Also, this physics grades 11-12 lesson on Young modulus notes gives the key ideas first, then worked examples and diagrams. After that, practise with the worksheet on Young modulus and check yourself with the quiz.
Quick answer: Hooke’s law says extension is proportional to force (F = kx) up to the limit of proportionality. stress σ = F/A and strain ε = x/L describe the material, and the Young modulus E = σ/ε is the gradient of the straight part of the stress–strain graph. Moreover, stored energy is ½Fx.
Level: Physics grades 11-12 (A Level · IB · Class 11-12) · Curriculum: A Level 3.4.2
Young modulus notes: the key ideas
What is Hooke’s law?
Moreover, the constant k is the force constant (or stiffness) of the spring, measured in N/m. Furthermore, the graph of force against extension is a straight line through the origin with gradient k. Also, beyond the limit of proportionality the line curves and Hooke’s law no longer applies. A stiffer spring has a larger k.
How much energy is stored in a stretched spring or wire?
The force grows from zero to F as the extension grows from zero to x, so the average force is F/2. Also, the work done, which is stored as elastic potential energy, equals the area under the force–extension graph.
E = ½Fx = ½kx² = F²/2k
E is the elastic potential energy (J), F the force (N), x the extension (m), k the force constant (N/m). Valid only inside the elastic limit and where Hooke’s law applies.
A spring has k = 250 N/m. In addition, find the energy stored at an extension of 0.060 m and the extra energy needed to stretch it to 0.100 m.
E at 0.060 m = ½ × 250 × 0.060² = 0.45 J. E at 0.100 m = ½ × 250 × 0.100² = 1.25 J. Extra energy = 1.25 − 0.45 = 0.8 J.
What are stress, strain and the Young modulus?
A thick wire stretches less than a thin wire of the same material, and a long wire stretches more than a short one. In addition, stress and strain remove these effects to describe the material itself.
Stress σ = F/A, force per unit cross-sectional area (Pa = N/m²). Strain ε = x/L, extension per unit original length (no unit). Moreover, the Young modulus E = σ/ε, measured in pascals, is the gradient of the straight part of a stress–strain graph.
| Material | Young modulus E / GPa | Typical behaviour |
|---|---|---|
| Steel | 200 | Stiff and ductile |
| Copper | 120 | Ductile |
| Aluminium | 70 | Ductile, lighter |
| Glass | 70 | Brittle |
| Nylon | 3 | Flexible polymer |
| Rubber | about 0.01 | Stretches a great deal |
E = (F/A) ÷ (x/L) = FL/Ax, and the force constant of a wire is k = EA/L.
F is the load (N), A the cross-sectional area (m²), L the original length (m), x the extension (m).
A steel wire 2.50 m long and 0.80 mm in diameter carries a 150 N load (E = 2.0 × 10¹¹ Pa). Furthermore, calculate the stress, strain and extension.
A = π(0.40 × 10⁻³)² = 0.000 m². Stress = F/A = 150 ÷ 0.000 = 298415518.297 Pa. strain = σ/E = 298415518.297 ÷ 2.0 × 10¹¹ = 0.00149. extension = strain × L = 0.00149 × 2.50 = 3.73 mm.
What does a stress–strain graph tell you?
Furthermore, on the straight part, up to the limit of proportionality (P), stress is proportional to strain. A little beyond is the elastic limit (E): up to here the material returns to its original length when the load is removed. After the yield point (Y) the metal stretches a lot with little extra stress and deformation becomes plastic, meaning permanent. Moreover, the highest point is the ultimate tensile stress (UTS). The wire then narrows (necks) and breaks at the fracture point F.
Moreover, different materials give very different curves. A brittle material such as glass or ceramic follows a steep straight line and breaks with almost no plastic region. A ductile material such as copper can be drawn into wire. A polymer such as nylon has a low gradient and a long, curved plastic region.
Students confuse stiffness with strength. Furthermore, the Young modulus (gradient) measures stiffness; the UTS (highest stress) measures strength. A material can be stiff and brittle, like glass, or flexible and strong.
Also, convert diameter to area early: A = π(d/2)². Half the diameter first, then square it. Also, doubling the diameter makes the area four times bigger.
- hooke’s law: F = kx, k in N/m, valid up to the limit of proportionality.
- Moreover, elastic potential energy E = ½Fx = ½kx², the area under the F–x graph.
- Stress σ = F/A (Pa); strain ε = x/L (no unit).
- Young modulus E = σ/ε = FL/Ax; steel is about 200 GPa.
- Furthermore, elastic deformation is reversible; plastic deformation is permanent.
Springs, wires and the stiffness of materials: frequently asked questions
What is the difference between the limit of proportionality and the elastic limit?
Also, the limit of proportionality is where force stops being proportional to extension. In addition, the elastic limit is the largest stress after which the material returns to its original length on unloading. In addition, they are usually very close in metals.
What is the unit of the Young modulus?
The unit is the pascal (Pa), because strain has no unit and the modulus equals stress divided by strain. Moreover, values for metals are usually quoted in gigapascals (1 GPa = 10⁹ Pa).
Why is the energy stored ½Fx and not Fx?
The force rises steadily from zero to F as the extension grows, so the average force over the stretch is F/2. Furthermore, the work done is the average force multiplied by the extension.
What is the difference between ductile and brittle?
A ductile material shows a large plastic region and can be stretched into wire without breaking. A brittle material breaks soon after the elastic limit with little or no plastic deformation.
How can you find the Young modulus in the lab?
Furthermore, hang increasing loads on a long thin wire and measure each extension. Plot force against extension, find the gradient and use E = (gradient) × L ÷ A, with the area from the measured diameter.
