Hooke’s Law of Elasticity

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Introduction

Elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. So, what is Hooke’s law of elasticity? Ok, let’s see. Hooke’s law was formulated by English scientist Robert Hooke. In the 19th century, Robert Hooke was researching about springs and elasticity. When he learned about the stress-strain relationship, he noticed that many materials exhibited a similar property. A linear region existed in which the force required to stretch the material was proportional to its extension. This is known as Hooke’s law.

Hooke’s Law Statement and Expression

Hooke’s law states that the strain of the material is proportional to the applied stress within the elastic limit of that material. Generally, the stress and strain are proportional for small deformations and this is called Hooke’s law. When elastic materials are stretched, i.e., when you apply stress on those materials, the atoms and molecules deform and when you remove the stress, the materials return to their initial state.

The mathematical expression for Hooke’s law is given below,

F= -kx

Where F is the force, x is the extension in length, and k is the proportionality constant known as the spring constant.

Also, within the elastic limit,

Stress/Strain= Constant

This constant is known as the modulus of elasticity or coefficient of elasticity. The modulus of elasticity depends on the type of materials used and is independent of stress and strain.

Types of Modulus of Elasticity

The modulus of elasticity is of 3 types,

• Young’s modulus of elasticity
• The bulk modulus of elasticity
• Modulus of rigidity

Young’s modulus of elasticity: Young’s modulus of elasticity is the ratio of longitudinal stress and longitudinal strain within the elastic limit. It is denoted by “y”.

Now, if L is the length of the wire, r is the radius and is the increase in the length of the wire by suspending a weight, mg at its one end, then Young’s modulus of elasticity becomes,

The bulk modulus of elasticity: The bulk modulus of elasticity is the ratio of the volume stress and volume strain within the elastic limit.

Rigidity modulus: The rigidity modulus is the ratio of shearing stress or tangential stress and shearing strain or tangential strain.

Applications of Hooke’s Law

Some of the important applications of Hooke’s law are listed below,

• The most common application of Hooke’s law is in springs as the law is applied in springs because of their elasticity
• It is used as a foundation for seismology, acoustics, and molecular mechanics
• It is used as the fundamental principle behind the manometer, the balance wheel of the clock, and a spring scale
• It is used in all branches of science and engineering to understand the behaviour of elastic material

Hooke’s Law Solved Example

Question: A spring displaced by 10 cm is held in place with a force of 1000 N. Calculate the spring constant of the spring.

Solution: First of all we have to convert the unit from cm to m as the spring is displaced by 10 cm but the unit of the spring constant is Newtons per meter. Hence,

10 cm= 0.1 m

Now, we know the Hooke’s law equation,

F= -kx

Given, F= 1000 N

x= 0.1 m

Now, substituting these values in the Hooke’s law equation, we have,

1000 N= -k x 0.1 m

1000 N/ 0.1 m= -k

To calculate the spring constant, negative sign should be neglected. Hence,

1000 N/ 0.1 m= k

Therefore, k= 1000 N/ 0.1 m

k= 10,000 N/m

Hence, spring constant, k= 10,000 N/m