Correct Answer - Option 3 : 18 hrs
Explanation:
In our case, the earth is a rotating sphere and its angular momentum L will be conserved and can be given as
\(L = I \times \omega \) hence option B is true now for angular velocity
let the present radius of earth = R, T1 = 24 hour
Radius after contraction, R1 = R/2, T2 = ?
By conservation of angular momentum,
\({I_1}{\omega _1} = {I_2}{\omega _2}\)
We can assume earth to be perfectly solid sphere of moment of inertia \(\frac{2}{5}M{R^2}\)
By conservation of angular momentum,
\(\begin{array}{l} {I_1}{\omega _1} = {I_2}{\omega _2}\\ \frac{2}{5}M{R^2} \times \frac{{2\pi }}{{{T_1}}} = \frac{2}{5}M{\left( {\frac{R}{2}} \right)^2} \times \frac{{2\pi }}{{{T_2}}}\\ \Rightarrow {T_2} = \frac{1}{4} \times 24\;hour = 6\;hour\;\\ Decrease\;in\;duration\;of\;day = 24 - 6 = 18\;hour \end{array}\)
