Area under a force-extension graph
- The work done in stretching a material is equal to the force applied multiplied by the extension created
- Therefore, the area under a force-extension graph is equal to the work done to stretch the material
Area under force-extension graphs
Work done is the area under the force - extension graph. This is true for graphs that obey Hooke's law and those which don't.
- For a region where a material demonstrates elastic behaviour and obeys Hooke’s law the work done is the area of the right angled triangle under the graph
- For a region where a material doesn’t obey Hooke’s law, the total area is the sum of the areas of the separate sections under the graph
Worked example
The graph shows the behaviour of a sample of a metal when it is stretched until it starts to undergo plastic deformation. What is the total work done in stretching the sample from zero to an extension of 13.5 mm?
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
Examiner 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.