Originally Posted by **Cyberrave** Ok, so the reason That the method 2 does not work is that before equilibrium is reached, the sring and mass ossilate up and down, where energy is lost due non conservative forces. |

Basically, yeah.

If you tweak method 2 to be

$\displaystyle \Delta EPE = \Delta GPE + \Delta KE$

then you get

$\displaystyle \frac{1}{2} k x^2 = mgx + \frac{1}{2} mv^2$

The main important point is that the velocity is non-zero at the point of the restoring force of the spring being equal to the weight, so there is energy wrapped up as kinetic energy. That explains the discrepancy between the two methods. Because the kinetic energy is non-zero at the point, you'll get oscillations. if you try and go on to solve it, you get:

$\displaystyle 10 x^2 = 9.8 x + \frac{1}{2} \left(\frac{dx}{dt}\right)^2$

$\displaystyle \frac{dx}{dt} - \left(10x^2 - 9.8 x\right)^{1/2} = 0$

This is a horrible non-linear 1st order ODE describing the oscillation. It's solvable though... Good luck!