Calculate work done in stretching (or compressing) a spring (up to the…

The idea

When a force stretches or compresses a spring, energy is transferred to the elastic potential energy store of the spring. The work done by the force equals the energy transferred, provided the spring has not been stretched beyond its limit of proportionality. For a spring obeying Hooke's law, force is proportional to extension, so the force rises steadily from zero to F. The average force is F ÷ 2, and work done = average force × extension, giving W = 0.5 × k × e².

For example: A spring with spring constant 200 N/m stretched by 0.10 m stores 0.5 × 200 × 0.10² = 1.0 J. Always convert centimetres to metres before squaring, and check that the extension used is within the straight-line region of the force–extension graph.

What earns the marks

• State that work done on the spring equals the energy transferred to its elastic potential energy store, so the same equation applies to stretching and compressing.
• Use W = 0.5 × k × e², where k is the spring constant in N/m and e is the extension or compression in metres.
• Explain that the factor 0.5 arises because the force increases uniformly from zero to F, so the average force is F ÷ 2.
• Convert all lengths to metres and square the extension, for example 5 cm becomes 0.05 m and 0.05² = 0.0025 m².
• Check that the spring remains within its limit of proportionality; beyond that point the equation no longer applies because force is no longer proportional to extension.
• Give the answer with the correct unit, the joule (J), and consider whether the magnitude is sensible for the spring constant and extension used.

Common mistakes and how to put them right

• Using the full force instead of the average force, giving W = k × e².

  Put it right by remembering the factor 0.5 from the average force F ÷ 2.

• Forgetting to convert centimetres to metres before squaring.

  Put it right by converting first, since 20 cm is 0.20 m and squaring 20 gives a value 10 000 times too large.

• Applying the equation beyond the limit of proportionality.

  Put it right by checking the force–extension graph and only using the equation where the line is straight through the origin.

Exam tips

• Write the equation, substitute values with units, then calculate; this makes method marks easier to award even if the final arithmetic slips.
• Convert every length to metres at the start and state the conversion explicitly, for example 8 cm = 0.08 m.
• If a graph is given, read the extension from the straight section and comment that the limit of proportionality has not been exceeded.