Question

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

• A. $\frac{24}{n}$ $hr$
• B. $24nhr$
• C. $\frac{24}{n^2}hr$
• D. $24n^{2}hr$

Answer 1

Answer: (C)

$\frac{24}{n^2}hr$

Let the mass and radius of the earth be $M$ and $R$ respectively.

$\therefore$ $I_1=\frac{2}{5}M{R}^2$

Also, time period of rotation of earth, $T=24$ $h$

$\therefore$ Angular velocity of rotation of earth, $w_1=\frac{2\pi }{T}=\frac{2\pi }{24}$

Radius of earth after it shrinks, $R^{\prime }=\frac{R}{n}$

$\therefore$ $I_2=\frac{2}{5}M(R^{\prime }{)}^2=\frac{2}{5n^2}M{R}^2=\frac{I_1}{n^2}$

Let the time period of rotation of earth be, $T_2$ $hr$

Using conservation of angular momentum: $I_2{w}_2=I_1{w}_1$

$\Rightarrow \frac{I_1}{n^2}\times \frac{2\pi }{T_2}=I_1\times \frac{2\pi }{24}$

$\Rightarrow T_2=\frac{24}{n^2}$ $hr$