buoyant force versus depth of water

Question

A box is partially submerged in water. It has a height H, but it is only submerged a depth D. A student graphs the relationship between buoyant force versus depth. The graph shows a linear line, that continues linearly even after the depth D surpasses the height H of the block.

Is the student correct or incorrect? Support your answer using the pressure exerted on the block.

Answer 1

The student is incorrect.

The buoyant force on the partially submerged box is the hydrostatic pressure at the bottom of the block multiplied by the area of the bottom of the box: Fb = D x ρ x g x A, where ρ is the density of water and A is the area. A plot of Fb vs. D will be linear until the box is completely submerged (D > H).

At this point the buoyant force will be constant because the pressure at the bottom of the box is

P b1 = s x ρ x g where s is the depth of water above the top of the box. At the bottom of the box, the pressure will be Pb2 = (s + H) x ρ x g since the bottom is at a depth of s + H.

The buoyant force is given by (Pb2 - Pb1) x A. Expanding we get (s + H) x ρ x g x A - s x ρ x g x A.

This simplifies to H x ρ x g x A. Since all the terms in the latter expression are constant, the plot of buoyant force vs. D will be a horizontal line.