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Density is mass per unit volume, ρ = m/V, in kg m⁻³, and the spring constant k from F = kΔL measures stiffness. Free AQA notes and 15 questions answered.
Density, Hooke's Law and Elasticity is part of Materials in A Level AQA Physics.
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A Level AQA
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Density is mass per unit volume, ρ = m/V, in kg m⁻³, and the spring constant k from F = kΔL measures stiffness. Free AQA notes and 15 questions answered.
Use this lesson to stay inside Materials while connecting the detail back to the wider Mechanics and materials topic.
Read the core explanations from Density, Hooke's Law and Elasticity before testing yourself from memory.
Picture two boxes of identical size. One is filled with lead shot, the other with polystyrene beads. Both occupy the same volume, yet one is dramatically heavier. That difference is captured by a single quantity: density.
The equation $\rho = \dfrac{m}{V}$ tells you how tightly matter is packed. A material with a high density has a large mass squeezed into a small volume — think steel at around 7800 kg m⁻³ compared with wood at roughly 600 kg m⁻³.
For a regularly shaped object (a cuboid, cylinder, or sphere), you calculate volume from its dimensions. For an irregularly shaped object, you use Archimedes' displacement method: submerge the object in water and measure the volume of water displaced. That displaced volume equals the object's volume.
Worked example: Calculating Density of a Metal Cylinder
A metal cylinder has a mass of 0.540 kg. Its radius is 0.020 m and its length is 0.215 m. Calculate its density. [3 marks]
Answer: ρ = 2000 kg m⁻³
Density is a property of the material itself, not the object's size. Doubling the volume of a steel block doubles its mass, so the ratio m/V stays the same. This is why density is described as an intensive property.
Now that you have a handle on density as a material property, the next question is: how do materials respond when you pull or compress them? The starting point is Hooke's law — one of the most tested relationships in A Level Physics.
Hang a mass from a spring and it stretches. Double the mass and the extension doubles. This proportional relationship holds — but only up to a point. That point is the elastic limit.
The spring constant $k$ is a measure of stiffness. A large $k$ means the spring is stiff — you need a large force to produce a small extension. A small $k$ means the spring is compliant — a small force produces a large extension.
The elastic limit is the point beyond which the material no longer returns to its original shape when the force is removed. Below the elastic limit, deformation is elastic (fully recoverable). Beyond it, deformation becomes plastic (permanent).
On a force–extension graph, the Hooke's law region is the straight-line section through the origin. The gradient of this straight line equals the spring constant $k$. Once the line curves, the elastic limit has been passed and the material no longer obeys Hooke's law.
Hooke's law applies to springs, wires, and many other materials in their elastic region. The key condition is always stated in the specification: provided the elastic limit is not exceeded. Include this condition in any exam definition for full marks.
Hooke's law describes a spring, but engineers need to compare different materials regardless of their dimensions. A thick steel rod and a thin steel wire are both steel — yet they behave very differently under the same force. To make fair comparisons, physicists use tensile stress and tensile strain.
Think of stress as the 'intensity' of the force — how concentrated it is across the material's cross-section. A force of 1000 N spread across a large area causes far less stress than the same force concentrated on a tiny area.
Strain tells you the proportional deformation. An extension of 2 mm in a 1 m wire is a strain of 0.002. The same 2 mm extension in a 10 mm wire is a strain of 0.2 — a very different situation, even though the raw extension is identical.
Worked example: Calculating Tensile Stress and Tensile Strain
A steel wire of original length 2.50 m and circular cross-section of diameter 0.80 mm is stretched by a force of 150 N, producing an extension of 3.0 mm. Calculate (a) the tensile stress and (b) the tensile strain. [4 marks]
Answer: (a) Tensile stress σ ≈ 2.98 × 10⁸ Pa (b) Tensile strain ε = 1.2 × 10⁻³ (no unit)
When you stretch a spring within its elastic limit, you do work on it. That work is stored as elastic strain energy — and it is fully released when the force is removed. This is the same energy that makes a stretched elastic band snap back, or a compressed spring launch a projectile.
On a force–extension graph, the elastic strain energy stored equals the area under the graph up to the point of interest. For the linear (Hooke's law) region, this area is a triangle, giving the familiar formula $E = \frac{1}{2}F\Delta L$. Substituting $F = k\Delta L$ gives the equivalent form $E = \frac{1}{2}k(\Delta L)^2$.
Beyond the elastic limit, the area under the graph still represents energy — but not all of it is recovered. Some energy is dissipated as heat during plastic deformation. This is why a permanently stretched spring feels warm after being over-extended.
Breaking stress is a critical design parameter. Engineers must ensure that the stress in any structural component remains well below the breaking stress of the material used, with an appropriate safety factor applied.
