Strain Energy
When a piece of wire is stretched, work is done on the wire. The wire gains potential energy. When the wire is released, the wire springs back, converting the potential energy to kinetic energy. This potential energy is due to the strain, and is called elastic potential energy.
The force and extension obey Hooke's law, provided the extension is small. So a graph of force against extension is a straight line. The work done on the wire is equal to the area under this graph.
To see that this is true, consider a very small increase, δe, in extension. Within this small change, the force F does not increase much. Taking F to be approximately constant, the work done is Fδe. On the F-e graph, this is the area of a rectangle with base δe, and height F. It stands on the e axis, just touch the straight line graph at its top.
We can draw a rectangle like this for each increment in extension. Over the full range of the extension, the rectangles cover the whole area under the graph.
Therefore, area under the force-extension graph is equal to the elastic potential energy. Since the area is a triangle, the potential energy is
Another common way to write this is to denote the extension by x, and the spring constant by k. The Hooke's law is written as
and the potential energy is
Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 2 October 2010.