Question

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in half-submerged state. If $\rho_{ C }$ is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

• A. more than half-filled if $\rho_{c}$ is less than $0.5$ .
• B. more than half-filled if $\rho_{c}$ is more than $1.0$ .
• C. half-filled if $\rho_{ C }$ is more than $0.5 .$
• D. less than half-filled if $\rho_{ C }$ is less than $0.5 .$

Answer 1

Answer: (A)

$\frac{V_{ m }+ V _{ a }+ V _{ w }}{2} \rho_{ w } g = V _{ m } \rho_{ c } \rho_{ w } g + V _{ w } \rho_{ w } g$
$V _{ w }= V _{ m }\left(1-2 \rho_{ C }\right)+ V _{ a } $
$\text { if } \rho_{ C }>\frac{1}{2} \Rightarrow V _{ w }< V _{ a } $
$\text { if } \rho_{ C }<\frac{1}{2} \Rightarrow V _{ w }> V _{ a }$
where, $V _{ w }=$ volume occupied by water in the shell
$V _{ a }=$ volume occupied by air in the shell
$V _{ m }=$ volume of the material in the shell