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, $\omega =\frac{2\pi }{T}$
$\therefore \left(\frac{2}{5}M{R}_1^2\right)\left(\frac{2\pi }{T_1}\right)$
$=\left(\frac{2}{5}M{R}_2^2\right)\left(\frac{2\pi }{T_2}\right)$
$\Rightarrow \frac{R_1^2}{T_1}=\frac{R_2^2}{T_2}$
$\Rightarrow \frac{R_1^2}{24}={\left(\frac{R_1}{2}\right)}^2\times \frac{1}{T_2}$
$\Rightarrow T_2=6h$