Stress, strain and Young modulus experiment
How stiff is a material? The Young modulus tells you how much a material resists stretching. This sheet covers the definitions, the wire experiment, the stress-strain graph, and how to calculate and interpret results.
Work through it in three passes
Learn the definitions
Distinguish stress, strain and Young modulus, and know their units.
Understand the experiment
See how to measure extension and why the gradient gives Young modulus.
Practise calculations
Use the calculator and worked examples to handle typical exam questions.
- Define stress, strain and Young modulus and give their units.
- Describe an experiment to determine Young modulus for a metal wire.
- Sketch and interpret a stress-strain graph up to breaking point.
- Calculate Young modulus from force, extension and wire dimensions.
- Identify the limit of proportionality, elastic limit and yield point on a graph.
Stress, strain and Young modulus
When you stretch a wire, two things control how much it extends: the force you apply and the wire's dimensions. Stress and strain normalise those effects so you can compare different materials fairly.
Stress σ
Force per unit cross-sectional area. Unit: Pa or N m-2.
Strain ε
Fractional extension. No unit — it is a ratio of two lengths.
Young modulus E
Stress divided by strain in the linear region. Unit: Pa or GPa.
A thick wire stretches less than a thin wire under the same load. A long wire stretches more than a short wire under the same stress. Stress and strain remove those size effects, so the Young modulus depends only on the material (and its condition), not on the sample shape.
| Quantity | Symbol | Definition | Unit |
|---|---|---|---|
| Original length | L | Length before loading | m |
| Extension | ΔL | Increase in length | m |
| Force / load | F | Tension in the wire | N |
| Cross-sectional area | A | Area of wire perpendicular to length | m² |
| Stress | σ | F / A | Pa |
| Strain | ε | ΔL / L | none |
| Young modulus | E | σ / ε | Pa or GPa |
Measuring Young modulus for a metal wire
The standard school or college method uses a long wire loaded with known weights. The key is to measure very small extensions accurately.
Long wire (usually 2 m or more), fixed support at the top, pulley or load hanger at the bottom, micrometer or digital callipers for diameter, metre ruler for original length, travelling microscope or vernier scale for extension.
Extensions are small — often less than a millimetre. A longer wire gives a larger, more measurable extension for the same strain. The control wire removes errors caused by the support sagging or the measuring scale moving.
Stress-strain graph for a metal wire
A typical stress-strain graph for a ductile metal has several distinct regions. The gradient of the initial straight part is the Young modulus.
| Feature | What it means |
|---|---|
| Limit of proportionality | The end of the straight-line region where σ ∝ ε and Hooke's law holds. |
| Elastic limit | Beyond this, the material no longer returns to its original length when the force is removed. |
| Yield point | The stress at which the material suddenly begins to stretch with little or no extra load. |
| Plastic region | The material permanently deforms. Strain increases faster than stress. |
| Ultimate tensile stress (UTS) | The maximum stress the material can withstand before necking begins. |
| Breaking stress | The stress at which the material fractures. |
Young modulus is the gradient of the initial straight section only. Do not use a point from the plastic region, and do not use the UTS or breaking stress.
Young modulus calculator
Enter the wire dimensions and the measured extension to calculate stress, strain and Young modulus. Use SI units (metres, newtons, square metres).
Worked examples
Example 1: calculating stress and strain
A steel wire of diameter 0.40 mm and length 2.5 m is stretched by a force of 45 N. Calculate the stress and strain.
Area:
Stress:
Strain:
Example 2: calculating Young modulus
A copper wire of length 2.00 m and diameter 0.56 mm extends by 1.2 mm when a load of 60 N is applied. Calculate Young modulus for copper.
Area:
Young modulus:
Example 3: from force-extension graph gradient
A force-extension graph for a metal wire has a straight-line gradient of 2.5 × 104 N m-1. The wire is 1.80 m long with cross-sectional area 3.0 × 10-7 m². Find Young modulus.
From F = (EA/L)ΔL, the gradient of a force-extension graph is EA/L.
Common mistakes and quick fixes
Forgetting to convert mm to m
Diameter and extension are often given in millimetres. Convert to metres before calculating area or strain.
Using diameter instead of area
Stress uses A = πd²/4, not d. A common error is to divide force by diameter.
Using the wrong part of the graph
Young modulus is the gradient of the linear region only. Do not use UTS or breaking stress.
Confusing extension with length
Strain is ΔL/L, not ΔL and not L/ΔL.
Self-check questions
1. A wire of length 3.0 m extends by 2.0 mm under load. What is the strain?
Method: Strain is the ratio of extension to original length. Convert both to metres.
ε = 2.0 × 10-3 / 3.0 = 6.7 × 10-4
2. A steel wire of diameter 0.80 mm supports a load of 120 N. Calculate the stress.
Method: First find cross-sectional area, then divide force by area.
A = π × (0.80 × 10-3)² / 4 = 5.03 × 10-7 m²
σ = 120 / 5.03 × 10-7 = 2.39 × 108 Pa
3. A material has Young modulus 200 GPa and strain 1.5 × 10-3. What is the stress?
Method: Rearrange E = σ/ε to σ = Eε.
σ = 200 × 109 × 1.5 × 10-3 = 3.0 × 108 Pa
4. Why is a long, thin wire used in the Young modulus experiment?
Answer: A long wire gives a larger extension for the same strain, making ΔL easier to measure accurately. A thin wire gives a larger stress for the same force, so measurable extensions occur at safe loads.
Final checklist for stress and strain questions
- Convert all lengths to metres before calculating area or strain.
- Calculate cross-sectional area using A = πd²/4 for a circular wire.
- Use σ = F/A and ε = ΔL/L; remember strain has no unit.
- Calculate Young modulus from E = FL/(AΔL) or the gradient of a force-extension graph.
- Only use the linear region of the stress-strain graph for Young modulus.
- Label the limit of proportionality, elastic limit, yield point, UTS and breaking stress clearly on sketches.