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> PEs = kx^2/2 belongs to the mass when the other end of the spring is
> anchored to earth.
May be you intended to say that kx^w/2 "belongs"
to the spring ?
In the case of the spring it is not clear in what is proposed -- Is it
contemplated that both the ball and the Earth be attached to the
spring? If so and it is a real spring, then the property of energy of the
spring will increase by the deltaPE of each atom of the spring