The pressure is measured at the bottom of a cylindrical container with cross-sectional area A that is filled to a height H by a liquid with density pL. A cube of volume Vc and density pc is placed in the liquid and floats partially submerged, casing a change in the height of the liquid (delta h). As a result, the pressure at the bottom of the container increases.

(a) In a coherent, paragraph-length response, describe the relevant forces acting on the cube and the liquid, and explain how they result in the increase in pressure at the bottom of the container.

(b) Write an expression for the pressure at the bottom of the container in terms of H, delta h, and physical constants, as appropriate.

(c) Determine delta h in terms of Vc and other given quantities and physical constants, as appropriate.

a) B and W, b) P = P₀ + ρ_liquid g (H + ρ_body / ρ_liqud V_body / A)

Explanation:

The pressure at the bottom of the cylinder is

P = P₀ + ρ g H

a) When the body is placed in the liquid, we apply Newton's equilibrium equation

B - W = 0

where B is the archimedean thrust

B = ρ_liquid g V_liquid

The weight is

W = ρ_body g V_body

in short, the forces in this system are the hydrostatic thrust of Archimedes and the attraction of gravity through the weight of the body.

Let's write all this force in function of the densities

c) ρ_liquid g V_liquid = ρ_body g V_body

V_liquid = ρ_body / ρ_liquid V_body

Volume is area by height, we substitute in the equations

A h = ρ_body / ρ_liqud V_body

h = ρ_body / ρ_liqud V_body / A

This is the height that has been added to the liquid

therefore the total height is

h_total = H + h

b) the pressure is

P = P₀ + ρ_liquid g (H + ρ_body / ρ_liqud V_body / A)