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Spring potential energy without Work (sort of)



Upon reflection I think its possible to get the spring potential energy
without introducing work (sort of)

First, comments and disclaimers:

1) I'll describe one dimensional situations only

2) Since work comes from N2 (Newton's 2nd law) as usually derived, it seemed
that you should be able to arrive at PE expressions directly w/o explicity
mentioning the Work intermediar, *though in practice it amounts to the same
thing*; I don't recommend this, as I think you lose some motivation and I
think some descriptional generality by doing so, but that is a matter of
taste.

3) The spring requires some calculus, which shouldn't surprise us since it
is an example of a force that produces a non-constant acceleration when it
is the only force acting on an object. And algebra based courses tend to
require some handwaving when treating such forces (or at least their
kinematics when working with instantaneous quantities).

Warm-up:

derive that
Delta KE = - mg h

for "weight" near the surface of an ideal planet-like object. h defined as
Delta x, having chosen x as the vertical coordinate with the positive
direction being up (feel free to use other choices and make corresponding
changes the Delta K result.

This can be done in an Algebra based class by utilizing F/m = g and constant
acceleration equations. Next replace h with Delta x and call mgx potential
energy. Then note that Delta(KE+PE) is zero and talk about conservation of
energy and the Feynman parable or your other favorites blah blah blah

Point out that a similar arguement utilizing spring forces works but
requires calculus as we no longer have constant acceleration equations to
work with:
______________________

Generic calculus versions: K==KE

infinitesimal version

-dK = d (1/2 m V^2) = m*V*dV = m*dx/dt*a*dt = m*a*dx (notice this last
term is really the infinitesimal work done!!)

Next integrate.

This naturally begs the question somewhat, as it really is the same
calculation that one does utilizing the calculation of work as an
intermediate step.
______________________________

I prefer to explicitly introduce the Work concept,

but I suppose one could do the above first to obtain a quantity that when
added to KE gives a conserved quantity (under the constraints obvious to
what is being discussed here - no non-conservative forces for the moment,
please). Then we could give this mysterious quantity a name Int m*a*dx,
i.e. PE. And then next point out that many books also give it the name Work
done by the force. The problem is that this means you'll have to blubber a
bit concerning non-conservative forces and as has been pointed out, the Work
concept has no such constraints, it can be defined for all forces for which
it makes since to do a Int F dot dr type calculation.