Buoyancy of Vertical Columns
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
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
Illinois Central College
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.
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Update: June 2012