Hooke’s Law
When you stretch a rubber band gently and release it, it snaps back to its original shape. When you compress a car spring and let go, it bounces back. This ability of a solid to return to its original form after a deforming force is removed is called elasticity. Hooke's Law is the foundational mathematical statement describing how elastic materials behave: within a certain limit, the deformation of a solid is directly proportional to the force causing that deformation. Named after the 17th-century English scientist Robert Hooke, the law establishes a linear relationship between stress — the internal force per unit area within a material — and strain — the fractional change in dimension. Think of a spring: double the weight you hang on it, and it stretches exactly twice as far (up to a point). This beautifully simple proportionality makes Hooke's Law one of the most useful relationships in all of engineering and materials science, governing everything from skyscraper design to the vibration of guitar strings. Crucially, Hooke's Law holds only up to the elastic limit of a material. Beyond this point, the material enters a plastic regime, where deformation becomes permanent and the linear relationship breaks down. Understanding this boundary is just as important as knowing the law itself — engineers must always ensure structures operate well within the elastic limit to guarantee safe, reversible behavior.
- Defining Stress
- Defining Strain
- The Linear Relationship
- The Elastic Limit and Beyond
- Spring Form of Hooke's Law
Defining Stress
Stress (σ) is the internal restoring force a material develops per unit cross-sectional area when an external force is applied. Imagine you are pulling a steel rod at both ends with a force F. Every tiny cross-section inside the rod is being pulled apart. Stress = F / A, where A is the cross-sectional area perpendicular to the force. The SI unit is the pascal (Pa = N/m²). Stress tells us how intensely the material is being loaded — a thinner rod under the same force has higher stress.
Defining Strain
Strain (ε) measures how much a material has deformed relative to its original size. For a rod of original length L₀ that stretches by ΔL, the longitudinal strain is ε = ΔL / L₀. Strain is dimensionless — it is a pure ratio. A strain of 0.01 means the object has changed length by 1% of its original length. Strain does not depend on the size of the object, making it a fair way to compare deformations across different objects.
The Linear Relationship
Hooke's Law states that, within the elastic limit, stress is directly proportional to strain: σ ∝ ε. The constant of proportionality is called the elastic modulus (E), also known as Young's modulus for longitudinal stress and strain. So σ = E × ε. On a stress–strain graph, this region appears as a straight line passing through the origin, with slope equal to E. The steeper the slope, the stiffer the material — steel (E ≈ 200 GPa) is much stiffer than rubber (E ≈ 0.01–0.1 GPa).
The Elastic Limit and Beyond
As stress increases past the elastic limit (also called the proportionality limit), the stress–strain curve bends away from the straight line. The material may still be elastic for a short while (returning to original shape), but the linear proportionality is lost. Beyond the yield point, permanent (plastic) deformation occurs. Eventually, at the ultimate tensile strength, the material fractures. Engineers design with a safety factor, ensuring actual stresses stay far below the elastic limit.
Spring Form of Hooke's Law
For a simple spring, Hooke's Law takes the familiar form F = −kx, where F is the restoring force exerted by the spring, x is the displacement from the equilibrium (natural) length, and k is the spring constant (in N/m). The negative sign indicates that the spring force always opposes the displacement — push the spring in, it pushes back out. A higher k means a stiffer spring. This form is mathematically equivalent to σ = Eε, because k encodes the material stiffness and geometry of the spring together.
The Key Equation
- \sigma — Stress (Pa (N/m²)): The internal force per unit cross-sectional area acting within the material.
- E — Young's Modulus (Elastic Modulus) (Pa (N/m²)): Material-specific constant measuring stiffness; the ratio of stress to strain in the linear region.
- \varepsilon — Strain (dimensionless): The fractional deformation; ratio of change in length ΔL to original length L₀.
- F — Applied Force (N): The external tensile or compressive force applied to the material.
- A — Cross-sectional Area (m²): The area of the material's cross-section perpendicular to the applied force.
- \Delta L — Extension (or compression) (m): The change in length of the material due to the applied force.
- L_0 — Original Length (m): The natural, unstretched length of the material before any force is applied.
Worked Example
Problem. A steel wire of length 2.0 m and cross-sectional area 4.0 × 10⁻⁶ m² is subjected to a tensile force of 8000 N. Young's modulus of steel is 2.0 × 10¹¹ Pa. (a) Calculate the stress in the wire. (b) Calculate the strain. (c) Find the extension of the wire.
Answer. Stress = 2.0 GPa; Strain = 0.010 (1.0%); Extension = 2.0 cm
Origins · 1 minRobert Hooke and how this was discovered
Robert Hooke first stated his law of elasticity in 1676, originally as a Latin anagram 'ceiiinosssttuv' — an encoded secret to establish priority — which he later revealed to stand for 'Ut tensio, sic vis' (As the extension, so the force). This succinct phrase captures the essence of the linear relationship that bears his name. Beyond elasticity, Hooke made pioneering contributions to microscopy (coining the word 'cell'), astronomy, architecture, and the study of fossils. Hooke had a famously bitter rivalry with Isaac Newton, and the two clashed over credit for the inverse-square law of gravity. No authenticated portrait of Hooke is known to survive — it is rumored that Newton, as President of the Royal Society, ensured Hooke's portrait was removed or lost after his death.
Related Concepts
- Young's Modulus — Young's modulus is the proportionality constant E in Hooke's Law, quantifying a material's stiffness; understanding it is the natural next step after grasping the stress–strain relationship.
- Elastic Potential Energy — When a spring or material obeys Hooke's Law (F = kx), the work done in stretching it is stored as elastic potential energy U = ½kx², directly derived from the linear force law.
- Simple Harmonic Motion — The spring form of Hooke's Law (F = −kx) is the defining force law for simple harmonic motion; a mass on a spring oscillates because the restoring force is proportional to displacement.
- Bulk Modulus — Bulk modulus extends the concept of Hooke's Law to three-dimensional compression (volume stress and volume strain), completing the picture of elastic moduli.