Re: ENERGY BEFORE Q

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

> Model 2a --> a little more realistic world with nonconservative forces, but
> no dissipation (no entropy production).  Example: Charged particle in
> time-varying magnetic field.
>
> 2) If OTOH we are talking about work in terms of F dot dx, and if we are
> willing to impose a number of restrictions and assumptions, and if we don't
> mind some slightly circular arguments, then we can make some
> pedagogically-useful connections to potential energy.
>
> 4) There is an intimate connection between force and momentum.  IF (big if)
> you know the relationship between momentum and kinetic energy, you can
> connect work to kinetic energy.  (Note we are calculating kinetic energy,
> not potential energy.  Also note we are calculating the KE, not defining
> it.)  Then if you want to connect this with potential energy, you have to
> make several major assumptions:
>    a) You need to _assume_ that KE+PE=constant.
>    b) You need to _assume_ that the force field is conservative, so that it
> is possible for there to be a PE function that depends only on position.
>    c) We are still assuming we know the KE as a function of momentum.
>
> Let us discuss each of these assumptions:
>
> A) I've seen way too many textbooks that claim they have "proved"
> conservation of energy, when in fact they just assumed it, as assumption
> (1a).  The alleged proof is blatantly circular.
>
> If you are going to _assume_ KE+PE=constant, why not be up front about it?
>
> B) Introductory texts typically restrict the discussion to conservative
> force-fields.  But this (like the previous item) seems circular to
> me.  Circles within circles.  Assuming the force-field is conservative is
> more-or-less tantamount to assuming it is the gradient of some
> potential.  So defining the potential in terms of a conservative force
> doesn't tell us anything beyond what we just assumed.

Hmm, this whole business is rather tricky. But I'm not yet convinced.
I agree we cannot prove conservation of energy in general. But I'm
not ready to say that means we have toss model 2 out of our intro
courses. In my opinion, model 2 is about mechanical energy including
the possibility of dissipation (or creation) thereof.

Here again is my basic (and very standard) outline:

1. Define work as the integral of force dotted with the displacement
of an object. (Technically, this is pseudowork in model 2. Don't tell
the students that. The adjective "pseudo" is off-putting and
unhelpful. Later we'll do thermo in which we do everything at the
particle level and at *that* level, pseudowork and "real" work mean
the same thing, so why cloud the issue? I disagree with Arons
completely on this point.)

2. Integrate Newton's second law to prove W_net = delta(K) where K is
defined in terms of the mass and overall velocity of the object.
(Technically this is the center-of-mass translational velocity. But
at this stage in our typical intro course, we have not introduced
internal motions of an object relative to the center of mass. So
again, we don't worry the students with piles of disclaimers they're
not ready for.) Do lots of examples using this equation, including
cases where there's friction (both kinetic and static), human applied
forces, gravity, springs, etc.

3. Now draw attention to the fact that with gravity, a ball regains
its speed when it falls, but with friction, you lose speed even if
you bounce backward off a bumper. Use this to introduce the idea of
conservative and nonconservative forces. Define them formally in
terms of whether the work is path-dependent. Point out that therefore
a conservative force does zero work over any complete loop in space.

4. Hence, in the conservative case, we are somehow storing up the
energy in a mechanically recoverable form. Call this PE. More
formally: Because the work is path-independent, the work is a
function of the final position only (for fixed initial position which
we'll choose as a reference position). Define the negative of that
function as the potential energy. (I see nothing circular about this
definition. Yes, mathematical sophisticates know that there are many
notions of conservative that can be shown to be equivalent:
curl-free, path independent, expressible as a potential, zero over a
closed path, etc. So? I choose ONE as my definition of conservative
and then prove the other ideas mathematically as *theorems*.)

5. Point out that PE involves an interaction between two objects:
ball and earth for gravity, or two masses for a diatomic spring. But
the PE doesn't belong to either. It belongs to the system of two
objects (plus their field, if you have introduced that concept
already in class).

6. So we must now discuss systems of objects. Include in your system
*at minimum* every relevant object which interacts via conservative
forces. Now redefine K as sum of K of each object in system. Redefine
W as sum of W on each object. Point out that the work on object i can
be due to either conservative or nonconservative forces. Split off
the conservative part (necessarily internal). That becomes U (sum of
all forms of PE for all pairs of objects).

7. We have therefore proved W_nc = delta(E_mech) for a system of
objects. Do another bunch of examples including rough blocks
connected to others by pulleys, Atwood's machine, figure skater
pushing off wall, etc. (Note that I have NOT assumed or required
KE+PE = constant.)

That's as far as model 2 goes. I have not yet introduced Q, internal
energy, a "correct" microscopic definition of W, etc. Why do I have
to? I'm still just doing mechanics problems. The bigger picture about
thermodynamics, nuclear energy, heat, etc comes later.

8. If however you're unsatisfied with leaving things hanging, I'm
willing to go one step farther. State the law of conservation as:
-delta(E_mech) = delta(E_int) for an *isolated* system (ie. every
relevant object is in the system now, not just the conservatively
interacting ones). Define internal energy as a sum of a bunch of
microscopic KE and PE terms of the atoms, electrons, etc. Use this to
explain how the figure skater gained mechanical energy, the sliding
block lost mechanical energy, 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/