Force-extension graphs and deformation

Hooke's law

ΔF = kΔx: up to the limit of proportionality, extension (or compression) is proportional to the force. k is the stiffness in N/m, equal to the gradient of the straight part of a force-extension graph. Hooke's law works for compression too: force-compression graphs have the same shape at first.

Key points on the graph

• Limit of proportionality: beyond this point force is no longer proportional to extension (the line curves).
• Elastic limit: the largest load after which the material still returns to its original length when the load is removed.
• Yield point: the material suddenly extends a lot for little or no increase in force.

Elastic and plastic deformation

• Elastic deformation: the material returns to its original shape and size when the force is removed.
• Plastic deformation: the material is permanently deformed and does not return to its original shape. This happens beyond the elastic limit.

On unloading after plastic deformation, the line comes back down parallel to the loading line but meets the axis at a permanent extension.

Worked example

A spring extends 4.0 cm under a load of 8.0 N. Find its stiffness, and its extension under 5.0 N (within the limit of proportionality).

1. k = ΔF / Δx = 8.0 / 0.040 = 200 N/m
2. Δx = 5.0 / 200 = 0.025 m

Answer: k = 200 N/m; extension = 2.5 cm

Check yourself

1. A force of 6.0 N stretches a spring by 0.030 m. Find its stiffness in N/m.
   Show answer

   200 N/m. k = 6.0 / 0.030 = 200 N/m.

2. A spring of stiffness 50 N/m has a 2.0 N force applied. Find its extension in cm.
   Show answer

   4 cm. Δx = 2.0 / 50 = 0.040 m = 4.0 cm.

3. A force of 30 N compresses a spring by 15 mm. Find its stiffness in N/m.
   Show answer

   2000 N/m. k = 30 / 0.015 = 2000 N/m.

4. What is elastic deformation?
   • Permanent change of shape
   • Deformation after which the material returns to its original shape when the force is removed
   • Deformation beyond the yield point
   • Breaking of the material
   Show answer

   Deformation after which the material returns to its original shape when the force is removed. Elastic means it springs back completely.

5. A wire is loaded beyond its elastic limit and then unloaded. What happens?
   • It returns to its original length
   • It becomes shorter than before
   • It snaps immediately
   • It is permanently extended
   Show answer

   It is permanently extended. Beyond the elastic limit, deformation is plastic and some extension remains.

6. What is the limit of proportionality?
   • The point at which the material breaks
   • The point beyond which force is no longer proportional to extension
   • The maximum force the material can take
   • The point where extension stops
   Show answer

   The point beyond which force is no longer proportional to extension. Up to here the graph is a straight line through the origin.

7. What happens at the yield point?
   • The material snaps
   • The material becomes stiffer
   • Extension becomes proportional to force
   • The material extends a lot with little or no increase in force
   Show answer

   The material extends a lot with little or no increase in force. The material 'gives', stretching considerably for no extra load.

8. What does the gradient of the straight section of a force-extension graph (force on the y-axis) represent?
   • The Young modulus
   • The elastic strain energy
   • The stiffness k
   • The breaking stress
   Show answer

   The stiffness k. Gradient = ΔF / Δx = k.

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Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (XPH11/YPH11), Issue 3
• Pearson qualifications: International Advanced Levels Physics (2018)

Last updated 7 November 2026. Spotted a mistake? Tell us.