Stretching transfers energy to a spring
Definition
Elastic potential energy
Energy stored when an elastic object is stretched or compressed.
- Work done by a stretching force transfers energy to the spring's elastic potential energy store.
- For a linear spring stretched from zero extension, E=12kx2E=\dfrac{1}{2}kx^2E=21kx2.
- Here, EEE is the energy transferred in joules, kkk is the spring constant in newtons per metre, and xxx is the extension in metres.
- The equation applies only while the force-extension relationship is linear.
Energy rises with extension squared
- Because E∝x2E\propto x^2E∝x2 for constant kkk, doubling the extension makes the stored energy four times larger.
- The factor 12\dfrac{1}{2}21 appears because the force increases uniformly from zero to FFF as the spring is stretched.
- The energy transferred also equals the area beneath a force-extension graph; for a straight line this area is the triangle 12Fx\dfrac{1}{2}Fx21Fx.
Example
Calculating stored energy
- A spring with k=240 N m−1k=240\,\text{N m}^{-1}k=240N m−1 is extended by 5.0 cm=0.050 m5.0\,\text{cm}=0.050\,\text{m}5.0cm=0.050m.
- E=12×240×(0.050)2=0.30 JE=\dfrac{1}{2}\times240\times(0.050)^2=0.30\,\text{J}E=21×240×(0.050)2=0.30J.
Rearrangement links energy to stiffness
- To find extension, use x=2Ekx=\sqrt{\dfrac{2E}{k}}x=k2E.
- To find spring constant, use k=2Ex2k=\dfrac{2E}{x^2}k=x22E.
- Square the extension in metres, not the centimetre value.
Common Mistake
Do not use E=12kxE=\dfrac{1}{2}kxE=21kx; the extension is squared.
Self review
- Which energy store increases when a spring is stretched?
- What equation gives the energy transferred to a linear spring?
- Why does doubling extension quadruple the energy?
- What does the area under a force-extension graph represent?
