A force-extension graph shows whether a spring obeys Hooke’s law. The straight part has a gradient equal to the spring constant, and the area under it is the stored energy.
This lesson is part of SPM Physics force and motion II. It applies the ideas from using Hooke’s law and elastic energy.
How do I read a dataset?
Here is an original set of readings for a spring.
| Force F (N) | 0 | 2.0 | 4.0 | 6.0 | 8.0 | 10.0 | 12.0 |
|---|---|---|---|---|---|---|---|
| Extension x (cm) | 0 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | 7.0 |
From 0 to 10 N, each 2.0 N adds exactly 1.0 cm. The extension is proportional to the force, so Hooke’s law applies. From 10 N to 12 N, an extra 2.0 N gives 2.0 cm, which is twice the expected 1.0 cm.
The limit of proportionality lies at about 10 N. Beyond that point, the graph bends.
Worked example: k and the stored energy
Find the spring constant from the straight part. Use the point (10.0 N, 5.0 cm), with x = 0.050 m.
Gradient = F ÷ x = 10.0 ÷ 0.050 = 200 N m⁻¹. The same value comes from any point on the straight part, for example 4.0 ÷ 0.020 = 200 N m⁻¹.
Energy stored at 8.0 N: x = 4.0 cm = 0.040 m. The area under the line up to this point is ½ × 8.0 × 0.040 = 0.16 J.
Why check the axes first?
Suppose the same data is plotted with extension on the vertical axis and force on the horizontal axis. The gradient of that graph is x ÷ F = 0.050 ÷ 10.0 = 0.0050 m N⁻¹.
That is not the spring constant. It is the reciprocal, 1 ÷ k. Taking the gradient of the wrong graph without checking the axes gives 0.0050 instead of 200 N m⁻¹, so always read the labels before you calculate.
The mistake to avoid
The common mistake is to read the spring constant from the whole curve, including the bent part. Compare the two.
| Method | Value of k |
|---|---|
| Point on the straight part (10 N, 5.0 cm) | 200 N m⁻¹ |
| Point on the bent part (12 N, 7.0 cm) | 171 N m⁻¹ |
The second value is lower because the spring has passed the limit of proportionality. Use only the straight part, and say so. The graph evidence comparison lab lets you compare how graphs respond to changes in the data.
Check yourself
Another spring gives the readings 0 N, 3.0 N, 6.0 N, 9.0 N at extensions of 0, 2.0 cm, 4.0 cm, 6.0 cm. Find k and the energy stored at 9.0 N.
Answer
k = 3.0 ÷ 0.020 = 150 N m⁻¹. The readings are proportional, so the graph is a straight line through the origin.
Energy at 9.0 N, x = 0.060 m: ½ × 9.0 × 0.060 = 0.27 J.
What to study next
Test the skill in the force and motion II practice set. If graph-reading is still uneven, revisit interpreting displacement, velocity and acceleration for the same gradient and area ideas.
Record axis errors with the mistake log and paper-error review tool. If you want a teacher to go through your graphs with you, see online one-to-one Physics tuition.