Q: A thin uniform cylindrical shell, closed at both ends, is partially filled with water is half-submerged state. If ρc 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 ρc is less than 0.5
(b) More than half-filled if ρc is less than 1.0
(c) Half-filled if ρc is more than 0.5
(d) Less than Half-filled if ρc is less than 0.5
Ans: (a)
Sol: V1 = material volume of shell
V2 = inside volume of shell
and, x = fraction of V2 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 ρc < 0.5