Re: why pseudowork (NOT)

[John Denker <jsd@MONMOUTH.COM>]

At 02:14 PM 10/28/99 -0800, John Mallinckrodt wrote:

> Here is a sketchy summary of three of the most commonly used definitions
> and what they turn out to be equal to:

This summary is helpful because it allows me to see (for the first time)
where JM is coming from.

In particular, I hereby subscribe (with minor reservations, as discussed
below) to definition #2, namely

> Sum of integrals of all external forces over the motions of the points of
> application calculated in an inertial frame

But (!) I disagree with the interpretation that immediately follows:

>   = change in total energy (bulk translational + internal)

I will show that my disagreement is not based on opinion ("I don't like it
nyah nyah nyah");  instead I will point out that its application to the
sliding-block problem is fraught with peril because it does not specify how
to handle some important additional aspects of the problem.  JM and I
obviously have a difference of opinion on how to handle these additional
aspects.

To be specific:

Since we are analyzing a problem where heat is generated, we need to be
able to draw a distinction between
 *) macroscopic forces and macroscopic displacements
versus
 *) microscopic forces and microscopic displacements

Otherwise (i.e. if we are going to follow all the forces at the microscopic
level) everything is reversible and there is no such thing as heat or
temperature or any of that.


In this particular case, I have adopted the interpretation that work is the

%Sum of integrals of all *macroscopic* external forces
% dotted into the *macroscopic* motions of the points of
% application calculated in an inertial frame

For non-dissipative problems, this is equivalent to JM definition #2 cited
above (with the dot product thrown in to make it a little more general).

For dissipative problems, I assert this version is superior, for reasons we
now discuss.

This notion of work leads to a work-energy theorem involving only the
macroscopic (nonthermal) energy of the object.

This notion of work has the great advantage that we can actually calculate
things with it.  In general we have no idea as to what the microscopic
forces or displacements are.  So it is really important to be able to throw
all the microscopic stuff "over the fence" into the thermal arena.


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


Just to see if I understand where JM is coming from, let me use his
definition to derive his result.  Here is what we may call the "stick/buzz"
model of friction:


                |               |
                |   block       |
                |   <----       |
                |_______________|
                  |  \  \  |  |
             ___________________
            |                   |
            |     table         |
            |                   |
            |                   |
            |                   |



where you can see little "fingers" (asperities) hanging down from the
bottom of the block.  In this model system, the lower tip of each finger
  *) temporarily welds to the table
  *) bends, storing spring-energy
  *) breaks free
  *) buzzes like crazy for a while, converting spring-energy to heat
  *) iterates this process

So, within the confines of this model, at the points of contact there is
never any velocity.  Therefore there is no work done on the finger-tips.  I
believe this exemplifies the point JM has been driving at.

======

Now let me say what I don't like about this approach to the problem:

  1) We have no reason to believe this stick/buzz model is a correct model
of the microphysics.  It could be that the fingertips dash over
imperfections in the tabletops like skiers over moguls, exerting forces
while moving.  If they have *any* tendency to exert a force while moving,
then the assertion that the the block and table undergo equal and opposite
work must be abandoned.

I'll even make a stronger statement: until proven otherwise, I hold the
opinion that the ultramicroscopic forces (electrical fields, chemical
bonds, and all that) are of the sort that exert force while moving.  That
is, the stick/buzz model is not a sufficiently correct model of *any*
physical system that I know about to support the claim that the sliding
block and stationary table undergo equal and opposite work.

  2) Even within the stick/buzz model, it is highly implausible to assume
that all the asperities are on the block and none are on the tabletop.  If
you assume that even a tiny percentage of the asperities are on the
tabletop, the assertion of zero work on the sliding block must be abandoned.

This item (2) does not militate against the assertion that the block and
table undergo equal and opposite work.  Work can balance because (within
the stick/buzz model, and using the JM definition of work) there is
non-zero work done on the table, because the table-fingers (considered as
part of the table-subsystem) are moving at the point of application.

  3) If you insist that we adopt the stick/buzz model, then there is a
simple explanation for our previous lack of understanding:  I was treating
the block as a simple object with a single well-defined macroscopic
velocity.  When I spoke of the velocity of the block (or the velocity of
the table) I was *not* talking about the velocity of the fingertips.

I find it bizarre to think of the sliding block doing work on the
stationary table.

This is for me sufficient motivation to adhere to a definition of work that
depends only on the macroscopic forces and motions.   From where I sit, the
practical and pedagogical advantages of such a definition appear overwhelming.


______________________________________________________________
copyright (C) 1999 John S. Denker             jsd@monmouth.com