Elastic potential energy

The formula

Elastic potential energy (EPE) = λx²/(2l), where x is the extension or compression. The formula is not in the booklet, so learn it.

Where it comes from

The tension at extension s is λs/l, which changes as the string stretches, so the work done is an integral: ∫ from 0 to x of (λs/l) ds = λx²/(2l). It is the area under the tension-extension graph, a triangle of base x and height λx/l.

Stretching further

The work done in stretching from extension x_1 to x_2 is λx_2²/(2l) - λx_1²/(2l). It is not λ(x_2 - x_1)²/(2l), which is a very common error.

Worked example

An elastic string has modulus 40 N and natural length 0.5 m. Find the work done in stretching it from an extension of 0.1 m to an extension of 0.3 m.

1. EPE at 0.3 m: 40 × 0.09/1 = 3.6 J.
2. EPE at 0.1 m: 40 × 0.01/1 = 0.4 J.
3. Work done = 3.6 - 0.4.

Answer: 3.2 J

Check yourself

1. An elastic string of natural length 1.25 m and modulus 25 N is stretched by 0.5 m. Find the elastic potential energy stored.
   Show answer

   2.5 J. EPE = 25 × 0.25/2.5 = 2.5 J.

2. An elastic string has natural length 0.6 m and modulus 30 N. Find the work done in stretching it from an extension of 0.2 m to 0.4 m.
   Show answer

   3 J. 30(0.16 - 0.04)/1.2 = 3 J.

3. A spring of natural length 0.4 m and modulus 80 N is compressed to a length of 0.3 m. Find the energy stored.
   Show answer

   1 J. x = 0.1, EPE = 80 × 0.01/0.8 = 1 J.

4. When an elastic string of natural length 0.8 m is stretched by 0.2 m it stores 1.5 J. Find its modulus.
   Show answer

   60 N. 1.5 = λ(0.04)/1.6, so λ = 60 N.

5. An elastic string of natural length 2 m and modulus 48 N is stretched to a length of 2.5 m. Find the energy stored.
   Show answer

   3 J. x = 0.5, EPE = 48 × 0.25/4 = 3 J.

6. Which expression gives the work done in stretching a string from extension a to extension b?
   • λ(b² - a²)/(2l)
   • λ(b - a)²/(2l)
   • λ(b - a)/l
   • λb²/(2l)
   Show answer

   λ(b² - a²)/(2l). Work done = final EPE - initial EPE.

7. On a graph of tension against extension for an elastic string, what does the area under the line represent?
   • The modulus of elasticity
   • The natural length
   • The work done in stretching the string
   • The tension at that extension
   Show answer

   The work done in stretching the string. Work = ∫T dx, the area under the T-x graph.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit M3
• Pearson Edexcel IAL Mechanics M3 (WME03) question paper, June 2025: front-cover instruction on g and accuracy

Last updated 8 November 2026. Spotted a mistake? Tell us.