3.7.3Forces and Motion
Force-Extension Graphs
5 min read 6 parts Lesson 60 of 66 Unit quiz Virtual lab
Section 1 of 6Anatomy of the Graph
A force-extension graph describes a particular sample, such as one spring or one wire of a fixed size.
Axes
- Force is normally plotted on the vertical axis and extension on the horizontal axis, but always check the labels.
Linear section
- A straight line through the origin shows Hooke’s law and the gradient equals the spring constant k.
Limit of proportionality
- The point where the line starts to curve marks the end of the proportional region.
Elastic limit
- The point beyond which a permanent extension remains after the force is removed;
- it can lie slightly beyond the proportional limit.
Area
- The area under the graph equals the work done in stretching, which is stored as elastic potential energy if the stretch is elastic.
Section 2 of 6Reading Values
Quantitative questions use the same short sequence of steps.
1
Identify the straight section starting at the origin and choose two points on it that are far apart.
2
Find the gradient by dividing the change in force by the change in extension, remembering to convert to N and m.
3
Use the gradient as k to predict forces and extensions within the linear region.
4
To find the work done up to the end of the linear region, calculate the area of the triangle ½Fx.
5
For a curved region, estimate the area by counting squares or splitting the shape into trapeziums.
Section 3 of 6Different Materials
Different Materials
Spring
- A spring gives a linear graph for small loads, and unloading retraces the loading line if it stays elastic.
Metal wire
- A wire is linear first and then curves as it yields, with a large permanent extension if loaded beyond the elastic limit.
Rubber band
- Rubber gives a curved graph, and the unloading curve lies below the loading curve, forming a hysteresis loop.
Hysteresis loop
- The area enclosed by the loop equals the energy dissipated as thermal energy in the rubber per cycle of loading.
| Material | Loading and unloading lines |
|---|---|
| Spring within limits | Same straight line |
| Metal beyond elastic limit | Unloading line parallel to the original line but shifted |
| Rubber | Different curves, forming a loop |
Section 4 of 6Springs Under Compression
Mirror graph
- A spring compressed by forces follows a similar line in the opposite direction, with negative force and extension.
Same constant
- Within the elastic range the spring constant is the same for stretching and compressing.
Coil binding
- When the coils touch, the spring becomes much stiffer and the graph rises steeply.
Section 5 of 6Worked Example
Question
- A spring obeys Hooke’s law up to a force of 8.0 N and an extension of 0.050 m.
- Find the spring constant and the work done in stretching it to this point.
- k = gradient = 8.0 / 0.050 = 160 N m⁻¹.
- Work done = area under the graph = ½ × 8.0 × 0.050.
- Work done = 0.20 J.
Answer k = 160 N m⁻¹; work done = 0.20 J.
Section 6 of 6Exam Checklist & Tips
3.7.3 – Key Points for the Exam
- Gradient Change in force divided by change in extension, equal to k in the linear region.
- Elastic limit The greatest force from which a material returns fully to its original length.
- Plastic deformation Permanent change in shape after the load is removed.
- Hysteresis The difference between loading and unloading curves for materials such as rubber.
Common Mistakes
- Calculating the gradient from raw readings in cm or g without converting to m and N.
- Using the whole area under the graph for ½Fx when the graph is curved.
- Confusing the limit of proportionality with the elastic limit; they are close but not identical.
- Saying that the area of a hysteresis loop is stored energy; it is energy dissipated as thermal energy.