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Re: The concept of potential energy.



At 08:09 AM 10/13/01 -0400, Ludwik Kowalski wrote:
I was objecting to what Dario posted on the basis of it being
"too advanced".

I agree with that, and I'll go even further: The equation
Delta W = - grad U dot ds
is too complicated to be useful, but not complicated enough to be correct.

Otherwise it was appropriate in the context
of our fictitious world of two forces, m*g and k*x.

No, it is not appropriate even then, unless you mean to exclude F=ma also.

The problem comes from attempting to identify -grad(U) with "the
force". That's just not right. In fact, -grad(U) is the _part_ of the
force associated with U; it leaves out many other contributions to the
force budget.

a) I can (with difficulty) imagine restricting attention to a
"fictitious world" where every force-field has the form m*g or k*x, which
can be derived from a potential. This is brutally fictitious, since it
excludes changing magnetic fields and other interesting force-fields.

b) But what about forces that aren't associated with a force-field at
all, such as F=ma? I cannot imagine doing physics in a world where F=ma
does not apply. What would be the point of that?????

As I wrote previously:
> Suppose I do work (F dot dx) on a free particle,
> imparting energy to it. Suppose all the energy winds up as _kinetic_
> energy in the particle.

In this case we have work that has nothing to do with potential energy.

MAIN POINT: Defining work in terms of potential energy is a
blunder. Defining potential energy in terms of work is an even bigger
blunder. The relationship between force and energy is fairly complicated,
so by the time you've got an even halfway-correct statement, it's too
complicated to be useful in an introductory pedagogical situation.

I can't imagine why anyone would want to attempt such definitions, when the
available alternatives are simpler _and_ more correct.

I recommend introducing energy as primary and fundamental. That's how I do
it. It serves my purposes quite nicely.
http://www.monmouth.com/~jsd/how/htm/energy.html

Energy is chapter 1. (Force doesn't show up until chapter 4.)

Defining "potential energy" in terms of "work" makes about as much sense as
defining "house" in terms of "bricklayer". The best you can say is that
some of the time it is part of the story.

BOTTOM LINE: Defining work in terms of potential energy is a
blunder. Defining potential energy in terms of work is an even bigger blunder.