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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).
Now consider the same spring stretched to the original x1.
We now define a new coordinate system whose origin coincides with x1.
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.
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.
Does the relaxed position of the spring define a unique origin for a spring or am Imissing something here?