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Name: Nic
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Question:
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?



Replies:
Nic,

As for the first point, the center of buoyancy is at the center of volume because it is essentially the average location. If you add up all the little bits of force on all the little bits of surface area, the effect is the same as applying all these bits of force at the center of volume. Calculus can show that the object will rotate as if the total force were at the center of volume. The center of volume works as a 'shortcut' to the correct predictions.

As for an object on the bottom, it is possible for two surfaces to stick together under water, thus "changing the rules", but it seldom happens without the correct equipment. You would need two surfaces so perfectly matched than no water molecules could sneak between them. An example is a plunger. You put the plunger under water and push out some of the water. The plunger tries to rise, but the water cannot get in because of perfect seal. The pressure on the underside is not as large as the pressure at that same depth outside the plunger. The center of buoyancy is not necessarily at exactly the center of volume, but it is extremely close. Horizontal pressure effects (pressures on the sides) decide the height of the center of buoyancy. Vertical effects (bottom, top) decide the horizontal positioning of the center of buoyancy.

As for the pillar on the bottom of a lake, there is still water within the material at the bottom. Even a pillar submerged in the sand has water pressing against its bottom. Water also presses on the sand from all sides, which then "transfers" the effect to the bottom of the pillar. Although sand molecules do not exert water pressure in the strictest sense, they do exert a normal force that has the same effect.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College


Nic:

In response to your first question, we often reduce a body to an equivalent point such as the center of mass to simplify calculations. In these cases, the forces acting on a body are assumed to act at the center of mass. It is the same with a center of buoyancy.

Second question. Were you to exclude all water or pressure from the bottom of the piling, it would not float. Specifically, the mud in the bottom is filled with water between the particles so a force is transmitted. In fact isolating the bottom of a body underwater is, in real life, is extremely difficult for as soon as the merest amount of water gets under the body, you are done for.

Robert Avakian



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