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.

JEE Advanced · 2012 · 08 Apr 2012 · Paper 2 · Fluid Mechanics

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

ρ_{c}

is the relative density of the material of the shell with respect to water, then the correct statement is that the shell is

Answer: (A)

For the shell floating half-submerged, applying Archimedes' principle:

\frac{V _{m} + V _{a} + V _{w}}{2} ρ_{w} g = V_{m} ρ_{c} ρ_{w} g + V_{w} ρ_{w} g

, where

V_{m}

is the volume of the shell material,

V_{a}

is the air volume, and

V_{w}

is the water volume inside. Simplifying:

V_{w} = V_{m} (1 - 2 ρ_{c}) + V_{a}

. If

ρ_{c} < 0.5

, then

V_{w} > V_{a}

, meaning the shell is more than half-filled. If

ρ_{c} > 0.5

, then

V_{w} < V_{a}

, meaning less than half-filled.

JEE Advanced · 2012 · 08 Apr 2012 · Paper 2 · Single correct option