A wire of cross-sectional area 2.0 × 10^-6 m^2 carries a tension of 50 N. Calculate the tensile stress. [1 mark]

AQA AQA 2018-style practice: A spring obeys Hooke's law and extends by 0.080 m when a force of 12 N is applied. Calculate the spring constant and the elastic strain energy stored at this extension.

Use Hooke's law to find the spring constant: k = Fx = 120.080 = 150 N m^-1. The elastic strain energy is the area under the force-extension graph, E = 12Fx = 12(12)(0.080) = 0.48 J. Equivalently E = 12kx^2 = 12(150)(0.080)^2 = 0.48 J. Markers reward correct use of F = kx for the spring constant and the area under the graph (or 12kx^2) for the stored energy.

EnglandPhysicsSyllabus dot point

How do solid materials respond when we stretch or compress them?

Density, Hooke's law and the spring constant, elastic and plastic behaviour, tensile stress and strain, the energy stored in a stretched material, and the difference between brittle and ductile behaviour.

A focused answer to AQA A-Level Physics 3.4.2.1, covering density, Hooke's law and the spring constant, elastic and plastic behaviour, tensile stress and strain, the elastic strain energy stored in a stretched material, and brittle versus ductile behaviour.

Generated by Claude Opus 4.810 min answerUpdated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

Have a quick question? Jump to the Q&A page

Jump to a section

What this dot point is asking
Density
Hooke's law and the spring constant
Elastic and plastic behaviour
Stress, strain and strain energy
Brittle and ductile materials
Try this

What this dot point is asking

AQA specification point 3.4.2.1 wants you to use density, apply Hooke's law and the spring constant, distinguish elastic from plastic deformation, define tensile stress and strain, calculate the elastic strain energy stored, and describe brittle and ductile behaviour from force-extension and stress-strain graphs.

Density

Density is a property of the material, independent of the size or shape of the sample. Water has a density of $1000 \text{ kg m}^{-3}$ , which is a useful benchmark; metals are typically several times denser.

Hooke's law and the spring constant

A stiffer spring has a larger $k$ , meaning a greater force is needed per unit extension. The force-extension graph is a straight line through the origin with gradient $k$ , but only up to the limit of proportionality; beyond this the line curves. Springs combine: in parallel the effective spring constant is the sum $k = k_1 + k_2$ (stiffer), while in series it is given by $\dfrac{1}{k} = \dfrac{1}{k_1} + \dfrac{1}{k_2}$ (less stiff).

Elastic and plastic behaviour

The limit of proportionality is where Hooke's law stops applying; the elastic limit (close by) is where permanent deformation begins. Below the elastic limit the deformation is reversible because the interatomic bonds stretch and spring back; beyond it, layers of atoms slide permanently past one another.

Stress, strain and strain energy

The strain energy equals the area under the force-extension graph. When a material is stretched elastically and released, this energy is returned; beyond the elastic limit some energy is dissipated as heat, shown by a loop (hysteresis) between the loading and unloading curves.

Brittle and ductile materials

Finding the energy stored from a force-extension graph

A wire stretches elastically by $5.0 \text{ mm}$ under a load of $40 \text{ N}$ , obeying Hooke's law throughout. Find the spring constant and the elastic strain energy stored.

step 1: Convert the extension to metres

$x = 5.0 \text{ mm} = 5.0 \times 10^{-3} \text{ m}$ .

step 2: Find the spring constant

$k = \dfrac{F}{x} = \dfrac{40}{5.0 \times 10^{-3}} = 8.0 \times 10^3 \text{ N m}^{-1}$ .

step 3: Use the area under the graph

The strain energy is the triangular area, $E = \tfrac{1}{2}Fx$ .

step 4: Evaluate

$E = \tfrac{1}{2}(40)(5.0 \times 10^{-3}) = 0.10 \text{ J}$ .

Try this

Q1. Define tensile stress. [1 mark]

Cue. Force per unit cross-sectional area.

Q2. A wire of cross-sectional area $2.0 \times 10^{-6} \text{ m}^2$ carries a tension of $50 \text{ N}$ . Calculate the tensile stress. [1 mark]

Cue. $\sigma = \dfrac{F}{A} = \dfrac{50}{2.0 \times 10^{-6}} = 2.5 \times 10^7 \text{ Pa}$ .

Q3. State what is meant by plastic deformation. [1 mark]

Cue. Permanent deformation that remains after the load is removed.

Exam-style practice questions

Practice questions written in the style of AQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AQA 20184 marksA spring obeys Hooke's law and extends by

0.080 \text{ m}

when a force of

12 \text{ N}

is applied. Calculate the spring constant and the elastic strain energy stored at this extension.

Show worked answer →

Use Hooke's law to find the spring constant: $k = \dfrac{F}{x} = \dfrac{12}{0.080} = 150 \text{ N m}^{-1}$ .

The elastic strain energy is the area under the force-extension graph, $E = \tfrac{1}{2}Fx = \tfrac{1}{2}(12)(0.080) = 0.48 \text{ J}$ . Equivalently $E = \tfrac{1}{2}kx^2 = \tfrac{1}{2}(150)(0.080)^2 = 0.48 \text{ J}$ .

Markers reward correct use of $F = kx$ for the spring constant and the area under the graph (or $\tfrac{1}{2}kx^2$ ) for the stored energy.

AQA 20214 marksUsing force-extension graphs, describe the difference between a brittle material and a ductile material, and explain what is meant by the elastic limit.

Show worked answer →

A brittle material (such as glass) has a force-extension graph that is a straight line up to the breaking point, with little or no curved (plastic) region; it breaks suddenly with almost no permanent deformation.

A ductile material (such as copper) shows a straight Hooke's law region followed by a long curved region of plastic deformation before it breaks, so it can be drawn into wires and stretches a great deal first.

The elastic limit is the point beyond which the material no longer returns to its original shape when the load is removed, marking the onset of permanent (plastic) deformation.

Markers reward the contrasting graph shapes, the sudden versus gradual failure, and a correct definition of the elastic limit.

Related dot points

Sources & how we know this

AQA A-level Physics (7408) specification — AQA (2017)