AS Physics: Deformation of Solids — Revision Notes

Condensed for the final weeks. For the full explanation, use the Deformation of Solids study guide.

Hooke’s law

F = k x

Force is proportional to extension up to the limit of proportionality. That qualifier is part of the law, and omitting it loses the mark.

Key points on the graph, in order: limit of proportionality → elastic limit → yield point → breaking point.

• Elastic deformation — the material returns to its original shape when the load is removed.
• Plastic deformation — a permanent extension remains.

The elastic limit is where behaviour changes from elastic to plastic. It comes after the limit of proportionality, not before — the two points are related but distinct, and for many materials the elastic limit sits only slightly beyond where proportionality ends.

Stress, strain and the Young modulus

stress   = F / A          Pa or N m^-2
strain   = x / L          no units
E = stress / strain = FL / Ax

The Young modulus is a property of the material, not of the object. Two wires of the same metal but different thickness have the same E; only their stiffness k differs. That distinction is examined nearly every series.

Determining E experimentally: use a long, thin wire — long to give a measurable extension, thin to give a large stress for a modest load. Measure the diameter at several points with a micrometer and take a mean, because the wire may not be uniform. A test wire is loaded with increasing known weights and its extension measured with a vernier scale or travelling microscope against an unstretched reference wire of the same material, run alongside it to cancel out effects like thermal expansion during the experiment. Plot stress against strain; E is the gradient of the straight-line region.

Worked example. A wire of length 1.5 m and cross-sectional area 2.0 × 10⁻⁶ m² extends by 1.2 mm under a load of 80 N. stress = F/A = 80 ÷ (2.0 × 10⁻⁶) = 4.0 × 10⁷ Pa. strain = x/L = (1.2 × 10⁻³) ÷ 1.5 = 8.0 × 10⁻⁴. E = stress/strain = (4.0 × 10⁷) ÷ (8.0 × 10⁻⁴) = 5.0 × 10¹⁰ Pa.

Elastic strain energy

E = 1/2 F x  =  1/2 k x^2

This is the area under the force–extension graph — which is why the ½ appears, and why for a non-linear graph you must find the area rather than use the formula.

For a material loaded beyond its elastic limit, the loading and unloading curves differ, and the area between them is the energy dissipated, usually as thermal energy.

Material types

Type	Behaviour
Brittle	Breaks at the elastic limit with no plastic deformation — glass, ceramics
Ductile	Large plastic deformation before breaking; can be drawn into wire — copper
Polymeric	Very large extensions; loading and unloading curves differ — rubber

Strong means high breaking stress. Stiff means high Young modulus. Tough means it absorbs a lot of energy before breaking. These are three different properties and questions rely on the distinction — glass is stiff and strong but not tough, because it shatters with almost no plastic deformation to absorb energy first.

Exam traps

• Stating Hooke’s law without “up to the limit of proportionality”.
• Treating the Young modulus as a property of the object.
• Using ½Fx for a non-linear graph instead of finding the area.
• Confusing strong, stiff and tough.
• Putting the elastic limit before the limit of proportionality.
• Forgetting to convert mm to m, or to use the radius when calculating area from a diameter.
• Confusing stress with pressure conceptually — they share the same unit (Pa), but stress specifically describes a force producing deformation in a solid, not a fluid pushing on a surface.

Self-test

1. State Hooke’s law in full.
2. Why do two wires of the same material but different diameters have the same Young modulus?
3. What does the area under a force–extension graph represent?
4. Distinguish strong, stiff and tough.
5. Why is a long thin wire used to measure the Young modulus?
6. Calculate the Young modulus of a wire of length 1.5 m and cross-sectional area 2.0 × 10⁻⁶ m² that extends 1.2 mm under an 80 N load.
7. Why is an unstretched reference wire run alongside the test wire in this experiment?

Answers: 1. The force applied is directly proportional to the extension produced, up to the limit of proportionality. 2. The Young modulus is defined using stress and strain, which account for cross-sectional area and original length, so it depends only on the material. 3. While the deformation stays within the elastic limit, the area under the loading curve is both the total work done stretching the material and the elastic strain energy stored, since all of it is recoverable. Once the material has been stretched beyond its elastic limit, this is no longer true: the total work done is still the full area under the loading curve, but only the smaller area under the unloading curve is recoverable elastic strain energy — the area between the two curves has instead been dissipated (mostly as thermal energy) in permanently deforming the material. 4. Strong means a high breaking stress; stiff means a high Young modulus; tough means absorbing a large amount of energy before fracture. 5. A long wire gives a measurably large extension and a thin wire gives a large stress for a modest load, reducing percentage uncertainty in both measurements. 6. stress = 80 ÷ (2.0 × 10⁻⁶) = 4.0 × 10⁷ Pa; strain = (1.2 × 10⁻³) ÷ 1.5 = 8.0 × 10⁻⁴; E = 4.0 × 10⁷ ÷ 8.0 × 10⁻⁴ = 5.0 × 10¹⁰ Pa. 7. It cancels out effects such as thermal expansion during the experiment, since both wires expand or contract by the same amount, isolating the extension due to the load alone.