Force-extension graphs: reading them, finding k and the energy stored

A force-extension graph shows how much a spring or other elastic object stretches as the force on it increases. For a spring obeying Hooke's law, force and extension rise in proportion, so the line starts at the origin and stays straight, and its gradient (force ÷ extension) is the spring constant k. Past the limit of proportionality the line curves, and force is no longer proportional to extension.

Hooke's law

• F is the force in newtons (N).
• k is the spring constant in newtons per metre (N/m). A stiffer spring has a bigger k.
• e is the extension in metres (m): the new length minus the original length.

AQA's GCSE Physics specification (4.5.3) says extension is directly proportional to the force applied, provided the limit of proportionality is not exceeded, and that you must recall and apply F = ke. It works for compression too, with e as the compression.

Reading a force-extension graph

Here force is on the vertical axis and extension (in metres) along the horizontal axis, for a spring with k = 25 N/m:

• The straight section through the origin shows a linear relationship: double the force, double the extension.
• The limit of proportionality is where the line starts to curve.
• The curved section is non-linear: force and extension are no longer proportional.

Some exam graphs put extension on the vertical axis instead. Then the gradient is 1/k, so read the axes before using a gradient.

Elastic and inelastic deformation

An object is elastically deformed if it returns to its original shape when the forces are removed, and inelastically deformed if it doesn't. AQA also points out that more than one force is needed to stretch, bend or compress a stationary object: a spring hanging from a clamp is pulled down by the load and up by the clamp.

Finding the spring constant from the graph

On the straight section, k is the gradient when force is on the vertical axis:

Then you can find other values: a 3.0 N force gives an extension of e = F ÷ k = 3.0 ÷ 25 = 0.12 m (12 cm), as long as you stay below the limit of proportionality.

Energy stored in a stretched spring

A force that stretches a spring does work, and that energy is stored as elastic potential energy. Up to the limit of proportionality:

AQA gives this equation on the Physics equation sheet. For the spring above, stretched by 0.20 m: Eₑ = ½ × 25 × 0.20² = 0.50 J. The same value is the area under the straight part of the graph: ½ × 0.20 m × 5.0 N = 0.50 J.

The force and extension practical

AQA's Required practical activity 6 investigates the relationship between force and extension for a spring.

1. Hang the spring from a clamp stand with a ruler beside it, and record the unstretched length.
2. Add masses one at a time. Each 100 g adds about 0.98 N of weight (W = mg, with g = 9.8 N/kg).
3. Record the new length each time and calculate extension = new length − original length.
4. Plot force against extension and draw a line of best fit. The gradient of the straight part is k.

Read the ruler at eye level, and wear eye protection in case the spring snaps. Make sure the masses can't fall on your feet.

Common mistakes

• Using length instead of extension. Extension is the change in length.
• Forgetting to convert cm to m. k is in N/m, so use metres.
• Using F = ke beyond the limit of proportionality. It only works on the straight part.
• Taking the gradient the wrong way round. If extension is on the y-axis, the gradient is 1/k.

Practise with a short timed paper

Forces and elasticity are in GCSE Physics. The daily GCSE Physics Higher paper is a short timed paper of 8 questions, with a new one every day. For the equation on its own, see our spring constant equation guide.

Read next

Physics · MechanicsSpring constant equation: k = F/e and Hooke's lawThe spring constant equation k = F/e from Hooke's law, units, finding k from a force-extension graph, the limit of proportionality, examples and practice.