# What is Hooke’s Law? Definition, formula, Graph, Applications

By Mohit Uniyal|Updated : October 14th, 2022

In the 19th century, while studying springs and elasticity, Robert Hooke, an English scientist, noticed that numerous materials exhibited an analogous property when the stress-strain correlation was studied. Robert Hooke formulated a mathematical expression for the same, termed Hooke's law.

Hooke's Law PDF

There was a linear region where the force required to extend the material was proportional to the extension of the material. This is referred to as Hooke's Law. In this article, let us thoroughly learn about Hooke’s law, its definition, formula, graph, and applications.

## What is Hooke’s Law?

Stress and strain take completely different forms in unlike situations. Normally, the stress and strain are proportional to each other for tiny deformations, and this is referred to as Hooke’s Law. Steel displays linear-elastic behavior in most engineering applications; Hooke's law is valid for it throughout its elastic span. Hooke's law is only valid for a part of the elastic range for some other materials, such as aluminum.

### Hooke’s Law Definition

Hooke’s Law states that: “Hooke’s law defines that the strain of the substance is proportional to the applied stress within the elastic limit of that substance.”

When the elastic materials are stretched or compressed, the atoms and molecules disfigure until the force is applied, and when the force is removed, they return to their previous state.

## Hooke’s law Formula

Mathematically, within the proportional limit of a material Hooke’s law formula, important for the GATE exam is expressed as follows-

### σ = Eε

Where in SI units

## Hooke’s Law Graph

The figure below shows the stress-strain curve for mild steel.

The material shows elastic behavior up to the proportional limit; beyond that, the material loses elasticity and shows plasticity.

From the emergence to the proportional limit nearing yield strength, the straight line suggests that the material follows Hooke’s law. Beyond the elastic limit between the proportional limit and yield strength, the material no longer has its elasticity and shows plasticity. Candidates can expect 1-2 questions based on this in the GATE question paper. The area from emergence to the proportional limit under the curve falls under the elastic span. The area under the curve from a proportional limit to the fracture point/rupture falls under the plastic span.

The material’s ultimate strength is based on the stress-strain curve's maximum ordinate value (from emergence to rupture). The value gives the rupture strength at a point of rupture.

## Hooke’s Law Applications

Hooke’s Law is used in all branches of science and engineering to understand elastic materials' behavior as per the GATE syllabus; there is no replacement for Hooke's law. Following are some of the applications of Hooke’s Law:

• It is used as a fundamental principle behind the manometer, spring scale, and the balance wheel of the clock.
• Hooke’s law sets the foundation for seismology, acoustics, and molecular mechanics.

Even though Hooke’s law is used widely in Engineering, it’s not a universal principle. The law is not applicable as soon as the elastic limit of a material is surpassed. Usually, for solid particles, Hooke’s law provides precise results when the deformations are small. Numerous materials deviate from Hooke’s law even well before reaching the elastic limit. Following are some of the disadvantages of Hooke’s Law:

• Hooke’s law comes to an end when applied beyond the elastic limit of a material.
• Hooke’s law is correct for solid bodies if the forces and deformations are small.
• Hooke’s law isn’t a universal principle and solely applies to materials as long as they aren’t stretched well past their capacity.
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## FAQs on Hooke’s Law

• Hooke’s law applies to any elastic entity of random complexity, as long as a single number can convey the deformation and the stress. Hooke’s law is applicable within the proportional limit.

• Hooke’s Law is linear. It states that the restoring force is proportional to the displacement within its proportional limit.  If shown on a graph, the line depicting force as a function of displacement should show a direct variation.

• Hooke’s law applies to a perfectly elastic material. It does not apply beyond the elastic limit of any substance. So we can state that Hooke's law holds good up to proportional limit.

• Objects that quickly regain their original shape after being deformed by a force, often obey Hooke's law. Hooke's law only holds good for some materials under definite loading conditions. Steel obeys Hooke's law throughout its elastic span. For aluminium, Hooke's law is only valid for a portion of the elastic span.

• Hooke’s Law is crucial because it helps us to acknowledge how an elastic object will behave when stretched or compacted within its proportional limit. It is important to enhance the technology by understanding the material behaviour properties.

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