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Fs = Fg

kx = mg

x = mg/k

Now, you can also calculate the energy of the initial and final states of this system. In the initial state, the spring is uncompressed and the mass is motionless, so both kinetic energy and the spring potential energy are zero. Therefore, the system has an initial energy of:

Ei = mgx

where x is the distance the spring will be compressed when the mass is released.

Now, if we calculate the energy of the system in it's final state, when the spring is fully compressed, we know that the potential energy is zero since we used our reference point for zero height as the point where the spring is fully compressed. We also know the kinetic energy is zero because the mass is motionless. This means that the final energy must be:

Ef = (1/2)kx^2

Using conservation of energy, we can say the initial energy of the system equals the final energy of the system. Now this is where my brain explodes. Setting the initial energy equal to the final energy gives:

mgx = (1/2)kx^2

2mg = kx

but i already showed earlier that because of the canceling of the force of gravity by the force of the spring, we know that:

mg = kx

This isn't possible! how can mg = kx AND 2mg = kx? Anyway, it obviously can't, this would mean that half the potential energy magically vanishes once the spring compressed and that can't happen.

Anyway, I know this can't be correct... if anyone can tell what is wrong with this argument, it would bring me great joy :)

Cheers.