# If all of a sudden, the earth is contracted to half its present radius, then the day will be reduced by

21 views
in Physics
closed

If all of a sudden, the earth is contracted to half its present radius, then the day will be reduced by

1. 12 hrs
2. 15 hrs
3. 18 hrs
4. 21 hrs

by (24.2k points)
selected

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, R= 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,

$\begin{array}{l} {I_1}{\omega _1} = {I_2}{\omega _2}\\ \frac{2}{5}M{R^2} \times \frac{{2\pi }}{{{T_1}}} = \frac{2}{5}M{\left( {\frac{R}{2}} \right)^2} \times \frac{{2\pi }}{{{T_2}}}\\ \Rightarrow {T_2} = \frac{1}{4} \times 24\;hour = 6\;hour\;\\ Decrease\;in\;duration\;of\;day = 24 - 6 = 18\;hour \end{array}$