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Elastic Potential Energy
Consider a spring being extended by a steadily increasing force. When the force has a magnitude F, the extension is x, as in the diagram below.  
   
     
A graph of force against extension is found to be a straight line passing through the origin, as long as the elastic limit of the spring has not been exceeded.   
Therefore we can write  
F α x  
   
This means that, during the process of extending the spring, the magnitude of the average force exerted was F/2.   
   
   
   
The slope of the graph is k, the elastic constant of the spring.   
Therefore we can write   
   
NB
In some situations, for example in the section on simple harmonic motion, you might see 
 
   
This is perfectly reasonable if we consider -F to represent the force with which the spring reacts when it is extended by some external agent.   
   
The work done stretching the spring depends on the average force exerted on it   
   
   
and, putting these two equations together gives   
   
The potential energy stored in a body, by virtue of its position or its state (in this case, its state of having been stretched) is simply equal to the work done putting it in that state or position.  
Therefore, we can write    
   
Notice, from the equation w = Fx, that the work done stretching the spring is represented by the area under the graph of F against x.    
   
This can be useful to know in cases where the graph is not a straight line  
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