Elastic Potential Energy

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 under graphs, downloadable AS & A Level Physics revision notes

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 WE - Work done area under graph question image, downloadable AS & A Level Physics revision notes 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, A_X , is a triangle

$A_{X} = \frac{1}{2} \times base \times height$

Calculate A_X, remembering to convert length to metres

$A_{X} = \frac{1}{2} \times (11.0 \times 10^{- 3}) \times 450 = 2.475 J$

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

$A_{XY} = (\frac{a + b}{2}) h$

- 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

$A_{XY} = (\frac{450 + 600}{2}) \times (2.5 \times 10^{- 3}) = 1.313 J$

Step 4: Calculate total area:

The total area, the total work done, is just the sum of these two areas

$work done = 2.475 + 1.313 = 3.79 J$

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.

Elastic potential energy is defined as the energy stored within a material (e.g. in a spring) when it is stretched or compressed
It can be found from the area under the force-extension graph for a material deformed within its limit of proportionality

Worked example

A spring is extended with varying forces; the graph below shows the results. WE - EPE area under graph question image, downloadable AS & A Level Physics revision notes What is the energy stored in the spring when the extension is 40 mm?

Answer:

Step 1: Recall how to determine energy sto

Energy stored in the spring is equal to area under the graph, A
This is a triangle, so can be calculated using

$A = \frac{1}{2} \times base \times height$

Step 2: Find the area under the graph

At 40 mm, the load is approximately 8.3 N
Converting length into metres gives an area of

$A = \frac{1}{2} \times (40 \times 10^{- 3}) \times 8.3 = 0.166 J$

This is equal to the energy stored in the spring, which should be given to 2 significant figures

$energy stored = 0.17 J$

Exam Tip

In your exam, a range of values will be allowed for the load at 40 mm. Any value from 8.1 N to 8.5 N will gain a mark. If you are struggling, draw horizontal lines on the graph to show the positions of 9.0 N and 8.5 N.

Calculating Elastic Potential Energy

A material within it’s limit of proportionality obeys Hooke’s law. Therefore, for a material obeying Hooke’s Law, elastic potential energy can be calculated using the following:
- Energy stored in a spring, i.e. elastic potential energy, is the area under a graph
- For a material obeying Hooke's law, this area is:

$EPE = \frac{1}{2} F x$

- Recall the equation for Hooke's law:

$F = k x$

- Substituting this into the equation for energy gives:

$EPE = \frac{1}{2} k x^{2}$

Where k is the spring constant (N m^-1) and x is the extension (m)

Exam Tip

The formula for EPE = ½ kx² is only the area under the force-extension graph when it is a straight line i.e. when the material obeys Hooke’s law and is within its elastic limit.

Elastic Potential Energy (CIE A Level Physics)

Revision Note

Area under a Force-Extension Graph

Area under Force-Extension Graphs

Loading and unloading

Loading and Unloading Graph

Worked example

Exam Tip