Inertia of a body is its inability to change by itself its state of rest or of uniform motion in a straight line. Similarly Moment of inertia of a body is its inability to change by itself its state of rest or of uniform rotatory motion about an axis.
The formula for Moment of Inertia of a rigid body is given as,
The moment of inertia of the rigid body (l) = Sum of moments of inertia of
all the particles present in the body.
Let us consider,

m₁, m₂, m₃ = masses of large particles in a rigid body

r₁, r₂, r₃ = distance of the masses from the axis
The formula for Moment of Inertia of a rigid body will be,

 l= m₁r₁² + m₂r₂² + m₃r₃² + …

The above formula can also be expressed as,

m_i = mass of i^th particle

r_i = distance between i^th particle and axis o rotation

Moment of inertia of a particle about an axis is equal to the product of the mass of the particle and square of its distance from the axis.

The formula for Moment of Inertia of a particle is given as,
Moment of inertia of the particle about the axis = mass of the particle (m) × distance of particle (r)
from the fixed axis

Formula for Moment of inertia of the particle about the axis =  mr²
The S.I. unit for moment of inertia is kg m²

Metal rod either expands or contract on heating or cooling. A stress is produced in rod if it is restricted from expanding or contracting. This stress is directly proportional to the thermal strain.

Thermal strain = Change in length / original length
Thermal strain = Δ L / L

Thermal strain = α L ΔT / L —— Linear thermal expansion (Δ L = α L ΔT).
Thermal strain = α ΔT

Young’s modulus of material = Stress / Strain
Y = Stress / α ΔT
So Stress = Y α ΔT

Therefore, Force = Y  A α ΔT

Y = Young’s modulus of elasticity
α = coefficient of linear expansion.
∆T = increase in temperature
A = Area

Ratio of lateral contraction to linear elongation is called Poisson’s ratio. It can also be defined as the ratio of unit transverse strain to unit longitudinal strain.

The formula for Poisson’s ratio ‘σ’ is given as,

Poisson’s ratio = σ =Lateral contraction / Linear elongation.

or

Poisson’s ratio = σ = – ε_t / ε_{l     ——— where} ε_t = transverse strain and ε_l = longitudinal strain

The decrease in thickness in lateral direction is lateral contraction and extension in length is known as linear elongation.

The ratio of tangential force per unit area to angular deformation occurred in body is known as Shearing (or) Rigidity modulus ‘n’ . It can also be defined as the ratio of shearing stress to shearing strain within elastic limit is Modulus of Rigidity ‘n’.
Rigidity modulus = T / θ   —- where T = the tangential force/unit area and θ is the angle of shear measured in radian,

Rigidity modulus = Force /Area  θ

It can also be  Rigidity modulus = Shear Stress / Shear Strain

Shear Stress = Force (F) /Area (A)

Shear Strain = Δ x / h  —— where Δ x is displacement and h is the separation between the top and bottom surfaces.

using the above formula we get,

Rigidity modulus = F h /A Δx

The unit for rigidity modulus is ‘newton metre-2 radian-1’ with symbol N m-2 rad-1.
The other unit which is widely used for rigidity modulus is pascal radian-1(Pa rad-1).