Spring constant calculator (Hooke's law)

Example: a spring with $k = 40$ N/m is stretched by 0.05 m. $F = 40 \times 0.05 = 2$ N.

Extension = stretched length − original length. A 10 cm spring stretched to 22 cm has an extension of 12 cm, not 22 cm.

Example: a force of 6 N gives that 12 cm extension. Convert to metres (0.12 m), then $k = 6 \div 0.12 = 50$ N/m. Using centimetres without converting gives 0.5, a hundred times too small.

Hooke's law only works up to the limit of proportionality. Beyond it, the force-extension graph curves and the extension is no longer proportional to the force. Past the elastic limit, the spring won't return to its original shape (inelastic deformation). On a graph of force against extension, the straight-line part has a gradient equal to $k$.

A stretched spring stores elastic potential energy: $E_e = \tfrac{1}{2} k e^2$ (given on the AQA equation sheet). For the 40 N/m spring stretched 0.05 m, $E_e = 0.5 \times 40 \times 0.05^2 = 0.05$ J. This works as long as the limit of proportionality hasn't been passed.

Hang the spring from a clamp next to a ruler, add masses one at a time and record the length each time. Weight in newtons is mass in kg × 9.8. Plot force against extension: the gradient of the straight part is the spring constant. Read the ruler at eye level to avoid parallax errors. Practise with our Hooke's law questions.

• Always use extension, not length.
• Convert cm to m before finding $k$ in N/m.
• On a force-extension graph, the spring constant is the gradient of the straight part: choose two points far apart to read it.
• If a graph curves at the end, say the limit of proportionality has been passed, so Hooke's law no longer applies there.