Question

A thin uniform cylindrical shell, closed at both ends, is partially filled with water. Let the thin cylindrical shell is vertically floating in water in a half-submerged state. lf ${{\rho }_{c}}$ is the relative density of the material of the shell with respect to water, hence the correct statement is that the shell will be,
A. more than half filled if ${{\rho }_{c}}$is less than $0.5$
B. more than half filled if ${{\rho }_{c}}$is more than $1$
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

The weight of the shell can be written as the half of the sum of volume occupied by the water in the shell, volume occupied by the air in the shell and the volume occupied by the material in the shell multiplied with the relative density of water and the acceleration due to gravity. This should be calculated. Hope these will help you to solve this question.

Complete answer:
Let us assume some of the variables in the shell,
The volume occupied by the water in shell is given as ${{V}_{w}}$, the volume occupied by the air in the shell can be given as ${{V}_{a}}$ and the volume occupied by the material in the shell is given as ${{V}_{m}}$.
Therefore the weight of the shell can be written as the half of the sum of volume occupied by the water in the shell, volume occupied by the air in the shell and the volume occupied by the material in the shell multiplied with the relative density of water and the acceleration due to gravity.
That is,
$\dfrac{{{V}_{m}}+{{V}_{a}}+{{V}_{w}}}{2}{{\rho }_{w}}g={{V}_{m}}{{\rho }_{c}}{{\rho }_{w}}g+{{V}_{w}}{{\rho }_{w}}g$
Let us rearrange the equation. First of all let group the ${{V}_{m}}$ terms together.
That is,
${{V}_{m}}{{\rho }_{w}}g+{{V}_{a}}{{\rho }_{w}}g+{{V}_{W}}{{\rho }_{w}}g=2\times \left( {{V}_{m}}{{\rho }_{c}}{{\rho }_{w}}g+{{V}_{w}}{{\rho }_{w}}g \right)$
Let us expand the bracket and simplify it,
$\begin{aligned} & {{V}_{m}}{{\rho }_{w}}g+{{V}_{a}}{{\rho }_{w}}g+{{V}_{W}}{{\rho }_{w}}g=2{{V}_{m}}{{\rho }_{c}}{{\rho }_{w}}g+2{{V}_{w}}{{\rho }_{w}}g \\\ & \Rightarrow {{V}_{W}}{{\rho }_{w}}g={{V}_{m}}g\left( {{\rho }_{w}}-2{{\rho }_{c}}{{\rho }_{w}} \right)-{{V}_{a}}{{\rho }_{w}}g \\\ \end{aligned}$
Let us cancel out the common terms in the equation,
That is,
${{V}_{W}}={{V}_{m}}\left( 1-2{{\rho }_{c}} \right)-{{V}_{a}}$
From this we can write that,
If ${{\rho }_{c}}>\dfrac{1}{2}$ ,
Then,
${{V}_{w}}<{{V}_{a}}$
Or if ${{\rho }_{c}}<\dfrac{1}{2}$ , Then,
${{V}_{w}}>{{V}_{a}}$
That is the shell is more than half filled, if ${{\rho }_{c}}$is less than $0.5$.

Hence the answer is given as option A.

Note:
The displacement method is also known as the submersion process. This method is helpful in accurate measurement of the volume of the human body and other oddly shaped objects. This is possible by measuring the fluid displaced when the body is submerged.