If we plot the elastic deformation of a spring we usually get a graph that looks like Graph 1:
NOTE: This graph shows the force applied to a string up to its elastic limit. If we continued to increase the force the string would deform plastically and the line would begin to curve.
From the graph we can see that the extension of the spring is directly proportional to the force applied. This relationship is described by Hooke’s Law:
Hooke’s Law – the extension of a spring (or other elastic object) is directly proportional to the force applied.
This law gives rise to the following equation:
Where F is the force measured in Newtons (N), k is the spring (force) constant measured in Newtons per metre (Nm-1) and x is the extension (or compression) of the spring measured in metres (m).
IMPORTANT: This only applies when a material is undergoing elastic deformation. It does, however, apply to other elastic objects (not just springs) such as a rubber band.
The spring (force) constant is the gradient of the force-extension graph.
The diagram below highlights what is meant by extension:
We need to remember that if we are given the stretched length of a material the original length must be subtracted to find the extension. For example, if the stretched length is 10m and the original length is 5m the extension is 10 – 5 = 5m.
Similarly, if we are given the compressed length of a material we need to subtract this from the original length to calculate the compression. For example if the compressed length is 2m and the original length is 5m the compression is 5 – 2 = 3m.
Work
The area beneath a force-extension graph for the elastic region of a material gives the work required to stretch (or compress) a material to a particular extension (x).
From Graph 1 we can see that this area is equal to the area of a triangle. This is shown clearly in Graph 2:
The work done (W) to reach a particular extension (x) can therefore be calculated using the following equation:
Where F is the force required to get a particular extension, x.
From Hooke’s Law we know that F = kx. Substituting this into the equation gives us the work done in terms of the spring constant:
Elastic Potential Energy
The work done to stretch (or compress) an object becomes stored in the object as elastic potential energy. This potential energy is released when the force being applied is removed. For example, if you compress a spring and then let go it jumps back to its original length by releasing this energy.
The elastic potential energy (E) stored by the material is equal to the work done to stretch (or compress) it:
Worked Examples
Example 1
The graph below shows the extension of a spring undergoing elastic deformation. Calculate the spring constant of the spring.
Example 2
A spring has a spring constant of 10,000Nm-1. Calculate the work done if the spring is compressed by 30cm.
Example 3
The length of a stretched spring is 0.25m. If the force applied to the spring is 500N and the spring constant is 5000Nm-1 calculate the length of the string (L0) when the force is removed. The force deforms the spring elastically.
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