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Re: [Phys-l] buoyancy on a submerged pole




On 11/3/2010 8:35 PM, John Clement wrote:
A simple gedanken experiment is to eliminate the glue and use something
different. Put a seal around the edge of the bottom of the box and then
through a small hole in the bottle of the aquarium pump out all of the
water
and after you have a vacuum close off the hole. There can only be
compressive forces between the two surfaces. And the bottom of the
aquarium
will flex less because the pressure on the top of the box is less than
the
pressure of the water when the box is absent. But that result can also
be
obtained by using a buoyant pseudoforce.

John M. Clement
Houston, TX

That's the trouble with pretend experiments: if you don't specify all
the conditions adequately, they can be controverted. If no water is
lost, the total weight on the base increases. As it applied closer to
the edges, the usual beam theory notes that the greater weight (usually)
provides less deflection. Wrong Conclusion, for your purposes??


This explanation is "obviously" looking at the empty box submerged being
held down vs the box glued to the bottom. But apparently this particular
hidden assumption did not get through. It was also mainly aimed at the
discussion over the stresses in the glue. As to the beam theory, whose
weight, and where is it applied?

If you push an empty box into the water and submerge it, the total force on
the bottom of the box has to increase of course when the water is conserved
and merely rises, and increses the pressure at the bottom of the aquarium.
But once the object is submerged, the total force on the bottom of the box
does not change with the distance between the box and the bottom of the
aquarium. Remember the pressure is determined only by the height of the
water. Of course the total force on the bottom does not change if you push
an empty box into a full aquarium, because the water level does not rise.
For the rest of a analysis we will assume the water is not lost.

Now if the empty box is attached to the bottom of the aquarium by a string
the string exerts an upward force on the bottom. At this point you are not
pushing on the box, so the total force on the bottom of the aquarium is
equal to the weight of the water + empty box. This is easily analyzed using
the idea of the buoyant force. We will make the usual physics assumption of
a massless empty box. So a submerged box attached by a string has the same
effect on the aquarium as no box, assuming no water is lost. We are
ignoring the effect that the points of application of the forces on the
bottom of the aquarium have been changed. The string will produce a dimple
in the bottom, which complicates the analysis.

Now if you do my gedanken experiment, you have not changed the total force
on the bottom of the aquarium, assuming the box is submerged all of the
time. But instead of a string pulling up on the bottom, you have eliminated
the pressure under the box. But now the box is pushing down on the bottom
of the aquarium. The downward push of the box is equal to the original
upward of the string and the downward push of the water on the area that is
now covered by the box. And the total force on the bottom of the aquarium
is the same as the total force of the box attached by a string, but less
than the case with the box held submerged by your hand.

Most of the arguments about the buoyant force have the difficulty that it is
actually in a sense a pseudo force or really a net force. It has conditions
such as the object must be in a fluid and the bottom part must be surrounded
by a fluid. So it can be used to analyze ships, balloons, fish and
submarines. But it can't be used to completely analyze objects glued to the
bottom of an aquarium. It can be used to calculate the total force on the
bottom of the aquarium, but not to analyze the stress in the glue.

I see no difficulty with introducing the buoyant force to intro physics
students even though it poses difficult questions for advanced analysis.
After all the normal force, and indeed many contact forces have the
difficulty that they are actually net forces. So when you push on something
the microscopic analysis is that you are applying a variety of forces to
each microscopic area. And even more microscopically the forces are due to
electrostatic forces and QM effects... Actually the buoyant force provides
a good introduction to the idea that you build up you understanding by
creating new models which refine or supplant the old ones. So first
understanding it as a "simple" force is OK. Then by looking at the "actual"
contact, the idea of pressure can be used to model this force.

Actually we use various models depending on the circumstances. So if you
have a submerged basketball, you would not use calculus to determine how
much force is needed to push it down in the water. You would use the simple
rule as if the buoyant force is just a simple force. But when you glue
something to the bottom, you need to use a different model to figure out the
internal stress in the glue, assuming the glue is rigid. If you have an
elastic glue, it will compress horizontally and eventually in the limit
could become a string. Depending on its elasticity it would merely have a
reduced contact area, and this would be truly an advanced problem. Then of
course there is the problem of a permeable glue!!! And a submarine stuck in
the mud is really stuck by a very elastic glue.

John M. Clement
Houston, TX