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Re: why pseudowork (NOT)

On Thu, 28 Oct 1999, John Denker wrote:

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

Thanks. There's much more in the paper, AJP, V60, 356-365.

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.

I will suggest that at least part of our difference of opinion
rests on what qualifies as "heat" and "work." From many of your
statements in previous messages I gather that you sometimes treat
heat and work as forms of energy that can be converted from one
to the other. I subscribe to what I think is the more conventional
attitude that work and heat quantify transfers of energy by
mechanical and thermal means respectively. As such one could
never "generate heat" although I would understand this
colloquialism to mean "causing the internal energy of a system to
increase." Of course, that can happen via work or heat.

I would say, for instance, that when two blocks collide
inelastically, they may or may not do work on each other
(depending on the definition you want to use, the reference frame,
etc.), but they *certainly* do not heat each other (beyond some
minor amount that might occur by conduction simply due to
contact.) Still, their internal energies increase and their
temperatures may rise measurably primarily as a result of the
conversion of bulk kinetic energy into internal energy.

I would say exactly the same thing in the case of frictional
interactions. Frictional interactions are irreversible and the
associated entropy changes may be calculated using a thermodynamic
path that involves heating, but I would not want to call friction
a thermal process. I find it more reasonable to think of friction
as an unknowably complex mechanical process.

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

I understand what you are trying to do here, but I don't see any
way of drawing an unambiguous distinction between what you are
calling "macroscopic forces" and "microscopic forces." At
precisely what level of "smallness" do macroscopic forces become
microscopic?

You go on to assert the superiority of this version, but I am left
wondering how to apply it in general. For instance, how would you
treat a glancing collision between the business sides of two
hairbrushes? Would it matter how many bristles each had? How
long they were?

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.

Basically. However, I imagined that the table also had finger-tips so
that positive work would be done on them (in the "lab frame") and an
equal amount of negative work on the block's finger tips (at least
under one pretty obvious definition of work.)

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.

This "rubbing of finger-tips" sounds like the beginning of an
infinite regress to me, but I'll grant you the likely unphysical
nature of the model for the reasons you go on to note in the next
paragraph.

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.

I didn't make that assumption and never expected to see zero work
done on the block.

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.

Right. (Except there is no "JM definition of work." I recognize
many different definitions.)

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 don't insist on the stick/buzz model. But you seem here to be
talking about a rigid block, one with no internal degrees of
freedom. If so, my definition #1--the pseudowork/kinetic energy
theorem--will be sufficient since there is no possibility for
internal energy changes. It will also have the advantage that you
will not have to try to distinguish between your microscopic and
macroscopic forces.

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

I take this to be your way of expressing a preference for a
definition of work that will allow you to say that no work is done
on an object that doesn't move. That's fine. I simply find it
needlessly limiting. I think you will have to live with
definition #1 and be willing never to treat internal energy
changes. Otherwise I'll bring out my hairbrush again and start
asking you some more difficult questions.

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.

Chacun a son gout

John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm