When a body rotating about a fixed axis, each particle of the body moves in a circle with linear velocity `v_(i)=r_(i)omega`, where `i=1,2,…,n`
The kinetic energy of motion of this particle is
`K_(i)=(1)/(2)m_(i)v_(i)^(2)`
`=(1)/(2)m_(i)r_(i)^(2)omega^(2) [because v_(i)=r_(i)omega]`
where every particle has mass `m_(i)` and distance from axis is `r_(i)` and `i=1,2,...,n` are no. of particles `omega` is constant for all particle.
`therefore K=(1)/(2)omega^(2)underset(i=1)overset(n)summ_(i)r_(i)^(2)`
Where K is the total energy of all particles.
Here `underset(i=1)overset(n)summ_(i)r_(i)^(2)=I` is known as moment of inertia.
If `m_(1),m_(2),...,m_(n)` are the masses of the particles of a rigid body and `r_(1),r_(2),...,r_(n)` are their perpendicular distance from a given axis, then the sum `m_(1)r_(1)^(2)+m_(2)r_(2)^(2)+....+m_(n)r_(n)^(2)` is calle the moment of inertia of the body corresponding to the given axis. Thus,
`therefore I=m_(1)r_(1)^(2)+m_(2)r_(2)^(2)+....m_(n)r_(n)^(2)`
`=underset(i=1)overset(n)summ_(i)r_(i)^(2)`
Defination of moment of inertia : The sum of the terms obtained by multiplying the masses of individual particles of a rigid body with the square of their respective perpendicular distance form a specified axis is called the moment of inertia of that body w.r.t. the selected axis.
Moment of inertia depends upon position and orientation of the axis of rotation, shape, size of the body and distribution of mass of the body about the axis of the rotation.
Moment of inertia is independent from the magnitude of angular velocity. Which the characteristics of motion of rigid body.
Mass is a inertia for linear velocity and moment of inertia is a inertia for rotational motion.
Equation of linear motion are `vecp=mvecv and vecF=mveca`, corresponding these equation, equation in rotational motion are `vecL=Ivecomega and vectau=Ivecalpha`.
The role of mass in linear motion is similar to the role of moment of inertia in rotational motion.
SI unit of moment of inertia is kg `m^(2)` or `Js^(2)` and dimensional formula is `[M^(1)L^(2)T^(0)]`.
