Vertical Oscillator

[Soaring Crane]

Show that the conservation of energy holds also for the vertical spring where x is measured from the vertical equilibrium position (x_0 = mg/k) for a mass m.

This is what I did so far. Where do I go from here?

f = mg - kx_0 = 0
kx_0 = mg
E = KE + PE_grav + PE_spring
= ([mv^2]/2) - mg(x + x_0) + [(x + x_0)^2]/2
= ([mv^2]/2) - mgx - mgx_0 + (k/2)[x^2 + 2xx_0 + x_0^2]
= ([mv^2]/2) - mgx - mgx_0 + [(kx^2)/2] + xx_0k + [(kx_0^2)/(2)]
E_x0 = ([mv^2]/2) - mgx_0 + [(kx_0^2)/2]

Thanks for helping.

[Integral]

Consider finding an expression for your velocity in terms of the position. Then compare energy at the equilibrium postion to that at maximum extension.

Also please consider reading through this thread (http://www.physicsforums.com/showthread.php?t=8997) to learn how to present equations in a more readable format.