Q: A thin uniform cylindrical shell, closed at both ends, is partially filled with water is 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

(a) More than half-filled if ρ_{c} is less than 0.5

(b) More than half-filled if ρ_{c} is less than 1.0

(c) Half-filled if ρ_{c} is more than 0.5

(d) Less than Half-filled if ρ_{c} is less than 0.5

Ans: (a)

Sol: V_{1} = material volume of shell

V_{2} = inside volume of shell

and, x = fraction of V_{2} filled with water

In case of floatation

Total weight = Upthrust

$\large V_1 \rho_c g + (x V_2)\times 1 \times g = (\frac{V_1 + V_2}{g}) \times 1 \times g $

On solving ;

$\large x = 0.5 + (0.5 -\rho_c )\frac{V_1}{V_2}$

x > 0.5 if ρ_{c} < 0.5