Re: ENERGY WITH Q

["Carl E. Mungan" <mungan@USNA.EDU>]

> Starting at the bottom page 1 it says:
>          "Represent the object as a small box moving along a curved
>          path, rather than as an abstract dot."
>
> A paragraph or so later it states the "Work / Kinetic Energy" theorem and
> says we should
>          "Derive this from N2 and kinematics."
> and goes on to apply the theorem to situations such as books sliding on
> tables under the influence of friction.
>
> The problem is that you cannot derive any such theorem that can be applied
> in this way.  The theorem as usually derived applies only to point
> particles ("dots") or objects with no internal degrees of freedom.
>
> You simply cannot apply the F dot ds formula to the total macroscopic force
> (F) on the book and its average macroscopic motion (ds).  If you haven't
> got the premises, you can't conclude the conclusion.
>
> In particular, suppose I have a cart.  I push it forward across the
> table.  I feel resistance, a force F.  I move it a distance ds.  When I let
> go, it comes to rest promptly.  Can you conclude that non-conservative
> forces are at work?  No.  The cart I had in mind has some clockworks
> attached to the wheels.  When you push it along it winds up the spring
> (using purely conservative forces).  When I let go the spring doesn't
> unwind because of a ratchet.  Every wind-up clock in the world works this
> way.  With a little more complexity I can arrange it so the spring winds up
> (always up) whether you push the cart forward or backwards.
>
> Bottom line:  Deriving thermodynamics in terms of "work" is a bad
> idea.  Exhibiting a large pile of books that do it this way isn't going to
> make it a good idea.

I assume this is simply a miscommunication and that you are too busy
to look over my long document more carefully.

The full name of the theorem referred to above is the
pseudowork-translational-kinetic-energy theorem. No thermodynamics is
involved. No mention has yet been made of nonconservative forces.

I'm confident you agree that for an object with or without internal
degrees of freedom and having total mass M, the net force (which is
of course external) on the object equals M times the acceleration of
its center of mass (CM). Now dot both sides with a differential
displacement of the CM and integrate. That gives the W-K thm.

Applied to your cart example. If F is the *only* external horizontal
force on the cart, I hereby publicly guarantee it will gain KE,
regardless of what clock mechanism is inside it. If you meant that
there are other external horizontal forces, please spell out a
free-body diagram for your clock-cart a bit more clearly.

That's it. There's nothing deeper to it than that at this stage!

I'm very nervous that this was not clear in my document. The *whole*
point of my document was to argue that we should *first* teach our
students about work and energy in a familiar mechanics context
*before* introducing any thermodynamic ideas. I later clearly state
that the first law is inductive and experimental, in contrast to the
work-energy theorem which is deductive. Perhaps it's what I'm calling
these equations that is throwing you. But I'm purposely trying to
stay reasonably close to standard textbook nomenclature.

I strongly encourage you to look over the examples in my
document--sliding blocks, ice skater pushing off wall, compression of
an ideal gas--so that we can discuss a situation where the forces are
(hopefully) clearly spelled out.

I know some texts imply that the W-K thm only applies to point
particles. They're simply wrong if W means pseudowork. The real
difficulty is that they're trying to take W to mean thermodynamic
work. That's a big mistake in my view. Why teach thermo in the
mechanics chapters? When we get to thermo, *then and only then*
should we explain the connection between pseudowork and thermodynamic
work. This is not hard to do once a microscopic picture has been
developed, because then as you say we *are* dealing with particles so
the two kinds of work mean the same thing.

I strongly object to texts which consider examples (in the mechanics
chapter) of say clay hitting the floor and then say (in passing, no
less!) that the floor does no work because it doesn't move. Is there
an introductory student alive who is supposed to make sense of this
before having been introduced to thermo? It's a case of books
worrying so much about chapter 20 that chapter 3 is virtually
unreadable because of all the qualifiers in it. When the students are
ready to understand the microscopic internal dissipation, then we can
discuss that. For now, we're just trying to understand accelerations,
collision times, kinetic energy, stopping distances, etc.
--
Carl E. Mungan, Asst. Prof. of Physics  410-293-6680 (O) -3729 (F)
      U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5026
mungan@usna.edu    http://physics.usna.edu/physics/faculty/mungan/