Energy stored in a spring

Work is done when a spring is extended or compressed. Elastic potential energy is stored in the spring. Provided inelastic deformation has not happened, the work done is equal to the elastic potential energy stored.

The elastic potential energy stored can be calculated using the equation:

elastic potential energy = 0.5 × spring constant × (extension)2

E_e = \frac{1}{2}~k~e^2

This is when:

  • elastic potential energy (Ee) is measured in joules (J)
  • spring constant (k) is measured in newtons per metre (N/m)
  • extension (e), referring to the increase in length, is measured in metres (m)
curriculum-key-fact
This equation also works for the reduction in length when a spring is compressed.

Example

A spring has a spring constant, (k), of 3 N/m. It is stretched until it is extended by 50 cm. Calculate the elastic potential energy stored by the spring, assuming it is not stretched beyond the limit of proportionality.

First convert centimetres to metres:

50 cm = 50 ÷ 100 = 0.5 m

Then calculate using the values in the question:

E_e = \frac{1}{2}~k~e^2

E_e = \frac{1}{2} \times 3 \times 0.5^2

E_e = 1.5 \times 0.25

E_e = 0.375~J

Question

A spring is compressed by 0.15 m. It has a spring constant of 80 N/m. Calculate the elastic potential energy stored by the spring.

E_e = \frac{1}{2}~k~e^2

E_e = \frac{1}{2} \times 80 \times 0.15^2

E_e = 40 \times 0.0225

E_e = 0.90~J