Question

A spherical shell of mass $m$ and radius $R$ filled completely with a liquid of same mass and set to rotate about a vertical axis through its centre has a moment of inertia $I_1$ about the axis$.$ The liquid starts leaking out of the hole at the buttom$.$ If moment of inertia of the system is $I_2$ when the shell is half filled and $I_3$ is the moment of inertia when entire water drained off$.$ then $:$

• A. $\frac{I_1}{I_2}\approx 1.5$
• B. $\frac{I_{{}_1}}{I_2}\approx 0.67$
• C. $\frac{I_1}{I_3}\approx 1.6$
• D. $\frac{I_2}{I_3}\approx 1.4$

Answer 1

The correct option is C $\frac{I_1}{I_3}\approx 1.6$

Moment of inertia of a shell$,$ $I_3=\frac{2}{3}M{R}^2$

Moment of inertia of a completely filled sphere$,$ $I_1=\frac{2}{3}M{R}^2+\frac{2}{5}M{R}^2=\frac{16}{15}M{R}^2$

Moment of inertia of a half sphere$,$ $I_2=\frac{2}{3}M{R}^2+\frac{2}{5}\frac{M}{2}{R}^2=\frac{26}{30}M{R}^2$

Hence,

option $\left(C\right)$is correct answer.