What is the maximum height a weighted spring can reach?

[arjwilliams]

Homework Statement

A spring is fixed at one end to the ceiling, with an ball of mass, m, and radius, r, hanging vertically at the other end of the spring. The system is in equilibrium. The spring has constant, k, and natural length, l₀.

I then pull the ball down to an extension, x₁, from the equilibrium position, x₀. How far will I have to pull the ball down for the ball to reach a distance, d, from the ceiling, ignoring dampening forces?

g = acceleration by gravity
x = extension at any time

Homework Equations

x0 = -mg/k
F = k(x-x₀) - mg
K.E = 0 at (x₀ + x₁) and d
both P.E (G.P.E and E.P.E) are maximum at these points.
K.E = 0.5*m*v^2 (where v is velocity of ball)
G.P.E = m*g*(x₀ + x)
E.P.E = 0.5*k*x^2
any others

The Attempt at a Solution

No idea where to start.

 

[Beaker87]

Try to set the potential energy at the bottom (where the ball is released) to the potential energy at the very top.