[Phys-L] Re: Help with Fluids....

[John Denker <jsd@AV8N.COM>]

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.
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