Springs and Potential Energy

[BlueEight]

Homework Statement

An ideal spring is suspended from a ceiling with a wombat attached to the end. The wombat is slowly lowered until the upward force exerted by the spring on the wombat balances the weight of the wombat. Show that the loss of gravitational potential energy equals twice the gain in the elastic potential energy of the spring-wombat-Earth system. WHY are these two quantities NOT equal?

Homework Equations

Change in gravitational potential = -mg(X_f - X_i)
Change in elastic potential = .5k(X_f² - X_i²)

The Attempt at a Solution

I realize that it is due to the fact that a person slowly lowers the wombat, thus acting as a non-conservative force, but do not know how to derive the solution.

 

[BlueEight]

I realize that the gravitational potential energy must be greater, but why is is exactly twice, no matter the mass of the object?