Re: [Phys-l] The Normal Force

[John Denker <jsd@av8n.com>]

On 05/24/2007 10:45 AM, Jeffrey Schnick wrote:

>  I'm just looking for a slightly more complete explanation, at
> the same level--one that might also account for why stepping onto the
> surface of a pond is so much different when the water is in its liquid
> state than it is when the water is in its solid state.

That's a good question.  Parts of the answer are easy, but
other parts are not so easy, especially for naive and
literal-minded students.

> Feynman says:
> > It is the fact that the electrons cannot all get on top of each other
> > that makes tables and everything else solid.

Well, he misspoke.  It's not true that "everything else" /is/
"solid", so any explanation of universal solidity must have
its wires crossed somewhere.

In fact, the argument about the exclusion principle says nothing
about directionality, so it applies to liquids as well as solids.
That is, it really explains near-incompressibility rather than
solidity.

>  I would expect that
> there might be some rearrangement of the electrons near the surface of
> both objects that would allow the ions to get close enough together for
> some electrostatic repulsion of positive ions by positive ions. 

I don't think that approach works.  It talks only about
electrostatic energy, which is potential energy.  First of
all, the electrostatic field is essentially 100% shielded,
i.e. the electrons shield the nuclei and vice versa.  Secondly,
the electrostatic energy does not vary with position (or
density) in the right way to explain why solids and liquids
are hard to compress.

The key point that Feynman was making is the importance of
/kinetic/ energy.  There are lots of things we take for granted
about the properties of materials that cannot be explained
unless you take into account the /kinetic/ energy.

  The fact that atoms and molecules and macroscopic objects
  have any size /at all/ is crucially dependent on KE.  Without
  the KE term in the equation of motion, everything would
  collapse to a point.

Now here's where naive students get confused:  What we call
the macroscopic potential energy of a compressed spring (or
compressed liquid) is in some sense due to microscopic kinetic
energy!

The following points may help with the explanation.  This is
not a complete explanation, but it's a start:
 *) The KE expression, p^2/(2m) is nonlinear.  Nonlinearity
  means, among other things, that the square of the average
  is not the average of the squares.  So there can be (and
  is) a lot of microscopic p^2 even where there is no
  macroscopic p^2.
 *) We define potential energy as energy that depends on
  positions, not velocities.  We define kinetic energy as
  energy that depends on velocities, not positions.
 *) At any /particular/ length scale, PE and KE are mutually
  exclusive, as should be obvious from the definitions.
 *) OTOH when you start mixing length scales such as by
  mentioning macroscopic lengths and microscopic velocities,
  KE and PE are no longer mutually exclusive.

Everything I've said so far deals with compressibility, and
applies more-or-less equally to liquids and solids.  We now
return to the main question:

> why stepping onto the
> surface of a pond is so much different when the water is in its liquid
> state than it is when the water is in its solid state.

This also returns to the "normal" keyword in the Subject: line.
Liquids don't support a non-normal force the way solids do,
and they don't even support a normal force unless it is a
/uniform/ normal force per unit area, i.e. a pressure.

So the question is, why are solids different from liquids.

The energy arguments in the previous section are non-directional,
and cannot begin to answer this question.

A partial explanation can be made using the Bragg-Nye model
(Feynman volume II chapter 30) which is probably good enough
for metals and metallic bonds.  It is also good enough for
a hand-wavy understanding of non-covalently bonded solids
such as ice.  In particular, Bragg-Nye explains why ice
crystals can crack by splitting along crystallographic
planes.

For an introductory discussion, stop here.  You should warn
the students that there is more to the story, but you don't
need to tell the rest of the story.



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

For the non-introductory audience, here is the next part of
the story.  Covalent bonds are directional in ways that cannot
be explained by Bragg-Nye.  For example, an ethylene molecule
is not free to rotate around its double bond (in contrast to
ethane, which is vastly freer to rotate around its single bond).

           \     /
            C = C           ethylene
           /     \




           \     /
          — C — C —         ethane
           /     \


It is possible to explain this directionality at the college
level or even the HS level, but it takes a week ... or more,
depending on how much foundation needs to be laid.  Remember
this whole thread grew out of Feynman's mention of the
connection between QM, KE, and the periodic table.  To explain
directionality, you go somewhat farther down that road.  You
need to explain why the periodic table has rows (energy
depends on N) and why the rows have limited length (l limited
by N).  Some possibly-helpful ideas on how to model and
visualize atomic orbitals can be found at
  http://www.av8n.com/physics/orbitals.htm