Buoyant Force
Introduction
The hydrostatic pressure increases with depth. Let's look at a cube underwater and calculate the buoyant force.
Consider a cube 1 meter on a side, completely underwater. The top of the cube is parallel to the water surface.
The upward force due to the pressure on the bottom of the cube is greater than the downward force on the top of the cube.
The pressure, P in N/m^2, versus depth, d in m, is
P = pgd
where g is the acceleration of gravity 10 m/s^2 and p is the density of water 10^3 kg/m^3.
The upward force on the bottom is F = P A where A is the area of the bottom 1m^2 , F = pgd A= 10^3 10 d 1 = 10^4 d
The downward force on the top is F = P A , F = p g t A where t is the depth of the top , F = 10^4 t
The difference is F = 10^4 (d  t) , now dt is 1 m so F = 10^4 N
The weight of a cubic meter of water is W = mg = 10^3 * 10 = 10^4 N
So, as Archimedes discovered, the buoyant force F equals the weight of the displaced water.
Scientific Explorations with Paul Doherty 

18 August 2005 