Elasticity

# Hooke’s Law

Robert Hooke, in 1676, studied experimentally the extension produced in a metallic wire by giving various tensions to it. From his experiment, he established a relation between strain produced in the wire and tension applied. He also studied elastic behaviour of coiled springs, metal rods, etc. and summed up his findings in a rule now known as Hooke’s law.

Hooke’s law states that within elastic limit, the elongation produced in a elastic body is directly proportional to the applied tension. $\text{i.e. Tension}∝\text{Elongation}$ $F∝x$ $F=kx$ where, $k$ is force constant or spring constant. It is defined as the force applied per unit elongation. Its value depends upon the shape and nature of the material. It may also be defined as the amount of force needed to extend the body by unit length. Its SI unit is $\text{Nm}^{-1}$.

If we consider restoring force (intermolecular force) instead of applied force, then $F=-kx$ where negative sign indicates that restoring force acts opposite to the elongation.

## Hooke’s Law in terms of Stress and Strain

Now, stress is a measure of the applied deforming force and strain is a measure of the distortion. Therefore, Hooke’s law can be restated in terms of stress and strain, which eliminates the dependency of proportionality constant on shape and size of the body.

We have, $F=kx$ Dividing both sides by area $(A)$, $\frac{F}{A}=\frac{kx}{A}$ $\frac{F}{A}=\frac{kx}{A}\cdot \frac{L}{L}$ $\frac{F}{A}=\frac{kL}{A}\cdot \frac{x}{L}$ $\text{stress}=\text{constant}×\text{strain}$ $\therefore \text{stress}∝\text{strain}$

Thus, Hooke’s law states that within elastic limit, stress produced in the body is directly proportional to strain.

$\frac{\text{stress}}{\text{strain}}=\text{constant }(E)$ where, the proportionality constant $(E)$ is known as modulus of elasticity. Its value depends upon the nature of the substance, the nature of deformation and temperature.

Since stress is measured in $\text{Nm}^{-2}$ and strain has no unit, the unit of modulus of elasticity is same as that of stress, i.e. $\text{dyne cm}^{-2}$ in CGS system and $\text{Nm}^{-2}$ or $\text{Pa}$ in SI system.

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