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

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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

## 1 Answer

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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,

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

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