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I am having trouble understanding why the point of application of a buoyancy force (center of buoyancy) is the center of volume for the submerged part of the body. Is there a proof for this statement somewhere? Also, it is nice to say that the buoyancy force is equal to the weight of displaced water but is this true for bodies that have their horizontal surfaces resting on a bottom? 1. Imagine a partially submerged cube. The pressure forces on the side walls balance out, leaving only the distributed pressure forces acting of the bottom face. The resolution of the distributed forces yields a single pressure force acting at the geometric center of the bottom face. In the vertical direction, is it not also acting at the bottom face? This is where the distributed pressure forces acted, but this is not the center of volume for the submerged section of the body. 2. Imagine a large, upright pillar submerged in a lake and that rests on the lake bottom with its top at the lake's surface. I would expect that, in theory, there should not be any vertical forces acting on the submerged body since the pillar's base presents no surface area to the water, and so there should be no buoyancy. All the pressure forces are acting along the pillar's circumference. So does this pillar really have no buoyancy despite being entirely submerged?

A long awaited reply, but here are the considerations.
1. The net point of the buoyancy force (the center of buoyancy) occurs because if the submerged object is in any other orientation, there will be a net force and the object will rotate until the net forces are zero. The geometric center and the buoyancy center need not coincide if the object does not have uniform density.

2. The argument that a submerged column has no buoyant force is flawed. Gravity is pulling the pillar down, and buoyancy is presenting a vertical force tending to lift the column. That does not depend on the pillar's base not presenting a surface area to the water. The buoyant force depends upon the weight of displaced water and acts in a direction opposite to the force of gravity.
Shape does not matter.

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