Hooke’s Law and Modulus of Elasticity

Hooke’s Law

According to Hooke’s law for a small deformation, the stress in a body is proportional to the corresponding strain.” i.e.,

stress ∝ strain

stress = (E) (strain)

Here, E = stress/strain is a constant called modulus of elasticity.

Depending upon the nature of force applied on the body, the modulus of the elasticity is classified in the following three types:

1. Young’s Modulus of Elasticity (Y)

When a wire is acted upon by two equal and opposite forces in the direction of its length, the length of the body is changed. The change in length per unit length (Δl/l)  is called the longitudinal strain and the restoring force (which is equal to the applied force in equilibrium) per unit area of cross-section of wire is called the longitudinal stress.

For small change in the length of the wire, the ratio of the longitudinal stress to the corresponding strain is called theYoung’s modulus of elasticity (Y) of the wire. Thus,

Let there be a wire of length ‘l’ and radius ‘r’. Its one end is clamped to a rigid support and a mass M is attached at the other end. Then

F = Mg and A = πr²

Substituting in above equation, we have

2. Bulk Modulus of Elasticity (B)

When a uniform pressure (normal force) is applied all over the surface of a body, the volume of the body changes. The change in volume per unit volume of the body is called the ‘volume strain’ and the normal force acting per unit area of the surface (pressure) is called the normal stress or volume stress. For small strains, the ratio of the volume stress to the volume strain is called the ‘Bulk modulus’ of the material of the body. It is denoted by B. Then

Here, the negative sign in formula implies that when the pressure increases volume decreases and vice-versa.

Compressibility
The reciprocal of the Bulk modulus of the material of a body is called the “compressibility’  of that material. Thus,

Compressibility = 1/B

3. Modulus of Rigidity (η)

When a body is acted upon by an external force tangential to a surface of the body, the opposite surfaces being kept fixed, it suffers a change in shape of the body, its volume remains unchanged. Then the body is said to be sheared.

The ratio of the displacement of a layer in the direction of the tangential force and the distance of the layer from the fixed surface is called the shearing strain and the tangential force acting per unit area of the surface is called the ‘shearing stress’.

For small strain in the ratio of the shearing stress to the shearing strain is called the ‘modulus of rigidity‘ of the material of the body. It is denoted by ‘η’.