On plant Mercury there is a special lake with two layers: dH2Oand dHg (liquid mercury). The liquid water layer floats on top ofthe liquid mercury layer. Let ρH2O and ρHg denote the densities ofwater and mercury respectively. The gravitational field near theplanet’s surface is gy, and the the atmospheric pressure near thesurface of the lake P0.

a.Determine an expression in terms of the gi variables for thepressure in the lake as a function of depth all the way to thebottom of the layer of mercury. Graph this function.

b. Suppose an object density ρ is dropped into the lake. AssumeρH2O < ρ < ρHg. What fraction of the object you think will besubmerged in the mercury after the object comes to rest in staticequilibrium in the limit ρ → ρH2O .What fraction of the object youthink will be submerged in the mercury after the object comes torest in static equilibrium in the limit ρ → ρHg?

c.Determine an expression for ρ in terms of ρH 2O and ρHg thatyou believe would result in the object being half-submerged in themercury layer and half-submerged in the water layer?Assume ρH2O< ρ < ρHg

d. Consider the general case where the density of the object issimply the unknown variable p. Determine an expression for thefraction of the object that will be submerged in the mercury whenthe object comes to rest in static equilibrium?