius R moment of inertia will be

${\mathrm{I}}_1=\frac{2}{5}{\mathrm{MR}}^2$

time period of the rotation of earth

${\mathrm{T}}_1=24\mathrm{hr}$

angular velocity of rotation of earth

${\omega }_1=\frac{\pi \mathrm{R}}{24}$

new radius of earth is ${\mathrm{R}}_2=\frac{{\mathrm{R}}_1}{\mathrm{n}}$

new moment of inertia is

${\mathrm{I}}_2=\frac{2}{5}\mathrm{M}\frac{{\mathrm{R}}_1^2}{{\mathrm{n}}^2}=\frac{{\mathrm{I}}_1}{{\mathrm{n}}^2}$

new angular frequency is ${\omega }_2$

by conservation of angular momentum,

$\begin{array}{l}{\mathrm{I}}_1{\omega }_1={\mathrm{I}}_2{\omega }_2\\ \left({\mathrm{I}}_1\right)\left(\frac{\pi \mathrm{R}}{24}\right)=\left(\frac{{\mathrm{I}}_1}{{\mathrm{n}}^2}\right)\left(\frac{\pi \mathrm{R}}{{\mathrm{T}}_2}\right)\\ {\mathrm{T}}_2=\frac{24}{{\mathrm{n}}^2}\mathrm{hr}\end{array}$