A container of a large uniform cross-sectional area `A` resting on a horizontal surface holds two immiscible, non viscous and incompressible liquids of densities `d` and `2d`, each of height `H//2` as shown in figure. The lower density liquid is open to atmosphere. A homogeneous solid cylinder of length `L(L lt (H)/(2))`, cross-sectional area `A//5` is immersed such that it floats with its axis vertical of the liquid-liquid interface with length `L//4` denser liquid. Determine
(a) density `D` of the solid and
(b) the total pressure at the bottom of the container. (Atmospheric pressure `= P_0`).
.

Explanation

(a) For equilibrium of cylinder
`W = F_B`
`((A)/(5)) Lg = d((A)/(5)) (3L)/(4) g + 2d ((A)/(5))(L)/(4) g`
`D = (3d)/(4) + (d)/(2) = (5d)/(4)`
(b) Total pressure at the bottom of cylinder
`P = P_0 + ("weight of liquids + weight of cylinder")/(A)`
=`P_1 + (dA((H)/(2)) g + 2d A((H)/(2)) g + D ((A)/(5))Lg)/(A)`
=`P_0 + d(H)/(2) g + dHg + (5 d)/(4) (1)/(5) Lg`
=`P_0 + ((6 H + L)/(4)) dg`.