Wait, What? A spring can return to its original length and still have stopped obeying Hooke’s law.

Students often collapse three different ideas into one: proportional behaviour, elastic behaviour and permanent deformation. They are not identical. A spring can leave the region where force is proportional to extension before it suffers permanent deformation. That distinction is exactly why the force-extension practical is more than “hang masses and draw a graph.”

The experimental question

The core job is to test how extension changes as force changes. For a spring behaving within its Hookean region:

F = kx

where F is applied force, x is extension and k is spring constant. A straight-line graph through the origin supports proportionality over the tested range.

Extension is not length

Measure the natural length before loading. Extension is:

x = loaded length − original length

Confusing total length with extension shifts the entire graph and destroys the physical interpretation of k.

Force is not mass

Hanging masses create force through weight:

F = mg

Using mass directly on a force-extension graph without conversion is dimensionally wrong unless the axis is explicitly mass and the interpretation is adjusted accordingly.

Reading technique matters

Use a fixed fiducial marker or pointer near the ruler scale rather than estimating from a moving coil edge. Read at eye level to reduce parallax. Let oscillations settle before recording. Add loads gently so the spring is not shocked into extra motion.

The Institute of Physics spring practicals emphasise careful measurement of extension against load and graph interpretation as the central evidence. See the IOPSpark spring investigation.

The graph tells you more than one number

In the linear region, the gradient of F against x is k. If you plot x against F instead, the gradient is 1/k. Axis choice changes the meaning of the gradient.

Curvature shows that proportionality is failing. A later return to the original length after unloading would still indicate elastic behaviour, but not necessarily Hookean behaviour throughout the whole loading range.

Loading and unloading can reveal hysteresis

If you increase the load stepwise and then decrease it, the unloading path may not exactly retrace the loading path. This can indicate internal energy dissipation or material effects. Rubber is especially likely to show obvious hysteresis; metal springs within modest loads may show much less.

Quantitative window

If a force increase from 1.0 N to 3.0 N produces extensions from 0.020 m to 0.060 m, then:

k = ΔF/Δx = 2.0 / 0.040 = 50 N m⁻¹

This is stronger than calculating k from one point because a gradient uses multiple measurements and reduces the influence of one reading error.

Uncertainty and sensible range

If ruler resolution is 1 mm and your smallest extension is 2 mm, uncertainty is a large fraction of the signal. Larger safe extensions improve fractional resolution. But loading too far may take the spring beyond the region you intend to study.

Observation versus inference

Observation: “At 2.0 N the extension was 0.041 m.” Inference: “The spring is consistent with proportional force-extension behaviour over this range.” Stronger inference: “The spring constant is approximately the graph gradient, assuming temperature and geometry remain stable.”

Failure modes

• Using total length instead of extension.
• Using mass values as force without conversion.
• Reading while the spring is oscillating.
• Starting with too large a load increment and missing where curvature begins.
• Ignoring ruler parallax.
• Assuming permanent deformation begins exactly where proportionality fails.

Unfamiliar transfer: elastic cord

An elastic cord may produce a curved force-extension graph from the start. The correct response is not to force a straight line. The method transfers: measure carefully, plot the relationship, identify the useful region and interpret the material behaviour actually observed.

Secondary → JC → deeper Physics

Secondary: measure extension, convert mass to force, plot force-extension graphs and identify proportional regions.

JC: distinguish proportional limit, elastic behaviour and permanent deformation; use gradients and uncertainty; analyse loading-unloading differences.

Deeper Physics: material testing extends to stress-strain curves, Young modulus, viscoelasticity, fatigue and energy dissipation.

Checkpoint

A spring gives a straight-line graph up to 3 N, curves between 3 N and 5 N, but returns to its original length after unloading from 5 N. What can you conclude?

Answer key and WHY reasoning

Hooke’s law is supported only in the straight-line region. The spring remained elastic up to 5 N in this trial because it recovered its original length, but proportionality had already failed above about 3 N.

How to study this practical

Revise from graphs, not apparatus lists. Given any force-extension graph, identify the proportional region, estimate k, state whether unloading evidence is needed to test elasticity, and predict which measurement dominates uncertainty at small extension.

Evidence boundaries

Your graph supports behaviour only over the tested loads and conditions. It does not establish that the spring remains Hookean at all forces, temperatures or after repeated fatigue cycles.

Authoritative next steps

• Institute of Physics: investigating simple steel springs
• SEAB O-Level syllabus directory
• SEAB A-Level syllabus directory

Teaching Guide

Ask students to explain why “elastic” and “Hookean” are not synonyms. Give one loading-only graph and ask what cannot yet be concluded about permanent deformation. This forces evidence boundaries rather than vocabulary recall.