An object of mass m is hung on the end of a spring of relaxed length l and spring constant k. The object is pulled down until the stretch of the spring is s_{o}. What is the length of the spring when the object next comes to rest? (i.e. how high does the object rise before it comes to rest? What is the maximum speed that the object reaches, and where does it occur?
Diagrams.
Initial state ("before") Object at max height Object at max speed
Mark the reference level you choose for gravitational PE clearly on the diagrams.
My reference level is the horizontal line drawn across all three
diagrams. It is at the end of the relaxed spring.
(Other choices are possible, of course.)
Setup of solution
The objects in the system are:
spring, earth, mass on spring
The forms of mechanical energy that this system possesses are :
kinetic, elastic potential and gravitational potential energy.
The kinetic energy is greatest when the total potential energy is least.
Fill in the first two columns of the table below. If you don't know the value of an entry in the top 3 rows, give it a name (algebraic symbol).
Quantity  Before(initial state)  Object at max height  Object at max speed 
stretch of spring 



height of object above reference level 



speed of object 



___________________  _____________________  _________________  _________________ 
Spring PE 



Gravitational PE 



Kinetic energy 



_____________________  _________________  _________________  
Total mechanical energy 


= mv^{2}/2  (mg)^{2}/2k 
Solution to first problem.
Set the total energy in the before state equal to the total energy in the "max height" state. You will get one equation so you can solve for only one unknown. If you have more than one unknown in your equation, go back and see if you can fix something in your table. Your answer should contain the given quantities:
m, k , g, l , s_{o}
ks_{0}^{2}/2  mgs_{0} = ks_{1}^{2}/2
+ mgs_{1}
k(s_{0}^{2 }^{ }s_{1}^{2})/2
= mg(s_{1 }+ s_{0})
Solution to second problem
The second part of the problem is harder. We need to fmd where the PE is least. So first write an expression for the PE in terms of the stretch of the spring s. Note this is not s_{o} or s_{f }but just some arbitrary, so far unknown, value of s. Then find the derivative dU/ds and set it equal to zero to find the value of s that gives the minimum PE. Check that it really is a minimum (not a maximum, for example). Look at the answer does it make sense? Once you have the answer, go back to your table, and fill in the final column. Set the totals in the bottom row equal, and solve for v_{max}.
The potential energy function is: