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Re: Elastic potential energy



----- Original Message -----
From: "Justin Parke" <FIZIX29@AOL.COM>
To: <PHYS-L@lists.nau.edu>
Sent: Friday, November 16, 2001 3:26 PM
Subject: Elastic potential energy


Consider an ideal spring attached to a wall and in a relaxed state.
Define x=0 to be the free end of the spring in this relaxed state.
The spring is then stretched so that the end of the spring is at x1.
Finally, it is stretched even further so that the end of the spring is at x2.
The difference in elastic potential energy in the spring between those two
points is 1/2 k(x2^2 - x1^2).

OK.

Now consider the same spring stretched to the original x1.
We now define a new coordinate system whose origin coincides with x1.

Now the spring "starts" with a potential energy 1/2 k*(-x1)^2 and a
tension k*(-x1) (negative because in "negative direction).

The spring is stretched to x2 in the old coordinate system which we now
simply call x, so that x = x2-x1. The difference in elastic potential
energy between these two points is now 1/2 k (x2 - x1)^2.

No. Now the difference is {1/2 k*[x-(-x1)]^2} - {1/2 k*(-x1)^2}. Which
is the same than before, of course.
The potential energy in x is not 1/2 k*x^2, because the (conservative)
work you're doing from O' to x is not:
integral from O' to x of k*x in dx
but
integral from O' to x of k*[x-(-x1)] in dx
which gives you 1/2 k*x^2 + k*x1*x. This is the right expression of PE
in the new RF.

I am confused by this result. The very fact that we even defined
a potential energy for the spring is because the spring force is
conservative. Thus the potential energy should be arbitrary to
within an additive constant and differences in potential energy
should be identical for different coordinate systems.

That's absolutely right. The mistake is in translating the parabola
1/2 k*x^2 in the new RF as 1/2 k*(x')^2. That's wrong. The relaxed
position of the spring define the origin of the RF where the parabola
is in reduced form.
The PE is really arbitrary to within an additive constant - but no
additive constant can cancel out the *linear* term k*x1*x that arises
because of the translation, and that you need to fix your problem.

It's a pity that arbitrariness of PE is always discussed only with
reference to the gravitational PE near the Earth. There the PE is
linear, so when you translate the origin only a constant term arises,
which can be cancelled out by an additive constant.

I hope this helps.

============================================================
Paolo Cavallo " I am a teacher at heart, and
there are moments in the classroom
when I can hardly hold the joy. "
P. J. Palmer, 1998
paolo.cavallo@iperbole.bologna.it
http://www.alberghetti.it/paolo.cavallo/pc.htm
============================================================

EMERGENCY - UNO STRACCIO DI PACE - www.emergency.it

Does the relaxed position of the spring define a unique origin for a spring or am I
missing something here?