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... Does the relaxed position of the spring define a unique
origin for a spring or am I missing something here?
John M responded:
Not at all, but the correct equation for any choice of
the "reference stretch" xo is
elastic potential energy = (k/2)x(x + 2*xo)
This reduces to the familiar (k/2)x^2 with the convenient (but
not mandatory) choice, xo = 0.
If I'm not mistaken, there is a minus sign error,
I get for (1dim motion along x axis, F=-k(x-xo);
put
elastic potential energy = (k/2)x(x-2*xo)
One can take advantage of the ability to add an arbitrary constant to
this into a more "elegant" formor
namely add (k/2)xo^2 to get
elastic potential energy = (k/2)(x-xo)^2 which is in a familiar form
compared to the usual
(k/2)x^2 , which is to say in words
"elastic potential energy = half k times (distance spring is stretched
compressed) squared" for any choice of coordinate origin.