Area under a Force-Extension Graph
- The work done in stretching a material is equal to the force multiplied by the distance moved
- Therefore, the area under a force-extension graph is equal to the work done to stretch the material
- The work done is also equal to the elastic potential energy stored in the material
Area under Force-Extension Graphs
Work done is the area under the force - extension graph. This is true for graphs obeying Hooke's law and those which don't. ½Fx can only be used to calculate area under a straight line graph passing through the origin.
- This is true for whether the material obeys Hooke’s law or not
- For the region where the material obeys Hooke’s law, the work done is the area of a right angled triangle under the graph
- For the region where the material doesn’t obey Hooke’s law, the area is the full region under the graph. To calculate this area, split the graph into separate segments and add up the individual areas of each
Loading and unloading
- The force-extension curve for stretching and contraction of a material that has exceeded its elastic limit, but is not plastically deformed is shown below
Loading and Unloading Graph
The unloading curve is beneath the loading curve. This is because it requires more energy to stretch the object than is released during loading, so the area under the contraction curve is lesser.
- The curve for contraction is always below the curve for stretching
- The area X represents the net work done or the thermal energy dissipated in the material
- The area X + Y is the minimum energy required to stretch the material to extension e
Worked example
The graph shows the behaviour of a sample of a metal when it is stretched until it starts to undergo plastic deformation.c What is the total work done in stretching the sample from zero to 13.5 mm extension?
Simplify the calculation by treating the curve XY as a straight line.
Answer:
Step 1: Recall how to determine work done from the graph:
- Work done is the area underneath the force-extension graph
Step 2: Calculate the area under the graph up to point X:
- To point X, the area under the graph, AX , is a triangle
- Calculate AX , remembering to convert length to metres
Step 3: Calculate the area between X and Y:
- Assuming the line XY is a straight line, the area under this region of the graph forms a trapezium
- Recall the equation for a trapezium of width h and side lengths a and b
-
- Here, h is the change in extension from X to Y, 2.5 mm
- a is the load at point X and b is the load at point Y
Step 4: Calculate total area:
- The total area, the total work done, is just the sum of these two areas
- The answer is given to 3 significant figures, as the data has been given to this number of significant figures
Exam Tip
Make sure to be familiar with the formula for the area of common 2D shapes such as a right angled triangle, trapezium, square and rectangles. If you do forget the equation for a trapezium's area, however, just split the shape up into rectangles and triangles.