A U-shaped tube with open ends contains three immiscible liquids of densities ρ₁, ρ₂ and ρ₃. If the liquids are in equilibrium, find the density ρ₁ as a function of the densities ρ₂ and ρ₃.

A right circular cylinder of height h = 30 cm and base area A = 10 cm², floats in water in a vertical position with 2/3 of its height immersed, a force F is applied in the axial direction of the upper base passing through the cylinder to have 5/6 of its height immersed. Determine:
a) What is the density of the cylinder relative to water?
b) What is the magnitude of the force F?
Data: g = 9.8 m/s², density of water = 1 g/cm³.