Hooke's law and force-extension graphsEdexcel International A Level Physics: Revision notes
Section 1
Hooke's law and stiffness
Hooke's law states that the extension of a spring (or wire) is directly proportional to the force applied, provided the limit of proportionality is not exceeded. As an equation:
F = kx
where F is the force in N, x is the extension (or compression) in m and k is the stiffness of the object in N m⁻¹. A large k means a stiff object: a large force is needed for a small extension.
The law applies to a compression as well as to an extension: pushing a spring so that it is compressed by x also needs a force F = kx.
Worked example: a spring has k = 40 N m⁻¹. A force of 6.0 N gives x = F/k = 6.0 / 40 = 0.15 m.
Stiffness k belongs to the whole object, so a longer or thicker spring has a different k. It is not a property of the material alone.
Section 2
Force-extension and force-compression graphs
Plot force on the vertical axis and extension on the horizontal axis. For a spring obeying Hooke's law the graph is a straight line through the origin, and the gradient equals the stiffness k.
A force-compression graph is drawn in the same way, with the compressive force plotted against the compression. For many springs the compression graph is a continuation of the extension graph through the origin, with the same gradient.
If the axes are swapped (extension against force) the gradient is 1/k, so always check which quantity is on which axis.
Section 3
Limit of proportionality and elastic limit
As the force is increased the graph eventually stops being a straight line.
- The limit of proportionality is the point beyond which force is no longer proportional to extension, so Hooke's law is no longer obeyed.
- The elastic limit is the point beyond which the material no longer returns to its original length when the force is removed.
The elastic limit can lie at or slightly beyond the limit of proportionality. Between them the material is still elastic but the extension is not proportional to the force.
Section 4
Elastic and plastic deformation, yield point
Elastic deformation: the material returns to its original length when the force is removed. The loading and unloading lines on the graph coincide.
Plastic deformation: beyond the elastic limit the material is permanently stretched. If the force is removed from the plastic region, the unloading line is parallel to the initial straight line and meets the extension axis at the permanent extension.
The yield point is where a ductile material, such as a metal, begins to stretch a large amount for little or no increase in force; the graph flattens. Beyond it the material deforms plastically.
Name the point on the graph, then say what it means. For example: beyond the elastic limit, a permanent extension remains.
Must Know
- F = kx, with k the stiffness in N m⁻¹
- Gradient of a force-extension graph = k
- Limit of proportionality: Hooke's law stops
- Elastic limit: permanent deformation begins
- Yield point: large extension for little extra force
- Unloading from the plastic region is parallel to the loading line
That's the notes covered.
Carry on to the next subtopic.
Exam questions on Hooke's law and force-extension graphs
- A student investigates a helical steel spring hung from a clamp. Masses are added one at a time to the lower end and the extension of the spring is measured after each addition. The spring obeys Hooke's law for all loads up to 8.0 N, at which load its extension is 0.20 m.The student adds further masses so that the force is 12 N. She finds that the extension is greater than the value predicted using the stiffness found from the first results. Explain this observation.2 marks
- Spring X has a stiffness of 60 N m⁻¹ and spring Y has a stiffness of 20 N m⁻¹. Both springs obey Hooke's law for all the forces used in a test.A force of 3.0 N is applied to spring Y. Calculate the force that must be applied to spring X to produce the same extension as in spring Y.2 marks
- A technician loads a metal wire, increasing the force steadily from zero. Up to 24 N the extension is directly proportional to the force, and at 24 N the extension is 1.2 mm. Between 24 N and 30 N the extension increases by larger amounts for each newton added, but the wire still returns to its original length when unloaded. Beyond 30 N the wire no longer returns to its original length when unloaded.Describe how the behaviour of the wire changes as the force is increased from zero to 30 N and beyond, using the terms limit of proportionality, elastic limit and plastic deformation.3 marks
Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).