Worked example: Calculating Elastic Strain Energy Stored in a Spring
A spring with spring constant k = 250 N m⁻¹ is extended by 0.12 m within its elastic limit. Calculate the elastic strain energy stored. [2 marks]
Answer: Elastic strain energy E = 1.80 J
The concepts in this topic are well-defined, but the marks are often lost in the details of method, units, and precise language. The guidance below targets the patterns that appear most frequently in AQA mark schemes.
When an exam question asks you to explain why a material obeys or does not obey Hooke's law, structure your answer as: observation + because + scientific principle. For example: 'The extension doubles when the force doubles, because the atoms in the material are displaced proportionally from their equilibrium positions, and the restoring force is proportional to displacement within the elastic region.'
For evaluate questions about material behaviour — for instance, comparing two materials from a stress–strain graph — always make a direct comparison using data from the graph, state which material has the higher breaking stress or greater stiffness, and give a conclusion. Vague qualitative statements without reference to values will not reach the top mark band.
Across this topic, every key quantity — density, spring constant, tensile stress, tensile strain, and elastic strain energy — is defined precisely, has a specific unit (or is dimensionless in the case of strain), and appears in at least one equation. Knowing which equation to reach for, and applying it with consistent SI units, is what separates a confident answer from a dropped mark.
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-09-12.
Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
The SI unit of density is kg m⁻³, because density is mass per unit volume, ρ = m/V, with mass in kilograms and volume in cubic metres. Steel has a density of around 7800 kg m⁻³, while wood is roughly 600 kg m⁻³.
Density is mass per unit volume, ρ = m/V, in kg m⁻³. For a metal cylinder of mass 0.540 kg, radius 0.020 m and length 0.215 m, V = πr²l = 2.70 × 10⁻⁴ m³, so ρ = 0.540 ÷ 2.70 × 10⁻⁴ = 2000 kg m⁻³. For an irregular object, the volume is the volume of water it displaces.
The spring constant k is a measure of stiffness, in N m⁻¹, from F = kΔL. A large k means a stiff spring that needs a large force for a small extension, and k equals the gradient of the straight-line section of a force–extension graph.
The elastic limit is the point beyond which a material no longer returns to its original shape when the force is removed. Within it, deformation is elastic and fully recoverable; beyond it, deformation becomes plastic and permanent, the material no longer obeys Hooke's law, and the force–extension graph curves.
The density of an irregular object comes from its mass and its volume, measured by Archimedes' displacement method: the object is submerged in water, and the volume of water displaced equals its volume. Density is then ρ = m/V.
The density of a cylinder is ρ = m/V, with volume V = πr²l. A metal cylinder of mass 0.540 kg, radius 0.020 m and length 0.215 m has V = π × 0.000400 × 0.215 = 2.70 × 10⁻⁴ m³, so ρ = 0.540 ÷ 2.70 × 10⁻⁴ = 2000 kg m⁻³.
Tensile stress is measured in pascals, where 1 Pa = 1 N m⁻², because stress is force per unit cross-sectional area, σ = F/A, with force in newtons and area in square metres.
Tensile stress is the force per unit cross-sectional area, σ = F/A, measured in pascals. Tensile strain is the fractional change in length, ε = ΔL/L₀, and has no units. A steel wire 2.50 m long and 0.80 mm in diameter stretched 3.0 mm by 150 N has a stress of about 2.98 × 10⁸ Pa and a strain of 1.2 × 10⁻³.
Strain has no units, because it is a ratio of two lengths, the extension over the original length, ε = ΔL/L₀. An extension of 2 mm in a 1 m wire gives a strain of 0.002, while the same extension in a 10 mm wire gives 0.2.
Density is an intensive property because it is a property of the material itself, not of the object's size. Doubling the volume of a steel block doubles its mass, so the ratio m/V stays the same.
The energy stored in a stretched spring is E = ½k(ΔL)², within the elastic limit. A spring with k = 250 N m⁻¹ extended by 0.12 m stores ½ × 250 × 0.0144 = 1.80 J; squaring 0.12 gives 0.0144, not 0.24.
Hooke's law states that the extension of a spring or elastic material is directly proportional to the applied force, provided the elastic limit is not exceeded: F = kΔL, where F is the force in N, k is the spring constant in N m⁻¹ and ΔL is the extension in m. The gradient of the straight section of a force–extension graph equals k.
Breaking stress is the maximum tensile stress a material can withstand before fracture. It is measured in pascals and is a fixed property of the material, independent of the sample's dimensions. Engineers keep the stress in any structural component much lower than the breaking stress of its material, with a safety factor applied.
Stress in a wire is σ = F/A, where the cross-sectional area is A = πr². A steel wire of diameter 0.80 mm has radius 0.40 × 10⁻³ m and area 5.03 × 10⁻⁷ m², so a 150 N force gives σ = 150 ÷ 5.03 × 10⁻⁷ = 2.98 × 10⁸ Pa.
Stress and strain are used instead of force and extension so that different materials can be compared regardless of their dimensions. A thick steel rod and a thin steel wire behave very differently under the same force, but stress and strain describe the material itself.