Question

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

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

Answer 1

The correct option is C $\frac{24}{n^2}$ hour

Conservation of angular momentum dictates that angular momentum before and aftef contradiction must be the same.

$I_1{w}_1=I_2{w}_2$

$I=\frac{2}{5}M{r}^2\to$ Momentum of intertia of earth

$w=\frac{2\pi }{T}\to$ Angular velocity of earth $=\frac{2\pi }{day}$

Let $r_2$ and $T_2$ be the new radius and new day duration

Therefore,

$\frac{{r_1}^2}{T_1}=\frac{{r_2}^2}{T_2}$

$\therefore T_2=\frac{{r_2}^2}{{r_1}^2}{T}_1$

When earth contracts, $r_2=\frac{1}{n}{r}_1$

$\therefore T_2=\frac{{\left(\frac{1}{n}{r}_1\right)}^2}{r_1^2}{T}_1$

$T_2=\frac{1}{n^2}{T}_1$

$T_1=24hours$

$\therefore T_2=\frac{24}{n^2}hours$