Work, energy, power and deformation

6. Deformation of Solids

5 min read 10 parts Lesson 17 of 17 Unit quiz

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6. Deformation of Solids — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Use Hooke's law, stress, strain and the Young modulus, distinguish elastic and plastic behaviour, and calculate elastic potential energy from force–extension graphs.Deformation — a change in th…

Tensile forces
Compressive forces
Natural length
Elastic behaviour
Limit of proportionality
Shapes for different materials

Section 1 of 10Tensile and compressive forces

Deformation
- a change in the size or shape of a body caused by forces.
Tensile forces
- a pair of forces that stretch a body, increasing its length.
Compressive forces
- a pair of forces that squash a body, reducing its length.
Natural length
- the length with no force applied.
- Extension = extended length − natural length.
- Compression = natural length − compressed length.
- For total length, add the extension to the original length.

Section 2 of 10Force–extension graphs

Elastic behaviour
- the extension disappears when the load is removed.
Limit of proportionality
- the point at which a force–extension graph changes from a straight line to a curve; beyond it the force is no longer proportional to the extension.
Shapes for different materials
- each material has its own characteristic graph.
- A metal gives a straight line, then a curve as it yields.
- A rubber band is non-linear: the gradient rises sharply when stretched near its limit,
  - because the long molecules align and cannot slide past each other.
- Glass is brittle: a straight line that ends suddenly at fracture.
- Every material has its own characteristic force–extension graph.

Section 3 of 10Hooke's law and the spring constant

Hooke's law
- the extension of a material is directly proportional to the applied force, up to the limit of proportionality: F = kx.
Spring constant k
- the force per unit extension, measured in N m⁻¹; a stiffer spring has a larger k.
From a graph
- k is the gradient of the linear part of a force–extension graph.
  - (force on the y-axis)
If axes are swapped
- with length L on the y-axis and load F on the x-axis, the gradient is x/F = 1/k, so k = 1/gradient.

Section 4 of 10Stress and strain

Tensile stress σ
- the force applied per unit cross-sectional area: σ = F/A, in pascals.
  What the symbols mean: σ = stress (Pa) · F = tensile force (N) · A = cross-sectional area (m²)
Ultimate tensile stress
- the maximum force per original cross-sectional area that a wire can support before it breaks.
Strain ε
- the extension per unit original length: ε = x/L, which is dimensionless.
  - (it may be given as a percentage)

Section 5 of 10The Young modulus

Definition
- the ratio of stress to strain: E = σ/ε = FL/(Ax), in pascals.
- It measures how stiff a material is: a larger E means a stiffer material.
Stress–strain graph
- a straight line through the origin in the elastic region; the gradient is the Young modulus.
Memory aid
- "when you are stressed, you show the strain": stress over strain.

Section 6 of 10Experiment: Young modulus of a wire

Clamp a long wire horizontally over a pulley with a load hanger at one end and attach a reference tape marker.
Measure the original length L with a metre ruler (from the clamp to the marker).
Measure the diameter with a micrometer or digital callipers at several places and average it.
Add masses one at a time, calculating the weight, and record the new length each time.
Subtract the original length to find the extension for each load; remove the load now and then to check the wire returns to its length.
Variables
- the independent variable is the load, the dependent variable is the extension.

Section 7 of 10Analysing the Young modulus experiment

Plot force against extension and draw a line of best fit.
Find the gradient, F/x.
Calculate the cross-sectional area from the mean diameter: A = πd2/4.
Calculate E = (gradient × L)/A.
Improvements
- measure diameter at several points, repeat readings and average, use a longer wire, and use a vernier scale for the extension.
- The extension is measured on the assumption that the wire stretches uniformly along its length.

Section 8 of 10Elastic and plastic deformation

Elastic deformation
- the material returns to its original length when the load is removed.
Plastic deformation
- the material does not return to its original length; it is permanently deformed.
Elastic limit
- the point beyond which the material no longer returns to its original length; plastic behaviour lies beyond it.
Careful distinction
- the elastic limit is at a slightly higher extension than the limit of proportionality; they are not the same point.
- A point on the straight part of the graph is elastic (not plastic).

Section 9 of 10Brittle and ductile materials

Type	Behaviour	Graph	Examples
Brittle	Breaks with little or no plastic deformation	Straight line through the origin, ending at fracture	Glass, concrete
Ductile	Large plastic region; stretches into a new shape before breaking	Straight line, then a curve towards the x-axis	Copper

Section 10 of 10Work done and elastic potential energy

Work done in stretching
- equal to the area under the force–extension graph, for any shape of graph.
- If the graph is a straight line (Hooke's law), the area is a triangle.
- For a non-linear graph, add the areas of separate sections (triangles, rectangles, trapeziums).
Elastic potential energy
- the energy stored in a stretched or compressed material; within the limit of proportionality it equals the work done.
Equations
- E = ½Fx,
  - and substituting F = kx gives E = ½kx2.
    What the symbols mean: E = elastic potential energy stored (J) · k = spring constant (N m⁻¹) · x = extension (m) · F = force (N)
- These equations apply only while the graph is a straight line; they do not apply beyond the limit of proportionality.
- Convert extension to metres (mm × 10⁻³) before calculating.

Last updated: 14 October 2026