Hooke's Law & the Young Modulus
A thick steel wire and a thin steel wire stretch by different amounts under the same force, despite being the same material. Force and extension alone can't separate "how stiff is this sample" from "how stiff is this material" — stress and strain can, and the Young modulus is what falls out of that separation.
Hooke's law
Stretch most solid materials by a small amount and the extension produced is directly proportional to the force applied. This is Hooke's law:
where $F$ is the applied force, $\Delta L$ is the extension, and $k$ is the stiffness (or spring constant) of the sample, in newtons per metre (N m−1). A larger $k$ means more force is needed for the same extension.
Hooke's law holds up to the limit of proportionality: beyond it, the graph stops being a straight line. This is a separate point from the elastic limit, which comes slightly after — up to it, the material still returns fully to its original length once the force is removed, even where the graph is no longer straight. Only past the elastic limit does deformation become permanent.
The limit of proportionality is where the force-extension graph stops being a straight line — $F$ and $\Delta L$ are no longer proportional beyond it. The elastic limit is a separate, slightly later point, beyond which deformation becomes permanent (plastic) rather than fully reversible. Between the two, a material can be non-linear — the graph curves — while still being fully elastic: remove the force anywhere before the elastic limit and the sample returns exactly to its original length, straight-line region or not.
"Elastic" in physics specifically means the deformation is fully reversible. A material stretched within its elastic limit stores the work done on it as elastic strain energy, all recoverable once the force is removed.
Why force and extension aren't enough
$k$ describes a specific sample, not the material. Cut a wire in half and the shorter piece needs more force for the same extension — effectively stiffer, though it's the same material. A thicker wire of the same length is stiffer too. $k$ is tangled up with both length and cross-sectional area, making it useless for comparing materials.
What's needed is a way of describing deformation that strips out sample geometry, leaving only a property of the material. That's stress and strain.
Tensile stress
Tensile stress is the force applied per unit cross-sectional area:
where $\sigma$ is stress, $F$ is the applied force, and $A$ is cross-sectional area. Stress has units of pascals (Pa), same as pressure, since both are force over area. A thicker sample spreads the same force over more material, so experiences less force per unit area.
Tensile strain
Tensile strain is the extension produced per unit original length:
where $\varepsilon$ is strain, $\Delta L$ is the extension, and $L$ is the original length. Being a ratio of two lengths, strain has no units — often quoted as a percentage. A longer sample stretches further in absolute terms for the same proportional deformation.
Stress and strain both divide out sample dimensions, leaving quantities that depend only on the material — exactly what's needed to compare materials on equal footing.
The Young modulus
For a material behaving elastically, stress and strain are directly proportional, and their ratio is a constant that depends only on the material — the Young modulus, $E$.
Because $E$ is a ratio of quantities with sample geometry already divided out, it's a genuine material property — the same whether you test a short thick sample or a long thin one, within their elastic limits. A large Young modulus means a stiff material needing large stress for a given strain. Steel's is around 200 GPa, against roughly 0.01–0.1 GPa for rubber — a difference of several thousand times.
Required Practical: Determination of the Young Modulus
This is AQA's Required Practical 4: determining the Young modulus of a wire.
Equipment
| Equipment | Purpose |
|---|---|
| Test wire (e.g. copper, known gauge) | The sample whose Young modulus is being measured |
| Control wire (same material, unloaded) | Cancels out thermal expansion/sag, since it experiences the same conditions as the test wire without the load |
| Two G-clamps | Fix both wires securely to the bench edge |
| Marker + fixed scale (or vernier scale/travelling microscope) | Measures the small extension of the test wire relative to the control wire |
| Micrometer screw gauge | Measures the wire's diameter, to calculate cross-sectional area |
| Metre rule | Measures the wire's original length |
| Mass hanger + slotted masses | Applies a series of known increasing loads |
| Safety goggles | Protects against a wire snapping under tension |
| Protective floor mat/sandbag | Cushions a falling mass if the wire does snap |
Method
Sources of error and how to minimise them
Room-temperature changes make a wire expand or contract slightly, which could be wrongly read as extension due to loading. Measure the test wire's extension relative to the unloaded control wire (mounted on the same support, experiencing the same conditions) rather than against a fixed external scale, so thermal effects cancel out.
Loading the wire beyond its elastic limit — even briefly, on a single trial — leaves permanent extension behind, so every subsequent "unloaded" length is wrong and the whole set of repeats is contaminated. Stay well within the known elastic limit throughout, and check the wire returns to its original length once all load is removed at the end.
A drawn wire is rarely perfectly uniform in cross-section. Measure the diameter at several points along the wire, and at two right angles at each point, then average — using a single measurement risks a biased area value.
Reading the marker position from an angle introduces error. View the marker at eye level, or use a vernier scale/travelling microscope for greater precision on the small extensions involved.
Wear eye protection throughout, and stand to the side of the wire rather than directly in front of or behind it, since a snapped wire under tension can whip back sharply.
Key takeaways
- Hooke's law: $F = k\Delta L$, with $k$ the stiffness of a particular sample (N m−1), valid up to the limit of proportionality.
- The limit of proportionality (graph stops being straight) and the elastic limit (deformation becomes permanent beyond this point) are different points — a material can be non-linear yet still fully elastic between them.
- Tensile stress $\sigma = F/A$ (Pa) and tensile strain $\varepsilon = \Delta L/L$ (no units) strip out sample geometry, leaving quantities that describe the material itself.
- The Young modulus $E = \sigma/\varepsilon = \dfrac{FL}{A\,\Delta L}$, measured in pascals, is a genuine material property, independent of a sample's length or cross-sectional area.
- A larger Young modulus means a stiffer material, requiring more stress to produce the same strain — steel's is several thousand times rubber's.
- The Young modulus is measured experimentally as the gradient of a stress-strain graph for a loaded wire, using a control wire to correct for thermal effects, with repeats kept well within the elastic limit.