Chapter 6.7, Problem 64E

Buoyancy Archimedes’ principle says that the buoyant force exerted on an object that is (partially or totally) submerged in water is equal to the weight of the water displaced by the object (see figure). Let ρ_w = 1 g/cm³ = 1000 kg/m³ be the density of water and let ρ be the density of an object in water. Let f = ρ/ρ_w. If 0 < f ≤ 1, then the object floats with a fraction f of its volume submerged; if f > 1, then the object sinks.

Consider a cubical box with sides 2 m long floating in water with one-half of its volume submerged (ρ = ρ_w/2). Find the force required to fully submerge the box (so its top surface is at the water level).

(See the Guided Project Buoyancy and Archimedes’ Principle for further explorations of buoyancy problems.)