Question

If the radius of the earth contracts to half of its present day value without change in mass, then the length of the day will be

• A. 24 h
• B. 48 h
• C. 6 h
• D. 12 h

Answer 1

Answer: (C)

From law of conservation of angular momentum, $J=I \omega=$ constant
$\therefore I_{1} \omega_{1}=I_{2} \omega_{2}$
where $I$ is moment of inertia and co the angular velocity.
Moment of inertia of earth assuming it be a sphere of radius
$R=\frac{2}{5} M R^{2}$
Also, $\omega=\frac{2 \pi}{T}$
$\therefore \left(\frac{2}{5} M R_{1}^{2}\right)\left(\frac{2 \pi}{T_{1}}\right)$
$=\left(\frac{2}{5} M R_{2}^{2}\right)\left(\frac{2 \pi}{T_{2}}\right)$
$\Rightarrow \frac{R_{1}^{2}}{T_{1}}=\frac{R_{2}^{2}}{T_{2}}$
$\Rightarrow \frac{R_{1}^{2}}{24}=\left(\frac{R_{1}}{2}\right)^{2} \times \frac{1}{T_{2}}$
$\Rightarrow T_{2}=6\, h$