Question

A container of large uniform cross-sectional area A resting on a horizontal surface, holds two immiscible, non-viscous and incompressible liquids of densities d and $2$d, each of height $\frac{H}{2}$ . The lower density liquid is open to the atmosphere having pressure $P_0$.
(i) A homogeneous solid cylinder of length $L(L<\frac{H}{2})$, cross-sectional area $\frac{A}{5}$ is immersed such that if floats with its axis vertical at the liquid-liquid interface with length $\frac{L}{4}$ in the denser liquid. Determine: (a) the density D of the solid, (b) the total pressure at the bottom of the container.

Answer 1

a) Let $\rho$ be the density cylinder

Volume of fluid having density 'd' displaced

$V_d=\frac{A}{5}\times \frac{3L}{4}=\frac{3AL}{20}$

Volume of '2d' displaced

$v_{2d}=\frac{A}{5}\times \frac{L}{4}=\frac{AL}{20}$

Total Buoyant force qererted

$F_{Buo}=V_d\times d\times g+V_{2d}\times 2d$ g

$=\frac{3AL}{20}dg+\frac{2AL}{20}dg$

$=\frac{1}{4}ALdg$

The cylinder remains in equilibrium

$F_{Buo}=F_{weight}$

$\frac{1}{4}ALdg=\rho \times \frac{A}{5}\times Lg$

$\frac{1}{4}d=\frac{\rho }{5}$

$\rho =\frac{5}{4}d$

b) The pressure at the bottom surface:

$P=\frac{\text{Total weight of liquid+Solid}}{\text{Area of crossection of Container}}$

$P=\frac{\frac{A}{6}(dgh/2+2dgh/2+\frac{5dgL}{4})}{\frac{A}{6}}$

$P=\frac{dg(6h-5L)}{4}$