Hooke’s Law Formula: Definition, Formula, and More

You have landed on the right page to learn about Hooke’s Law Formula. Have you ever stretched or compressed any spring? You need to apply a tensile or compressive force to elongate or compress it. But how will you know that how much force you need to compress or elongate the spring by a given amount? To get the answer to the above questions, we will have to learn about Hooke’s law.

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The opposing force developed by spring is governed by Hooke’s law. Hooke’s law of elasticity was developed by the English scientist Robert Hooke. Does this law apply to any type of material and any amount of deformation? The answer is no. Hooke’s law is only applicable in the elastic region (up to the proportional limit) of elastic material. So, if any elastic material is stretched or compressed beyond the proportional limit, Hooke’s law will not be valid. In this article, we will learn more about elastic material and Hooke’s law.

Elastic Material

A material is said to be elastic if it regains its original shape and size after the load acting on it is removed. If the load vs. deformation graph is drawn for an elastic material, it retraces its path in the opposite direction when the load is removed. This property of the material is called Elasticity. But in inelastic material, the material does not regain its shape, or the graph will not retrace its path. To understand whether the material is elastic or inelastic, we use the stress vs. strain graph.

To some extent, most solid materials exhibit elastic behaviour and recover up to a certain limit of load or applied force. This limit is called the elastic limit. It is defined as the maximum stress or force per unit area inside the material that can arise before the onset of permanent plastic deformation. Stress beyond the elastic limit causes the material to yield or flow. For such materials, the elastic limit marks the end of elastic behaviour, and plastic behaviour starts from there. But most brittle materials do not deform much even beyond the elastic limit.

So, even with elastic material, Hooke’s law will not be valid after a certain amount of deformation. It is valid only up to the proportional limit, and beyond this, Hooke’s law does not give a true result.
In the below graph,

1. Point \(A\) is the proportional limit.
2. Point \(B\) is the elastic limit or upper yield point.
3. Point \(C\) is the lower yield point.
4. Point \(D\) represents ultimate strength.
5. Point \(E\) represents the fracture point.
6. The region under the curve \(O\) to \(B\) is the elastic region, and the region under the curve \(B\) to \(E\) is the plastic region.

Hooke’s Law

According to Hooke’s law, the magnitude of deformation (strain) is exactly proportional to the deforming force or load for small deformations of the material. Under these conditions, the deformation will be within the elastic limit, and the object regains the original shape and size upon removal of the load. Elastic behaviour of the material, according to Hooke’s law, can be explained by the fact that small displacements of their constituent molecules, atoms, or ions from normal positions is also proportional to the force that causes the displacement.
Mathematically, Hooke’s law states that the applied force \(F\) equals \(k\) (Stiffness constant) time the displacement or change in length \(x\),
\(F = -kx\)
The value of \(k\) depends on the type of elastic material, its dimensions, and shape. The negative sign represents that the direction of force is opposite to that of \(x\).

Learn Strain Energy Formula here

Spring Force

When the spring is in natural length, it will not exert any force. But when it is elongated or compressed from its natural length, it will exert force. According to Hooke’s law, the spring force is proportional to the displacement of the object from the equilibrium position (natural length).
Spring force, \(F = -kx\)
Where,
\(k\) is the spring constant,
\(x\) is the displacement from the natural length of the spring.
Here negative sign signifies that the spring force will be developed in the opposite direction to deformation from natural length. The equation also tells that the force acting at each instant during the compression and extension of the spring is varying with displacement from natural length.

Stress-Strain Relation

Hooke’s law can be expressed in the form of stress and strain. Stress is the force on unit areas inside a material that develops as a result of the externally applied load. Strain is the deformation per unit length produced by stress. For relatively small stresses (up to the proportional limit), stress is proportional to strain.
Thus, according to Hooke’s law,
Stress \(\propto \) Strain
Or
Stress \( = K \times {\text{Strain}}\)
Here \(K\) is the proportionality constant and is known as the modulus of elasticity.

Sample problems

Q.1. A spring is elongated by \(5\;\rm{cm}\) by application of force of \(500\;\rm{N}\) and held at that place.
(a) What is the spring constant of the spring?
(b) How much force is required to elongate the spring by \(8\;\rm{cm}\)?
Ans: (a) Given,
Deformation in spring, \(x = 5\;\rm{cm} = 0.05\;\rm{m}\)
The magnitude of external force, \(F = 500\;\rm{N}\)
Now, according to Hooke’s law,
\(F = kx\),
Here \(k\) is the spring constant. Then,
\(k = \frac {F}{x}\)
\( \Rightarrow k = \frac{{500}}{{0.05}} = 10 \times {10^3}\,{\text{N}}\,{{\text{m}}^{ – 1}}.\)
(b) In this case, the elongation is \(8\;\rm{cm}\). Thus we have,
Also, The value of the spring constant, \(k = 10 \times {10^3}\,{\text{N}}\,{{\text{m}}^{ – 1}}\)
Now, according to Hooke’s law,
\(F = kx\),
\(\Rightarrow F = 10 \times {10^3}\,{\text{N}}\,{{\text{m}}^{ – 1}} \times 0.08\,{\text{m}}\)
\( \Rightarrow F = 800\,{\text{N}}.\)

Summary

According to Hooke’s law, for the small deformation of elastic material, the amount of deformation is proportional to the applied force. Hooke’s law is valid for small deformation (up to the proportional limit) in elastic material. Any material is said to be elastic if it regains its original shape and size if the load acting on is removed. The property of material due to which it regains its original shape and size after small deformation due to applied load is called elasticity.
According to Hooke’s law, the applied force \(F\) equals \(k\) (Stiffness constant) times the displacement or change in length \(x\), \(F = -kx\). The negative sign represents that the direction of force is opposite to that of \(x\). The value of \(k\) depends on the kind of elastic material under consideration, dimensions of material, and shape. The constant \(k\) is also called the stiffness constant. When any external stress is applied to the material, the material will deform, and strain will be produced. Hooke’s law can be expressed in the form of stress and strain. For small stresses (under the proportional limit), stress is proportional to strain.
\({\text{Stress}} = K \times {\text{Strain}}\), where \(K\) is the proportionality constant and is known as the modulus of elasticity.

FAQs

Q.1. What is Hooke’s law?
Ans: Hooke’s law states that the displacement or magnitude of deformation is exactly proportional to the deforming force or load for relatively minor deformations of an object.

Q.2. Does all the substance obey Hooke’s law?
Ans: No, certain materials do not obey Hooke’s law. Some of them are rubber, plastic, mud, etc. Hooke’s is valid only for small deformation of elastic material.

Q.3. What does the slope in stress vs. strain graph of elastic material within proportional limit represent?
Ans: Within proportional limit, the stress is directly proportional to strain. The slope represents the stiffness constant described by Hooke’s law. It is also called the modulus of elasticity.

Q.4. Does Hooke’s law is only valid for longitudinal deformation?
Ans: Hooke’s law is valid for all such deformation \((2-D,\;3-D)\) in which the deformation is proportional to force. It is valid for twisting (for small-angle), hydrostatic compression, shear stress vs. shear strain, etc.

Study Stress-Strain Curve here