Elastic strain energyEdexcel International A Level Physics: Revision notes
Section 1
Energy stored in a stretched material
When a material is deformed elastically, work is done by the force and is stored as elastic strain energy (E_el). It is released when the force is removed.
The work done is force × distance moved. Because the force rises from zero to F as the extension grows to x, the average force is F/2 (when force is proportional to extension), so:
E_el = ½Fx
With Hooke's law, F = kx, this is also E_el = ½kx².
Worked example: k = 250 N m⁻¹, x = 0.080 m. F = 20 N, so E_el = ½ × 20 × 0.080 = 0.80 J.
Do not forget the ½. Fx alone would be the work done by a constant force. Also, the energy for a further extension is the difference of two energies, not ½kx² of the extra extension.
Section 2
Area under a force-extension graph
On a graph of force against extension, the area under the graph equals the elastic strain energy (work done).
- For a linear graph the area is a triangle: ½ × base × height = ½Fx.
- The energy change between two extensions is the area between those two extensions on the graph, so the extra energy is E(x₂) − E(x₁).
This works for any shape of graph, as long as the material is deformed elastically.
Section 3
Non-linear graphs: estimating the area
If force is not proportional to extension, the graph is curved and ½Fx cannot be used. Estimate the area instead by:
- counting squares (and deciding the value of one square in joules), or
- dividing the area into strips or trapeziums and adding them.
Worked example: forces 0, 3.0 and 5.0 N at extensions 0, 0.050 and 0.100 m. Area = ½(0 + 3.0)(0.050) + ½(3.0 + 5.0)(0.050) = 0.075 + 0.20 = 0.28 J.
State that the result is an estimate; more, narrower strips give a better estimate.
Section 4
Using strain energy in problems
Elastic strain energy is often transferred to or from other stores. Use conservation of energy:
- Spring launcher: ½kx² = ½mv² (if no energy is lost)
- Bungee: gravitational potential energy lost = elastic strain energy gained (jumper momentarily at rest)
If the speed measured is lower than predicted, the difference is energy transferred as thermal energy (friction, internal energy of the spring) to the surroundings.
Efficiency = useful energy output / energy input.
Check that the extension stays within the limit of proportionality before using ½kx².
Must Know
- E_el = ½Fx = ½kx² (linear graph only)
- Energy = area under a force-extension graph
- Non-linear graph: estimate the area (squares or trapeziums)
- Energy change = difference between the two areas
- Energy is stored elastically and recovered on unloading
That's the notes covered.
Carry on to the next subtopic.
Exam questions on Elastic strain energy
- A spring of stiffness 250 N m⁻¹ is stretched by 0.080 m from its natural length. The spring obeys Hooke's law throughout.The spring is now stretched by a further 0.040 m. Calculate the additional elastic strain energy stored.2 marks
- A student stretches a strip of polymer, keeping it within its elastic limit. She records the force at each extension: 0 N at 0 m, 3.0 N at 0.050 m, 5.0 N at 0.100 m, 6.0 N at 0.150 m and 6.5 N at 0.200 m. The force is not directly proportional to the extension.Estimate the elastic strain energy stored in the strip when its extension is 0.100 m.2 marks
- A toy launcher contains a spring of stiffness 600 N m⁻¹. The spring is compressed by 0.050 m and then released, firing a ball of mass 0.025 kg along a horizontal barrel. The spring obeys Hooke's law throughout.Calculate the elastic strain energy stored in the compressed spring and hence the maximum speed of the ball, assuming all of the energy is transferred to the ball as kinetic energy.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).