1. Concept Overview
When a material is stretched or compressed slightly, the deformation produced is directly proportional to the force applied. This simple and beautiful linear behaviour is described by Hooke’s Law. It works only for small deformations where the material behaves elastically and returns to its original shape once the force is removed.
This law forms the backbone of elasticity and helps define important concepts like Young’s modulus, bulk modulus, and shear modulus.
2. Definition
3. Understanding Hooke’s Law
3.1. Linear Relationship Between Stress and Strain
Hooke’s Law states that if you double the stress, the strain also doubles—provided the material has not crossed its elastic limit. This straight-line relation means the material behaves like a perfect spring for small deformations.
The general form is:
\( \sigma = E \epsilon \)
Here:
- \( \sigma \) = stress
- \( \epsilon \) = strain
- \( E \) = elastic modulus (Young's modulus for stretching/compression)
3.2. The Material Acts Like a Spring
Inside a solid, atoms behave somewhat like tiny springs. When a force stretches or compresses the material, these internal springs pull back, creating a restoring force. This restoring force is responsible for the linear relation between stress and strain.
3.2.1. Spring Form of Hooke’s Law
For a simple spring, Hooke’s Law is written as:
\( F = -k x \)
where:
- \( F \) is the restoring force
- \( x \) is the extension or compression
- \( k \) is the spring constant
The negative sign shows that the restoring force acts in the opposite direction of the deformation.
3.3. Elastic Limit and Limit of Proportionality
3.3.1. Elastic Limit
The maximum stress up to which a material returns completely to its original shape after removing the force is called the elastic limit. Beyond this point, permanent deformation begins.
3.3.2. Limit of Proportionality
This is the point up to which stress is exactly proportional to strain. Hooke’s Law works only up to this point. After this, the stress–strain curve begins to bend and the relationship becomes non-linear.
4. Graphical View of Hooke’s Law
On a stress–strain graph, the region where Hooke’s Law is valid appears as a straight line passing through the origin. The slope of this line gives the elastic modulus of the material.
A steeper slope means the material is stiffer.
5. Importance of Hooke’s Law
5.1. Basis of Elastic Moduli
Hooke’s Law forms the foundation for defining Young’s modulus, shear modulus, and bulk modulus. All these moduli describe how much a material resists different types of deformation.
5.2. Used in Everyday Applications
Springs, weighing machines, shock absorbers, bridges, and many mechanical systems rely on Hooke’s Law. Even small parts of buildings and vehicles are designed using this relation to ensure stability under load.
6. Examples to Build Intuition
6.1. Stretching a Small Spring
When a light weight is hung on a spring, it extends by a small amount. If the weight is doubled, the extension also doubles. This is a direct example of Hooke’s Law at work.
6.2. Pulling a High-Quality Metal Wire
A thin metal wire elongates by a very small amount when pulled. This elongation is proportional to the applied force as long as the wire stays inside its elastic region.
6.3. Small Deformation in a Bridge
A bridge bends slightly under the weight of vehicles. The bending is proportional to the load, provided the stress remains within the elastic limit. Engineers use this behaviour to ensure safety.