Re: [Phys-L] Nice question on buoyancy and balance

[John Denker <jsd@av8n.com>]

On 01/31/2014 11:09 AM, Carl Mungan wrote:
> The quick answer is the ball rises because... before it was only
> partly immersed in liquid and now it's fully immersed in liquids.

For this one I have a picture:
  http://www.av8n.com/physics/img48/buoyancy-multi.png

Start with situation A.  There is an /imaginary/ boundary 
containing a certain amount of water plus a certain amount
of air.  This is a hydrostatic equilibrium situation.

In situation B, we have replaced the imaginary thing with
a real thing of the same weight, the same shape, and the
same position.  Since all the forces (and all the momentum
flows!) are the same, we are still in hydrostatic equilibrium.
This illustrates Archimedes's principle.

Situation D is similar to A, except that we now have water
plus oil instead of water plus air.  In situation E, we
replace the imaginary object with a real object of the 
same weight, size, and position.  Since oil is heavy, we
need a lot less water to make up the required weight.

This extends Archimedes's principle to multiple fluids:
The weight of the displaced fluids (plural) must equal the 
weight of the floating object.

Technically the same rule applied previously:  When weighing
something you really ought to account for the buoyancy due
to the displaced air.

=========================

I gave the object a peculiar shape because I'm still on
my soapbox about momentum flow.

Calculating the forces in any detail would be a nightmare.
The hydrostatic pressure acts everywhere perpendicular to 
the local surface, and each local patch of surface has some
screwy orientation.  This is complicated.  I don't want to
teach students that physics is complicated.  I want to teach
them that it is simple and elegant and powerful.

Now, if you're a smart physicist you know the only thing 
that matters is the total force.  So, how do you get from
the fundamental laws of motion to the total force?  The
fundamental force laws speak of point-by-point force balance.
There's gravity here and pressure there and it's a big mess.
Sure, you can write a big summation sign and it has to work
out, but that strikes me as too mathematical and not as
pictorial as it should be.

In terms of momentum flow, it's easier to see that the total
is super important.  Momentum is a conserved quantity!  If
there is an imbalance, momentum is going to accumulate, and
that's a problem.  I don't know about you, but in situation
A I can visualize momentum flowing in (via gravity) and the
same amount of momentum flowing out (via buoyancy).  In
situation B the same amount of momentum is flowing out, so
there had darn well better be the same amount flowing in.

Because it is a conserved quantity, once it gets inside the
boundary I can rearrange it at will within the boundary
without changing anything.  The means I don't care about
the details of where on the boundary the buoyant force is
acting.

I'm not saying you can't figure this out in terms of forces.
I'm mostly just saying that the momentum-flow formulation 
automagically draws attention to the things that matter.

I'm also saying that when you have a force /field/ the force
approach is always a nightmare.  You can't talk about the force 
"at a point" because there are always /two/ equal-and-opposite 
forces at each point, since parcel X is acting on parcel Y 
/and vice versa/.  You would have to draw the free body diagram 
for parcel X separately, and then draw the FBD for parcel Y 
separately, and even then it's hard to diagram the relationship 
between the two of them.

In contrast, there is absolutely no problem diagramming the
momentum flow across the boundary at a given point.

The momentum-flow approach is a win for hydrostatics problems,
and it is huuuuge win for hydrodynamics problems.  I reckon
you "could" do fluid dynamics in terms of forces instead of
momentum, but it would be sheer masochism.