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von Philp wrote:_______________________________________________
Okay....
Imagine 3 vessles filled with water to the same height resting on a
table.
One vessle is a vertical cylinder, one is shaped like a "V", and one is
shaped like an "A". They all have the same area for the base. You've
see
the situation before....Pascal's Vases?
Now, I understand that the pressure on each base would be the same,
True, if given a favorable interpretation.
and
therefore the total force on each base would be the same.
False.
We need to be careful about "the" pressure and "the" total force.
-- Pressure*area tells us the force of the fluid acting agains the
_inside_
surface of the base; call this the fluid/base force.
-- The total base/table force is something else entirely. It includes
contributions carried by the stiffness of the walls of the vessel.
(The fluid _per se_ has no stiffness, but the vessel does.)
I also understand that the "V" shape would contain more total water, and
therefore would weight more, and therefore would exert more total force
on
the table.
Yes, the V has more total force.
What I don't understand is how the force on each base is the same, but
the
force on the table is not.
It is good to not understand that, because it's not true.
Is the additional weight of the water
transmitted through the walls of the container to the table via the
"edges"
of the base,
It is.
and that does not count as the force on the base due to the
fluid pressure?
It does count. We know it *must* count. We know this because of Newton's
third law, aka conservation of momentum.
What would be an effective way to explain this apparent paradox to
students?
1) I would start with Newton's third law. As always, when in doubt about
how
to interpret the third law, it is helpful to recast it as a conservation
of
momentum proposition. Downward momentum flows from the fluid into the
walls;
it has to flow out somewhere.
2) At the next level of detail, draw the force diagram. For students who
can't
handle calculus or even trigonometry, approximate the V-shaped vessel by
a
jaggy V with (say) 5 steps:
|_ _|
|_ _|
|_ _|
|_ _|
|_|
Then it's just a simple sum with 5 terms.
Show that there is a "coincidence" that the 5-term expression for the
total force
looks very similar to the 5-term expression for the total mass. Of course
it's
not really a coincidence; at each level, part of the fluid is supported
by
lower-lying fluid (making pressure) and the rest is supported by the wall
(making force in the wall). Each bit of fluid has to be supported by
*something*.
It's just conservation of momentum, aka third law.