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