3.7.4Forces and Motion
Elastic Potential Energy
5 min read 6 parts Lesson 61 of 66 Unit quiz Virtual lab
Section 1 of 6Energy in a Stretched Spring
Stretching a spring needs work, and that work is stored for later release.
Definition
- Elastic potential energy is the energy stored in a deformed object that returns to its shape when the force is removed.
Basic equation
- E = ½Fx, where F is the final force and x is the extension, valid when Hooke’s law applies.
Spring form
- Substituting F = kx gives E = ½kx², which needs only the spring constant and the extension.
Area link
- The energy equals the area under the force-extension graph, which is a triangle for a linear spring.
Squared extension
- Doubling the extension of a spring stores four times as much energy.
Section 2 of 6Deriving the Formula
The factor of one half comes from the force rising gradually rather than being constant.
1
The force on a spring grows steadily from zero to its final value F as it is stretched.
2
The average force over the extension x is therefore ½F for a linear spring.
3
Work done equals average force multiplied by distance moved, so W = ½Fx.
4
This work is stored as elastic potential energy, so E = ½Fx.
5
Using F = kx for the final force gives E = ½kx², the form most often tested.
Section 3 of 6Energy Transfers
Energy Transfers
Catapult
- A released spring converts stored energy to kinetic energy, so ½kx² = ½mv² if no energy is lost.
Bungee cord
- At the lowest point the jumper is momentarily at rest,
- so the gravitational potential energy lost has all become stored energy in the cord.
Non-linear materials
- For rubber or yielding metals the stored energy is found from the area under the actual curve.
Losses
- Some of the work done on a plastic or hysteretic material becomes thermal energy and is not recoverable.
Section 4 of 6Practical Considerations
Measuring k
- Plot energy against extension squared, and the gradient of the straight line equals one half of k.
Compression
- A compressed spring stores energy in the same way, using the compression distance for x.
Series and parallel
- Combinations of springs change the overall spring constant, and so the energy stored for a given force.
Section 5 of 6Worked Example
Question
- A spring of constant 200 N m⁻¹ is stretched by 0.060 m, then used to launch a 0.050 kg trolley along a frictionless track.
- Find the stored energy and the trolley’s speed.
- E = ½kx² = 0.5 × 200 × 0.060² = 0.36 J.
- All the stored energy becomes kinetic energy: 0.36 = ½ × 0.050 × v².
- v² = 0.72 / 0.050 = 14.4, so v = 3.8 m s⁻¹.
Answer
- Stored energy = 0.36 J;
- speed = 3.8 m s⁻¹.
Section 6 of 6Exam Checklist & Tips
3.7.4 – Key Points for the Exam
- Elastic potential energy Energy stored in an object that is deformed elastically.
- Spring constant The force per unit extension for a spring, in N m⁻¹.
- Extension The change in length from the natural length.
- Elastic deformation A change in shape that disappears when the force is removed.
Common Mistakes
- Using E = Fx instead of ½Fx, which doubles the answer.
- Squaring k or the product kx rather than squaring only the extension x.
- Forgetting to convert extension to metres before using ½kx².
- Applying the formula to a material beyond its limit of proportionality, where the graph is no longer a straight line.