20th July --- Notes for IB (Lectures)

If you pull on a spring and stretch it, then you do work. That is because you are applying a force over a displacement. Since work is the transfer of energy, we must account for to what the energy was transferred. We say that the energy was transferred into the spring. The work becomes stored energy in the spring. The work becomes potential energy in the spring.
A spring can be stretched or compressed. The same mathematics holds for stretching as for compressing springs. We will be primarily discussing energy as it is stored in a spring when it is stretched here; however, the same physics would work for energy as it is stored in a spring when it is compressed.

The area under this graph of force vs. extension is in Joules, units of energy. 
Do not forget that units of work are units of force times units of displacement, or units of Newtons times units of meters. And units of work are units of the transfer of energy, that is, they are units of energy, or Joules.
So, the area under this graph symbolizes energy. It is the work done to stretch the spring.
Now, work is the transfer of energy. After the spring has been stretched, and work has been done, where has the energy been transferred to? We say that it has become potential energy in the spring. That is the energy has been stored in the spring. Therefore, the amount of energy symbolized by the area under the above graph is the energy that has been stored in the spring. It is the potential energy of the spring.
This area can be calculated. It is shaped like a triangle; so, its area is one half times its height times its base. We have:

Area under graph = (0.5)(F)(x)

This area is the energy stored in the spring. The symbol for the energy stored in the spring could be U_s. The U stands for potential energy and the subscript s stands for spring. So, now we have:

U_s = (0.5)(F)(x)

The spring is a linear spring where the stretching force is directly proportional to the extension, as mentioned above. This, again, can be stated as:

F = kx

Placing this substitution for F in the above formula for U_s we get:

U_s = (0.5)(kx)(x)

Removing the parentheses and noticing that x times x is x², we have:

U_s = 0.5kx²

This last formula reads: The potential energy of a spring, or the energy stored in a spring, equals one half times the spring constant times the square of the extension. This is how to calculate how much energy is stored in a spring.


Source
http://zonalandeducation.com/mstm/physics/mechanics/energy/springPotentialEnergy/springPotentialEnergy.html