Eduprobe
Question:
A thin uniform cylindrical shell, closed at both ends, is partially filled with water. It is floating vertically in water in a half-submerged state. If ρc is the relative density of the material of the shell with respect to water, then the correct statement is:
less than half-filled if ρc is less than 0.5
more than half-filled if ρc is less than 0.5
more than half-filled if ρc is more than 1.0
half-filled if ρc is more than 0.5
Solution:
Let Vw be the volume occupied by water in the shell
Va be the volume occupied by air in the shell
Vm be the volume occupied by material in the shell
Vm + Va + Vw = total volume of the shell
The weight of the shell and the water inside it is equal to the buoyant force.
ρwg(Vmρc + Vw) = (Vm + Va + Vw)ρwg
where ρw is the density of water and g is the acceleration due to gravity.
Since the shell is half-submerged, the buoyant force is equal to half the weight of the shell plus the water inside it:
(Vm + Va + Vw)/2 ρwg = Vmρcρwg + Vwρwg
Vm + Va + Vw = 2Vmρc + 2Vw
Va + Vm(1 - 2ρc) = Vw
If ρc > 1/2, then 1-2ρc is negative. In this case, Vw < Va, which means that the shell is less than half-filled.
If ρc < 1/2, then 1-2ρc is positive. In this case, Vw > Va, which means that the shell is more than half-filled.