Changes in energy: The amount of elastic potential energy stored in a stretched spring can be calculated using the equation:

The idea

When a spring is stretched, work is done against the spring's stiffness and energy is transferred to the elastic potential store of the spring. The amount stored depends on how stiff the spring is and how far it has been stretched from its natural length. The equation that links these quantities is Ee = 0.5 × k × e², where Ee is elastic potential energy in joules (J), k is the spring constant in newtons per metre (N/m) and e is the extension in metres (m). Extension means the increase in length beyond the unstretched length, not the total length.

For example: A spring with k = 200 N/m stretched by 0.10 m stores 0.5 × 200 × (0.10)² = 1.0 J. The equation applies provided the spring has not been stretched beyond its limit of proportionality, so the force–extension graph remains a straight line through the origin.

What earns the marks

• State the equation as Ee = 0.5 × k × e² and identify each symbol with its unit: Ee in joules (J), k in newtons per metre (N/m), e in metres (m).
• Explain that extension e is the change in length from the spring's natural length, so e = stretched length − original length, not the total stretched length.
• Substitute values correctly, including squaring the extension before multiplying by 0.5 and by the spring constant.
• Recognise that the equation applies only while the spring obeys Hooke's law, that is up to the limit of proportionality.
• Interpret the factor 0.5 as arising because the force increases from zero to k × e as the spring stretches, so the average force is half the final force.

Common mistakes and how to put them right

• Using the total stretched length instead of the extension.

  Put it right: Subtract the original length first so e is the increase in length.

• Forgetting to square the extension or squaring the whole expression.

  Put it right: Square only e, then multiply by 0.5 and k.

• Mixing units, such as using centimetres for extension or grams for mass.

  Put it right: Convert all lengths to metres and use k in N/m before calculating.

Exam tips

• Write the equation, then substitute values with units before evaluating, so method marks are visible even if the final arithmetic slips.
• Check whether the question gives total length or extension; if it gives total length, calculate the extension first and show that step.
• Give the final answer in joules with an appropriate number of significant figures, and include the unit.