Re: Definitions

[Brian McInnes <bmcinnes@PNC.COM.AU>]

Hi All,
In his contribution on "definitions" Tim has raised again the question
of the work-energy theorem.

> Take "the work-energy" theorem.  We have had discussions here where we
> discover that different people have different definitions, and often don't
> realize it!  I think it would be useful to be able to say "there are
> several theorems that look like the work-energy theorem, but the official
> definition is ...."
>
>
> But who can tell me the "official" interpretation of the work-energy
> theorem?

I would like to propose the thesis that
(1) we all agree on a "work-energy" theorem for a particle;
(2) we disagree on a "work-energy" theorem for an extended body' and
(3) we must look at the consequence of the different definitions on
applications of the First law of thermodynamics.

(1)  For a particle the work done by any force acting on the particle
is defined as the line integral of that force during the interaction.
Further the total work done on the particle is defined as the line
integral of the net force during the interaction.  All agree?
Still further we can use Newton's second law to show that the increase
in kinetic energy of the particle is equal to the total work done on
it.  That, in effect, is the work-energy theorem for a particle.
Again, do all agree?
There is a side-track we can take here and define the concept of
conservative force: link the work done by conservative forces with the
decrease in potential energy of the particle and associate the work
done by non-conservative forces with the change in mechanical energy
of the body.  Let's leave that for now.

(2) When we consider forces acting on an extended body, we come to the
situation stated by Tim:"different people have different definitions".
 The different definitions are those of work.

Some define work done by any force as the integral of the dot product
of the force and the displacement of the centre of mass of the body.
Later I refer to this as the first model.  This gives a non-zero value
for work done by the push of wall on the roller-skater or the push of
the ground on the high-jumper or the frictional force on a body
sliding on a rough surface.  This non-zero value is then associated
with the change in kinetic energy (work-energy theorem).  The changes
in kinetic energy are quite obvious in all three of these examples.
I'm fairly sure all who build this model will agree I'm describing it
properly.  You say to me :"What are you worried about; isn't this a
nice, neat way of modeling the real world?"  I reply " Yes, as far as
it goes but first, remember (or realise) that what you have described
as work is not the line integral of the applied force and second, as
we shall see, you are going to meet a major problem in linking this
model with the First law of Thermodynamics".

Others define work as the line integral of the force.  Later I refer
to this as the second model.c (By the way, there is no difference in
the two definitions for a particle but there can easily be a
difference for an extended body.) Under this definition, no work is
done by any of the three forces referred to in the examples cited in
the previous paragraph. (The frictional force is not, in my model, a
single force moving its point of application as the block slides along
the surface, but a large number of extremely localised forces,
associated with the breaking of cold welds at different places; none
of these forces are displaced and no work is associated with them.)
So, work done in these examples is zero.  Obviously the change in
kinetic energy is not zero.  That change can be calculated, using
Newton's second law and, as in the previous model, finding the
integral of the dot product of the force and the displacement of the
centre of mass of the body.  Both models agree that this is the change
in kinetic energy; the two models disagree as to whether this is the
net work done on the body.

(3)  Now, let's look at the First law of Thermodynamics.  Go back to
where we were on that side-track dealing with a particle and
conservative and non-conservative forces.  What we had then was a
statement that the work done by non-conservative forces on a particle
is equal to its change in mechanical (kinetic plus potential) energy.

In taking this to extended bodies and moving to the first law there
are a number of developments. In place of a particle we deal with a
system, which must be carefully defined; the concept of work (W) will
cover only the mechanical ways of changing the total energy of the
system; non-mechanical ways are called heat (Q); the total energy of
the system (E ) or (U)  will comprise not only the kinetic and various
potential energies of the body but other energies which can be labeled
thermal, chemical, etc.  The law is written as W + Q = delta U.

I wrote earlier that there is "a major problem in linking this model
(the first model) with the First law of Thermodynamics"
Take the example of the block sliding to rest on the rough floor.
Suppose the system is the block. The first model gives a negative
value for W and a real, observed decease in kinetic energy of the
system.  It looks good.  But hang on a moment.  Doesn't the block and
the surface get warm; isn't there immediately some sort of increase of
thermal energy of the block; isn't there a non-zero value for Q?
I think the only way out of this with the first model is to redefine
the system as the block and the surface and there are no more problems
with the first law, which then tells you that the system has lost some
kinetic energy and there has been a heat transfer to the surroundings
(which, in the fullness of time, happens).  Then the frictional force
is internal to the system and the question of whether or not it does
work is superfluous.  There are no such accounting problems with the
second model.
In the example of the high jumper what the first model won't account
for is the reduction of chemical energy involved in a jumping process.
 The first model has a positive W, a zero Q, an increase in
gravitational potential energy and a usually ignored loss in chemical
energy; the second model has a
zero W, a zero Q and compensating increase in gravitational potential
energy and decrease in chemical energy.  (A nice point is what is the
definition of the system here: air is part of the biochemistry
involved in the decrease in chemical energy.)

Yes, I agree with Tim that definitions are important. so are the ways
we model the real world.  It would be nice to tie up the work-energy
theorem but the integrity of the First law of Thermodynamics is far
more important.

Brian McInnes