[Phys-L] connecting buoyancy to force +- density gradient

[John Denker <jsd@av8n.com>]

On 01/30/2014 08:32 AM, Jeffrey Schnick wrote:

> we cause more downward momentum to flow into it by bringing a steel 
> ball (part of the surroundings) into contact with the water.

Slight nitpick:  The beauty of the momentum-flow approach is that
it doesn't directly depend on the ball being in contact with the
water.  The key idea is momentum flow /across the boundary/.  So
one of the crucial steps is bringing the steel ball inside the
boundary of the RHS subsystem.

> we cause .... by bringing a steel
> ball ... into contact with the water.

I knew the discussion would eventually encounter this point. 
As soon as you start talking about contact with the water, some 
folks will want to know how that produces a force, and how much 
force.  This assumes we are using the force-based approach, which 
is not my first choice, but it's not unreasonable.

   It is super-easy for students to get the forces wrong.

Here's how I analyze it.  

The first step is to replace the steel ball with a /cylinder/
with the axis aligned vertically.  Pressure on the sides contributes
nothing to the force, by symmetry.  (I like a good symmetry argument
just as much as a conservation argument.)

Now, the force on the cylinder depends on the difference in pressure
between the top face and the bottom face.  
   This is where students go wrong.  They think of "the" pressure 
   of the water as a constant, forgetting that it is a strong 
   function of depth. 
The right answer is that net force is proportional to area times ΔP.
The displaced volume of water is proportional to area times Δh.
Assuming compressibility is negligible, we have ΔP proportional to 
Δh, so it's all nice and consistent.

This part of the story is no different from any other buoyancy problem.
I mention it because I'm still on my soapbox about the simplicity and
unity of physics.  I have a terrible memory, and I can't be bothered
to remember the details of the buoyancy law, certainly not in enough
detail to be sure of the limitations to its validity.  By rederiving
it every so often, I can be sure I am remembering it correctly, including
the limitations.

The same argument in reverse explains how the steel object exerts
a net force on the fluid.  You could infer this force from the
buoyancy law plus the third law of motion ... but it's nice to see
how it works microscopically.  Again it is nice to see the connection
between the force and the pressure /gradient/ ... something that is
not 100% obvious from the usual statement of the buoyancy law.

The extra detail comes in handy in any situation where something is 
messing with the pressure gradient ... e.g.
 -- a centrifugal field, which is typically quite non-uniform;
 -- the non-uniform gravitational field inside a star or other
  self-gravitating fluid;
 -- a stratified fluid such as layers of air/oil/water or cream/milk.