Moment of inertia

Moment of inertia:

Expressions of linear momentum, force (for a fixed mass) and kinetic energy include mass as a common term. In order to have their rotational analogues, we need a replacement for mass. If we open a door (with hinges), we give a certain angular displacement to it. The efforts needed for this depend not only upon the mass of the door, but also upon the (perpendicular) distance from the axis of rotation, where we apply the force. Thus, the quantity analogous to mass includes not only the mass, but also takes care of the distance wise distribution of the mass around the axis of rotation. To know the exact relation, let us derive an expression for the rotational kinetic energy which is the sum of the translational kinetic energies of all the individual particles. 

Moment of Inertia of a Uniform Ring:

An object is called a uniform ring if its mass is situated uniformly on the circumference of a circle. It is a two dimensional object of negligible thickness. If it is rotating about its own axis (line perpendicular to its plane and passing through its centre), its entire mass M is practically at a distance equal to its radius R form the axis. Hence, the expression for the moment of inertia of a uniform ring of mass M and radius R is I = MR2. 

Moment of Inertia of a Uniform Disc:

Disc is a two dimensional circular object of negligible thickness. It is said to be uniform if its mass per unit area and its composition is the same throughout. The ratio σ = m / A is called the surface density. Consider a uniform disc of mass M and radius R rotating about its own axis, which is the line perpendicular to its plane and passing through its centre σ = M/(π)R2. As it is a uniform circular object, it can be considered to be consisting of a number of concentric rings of radii increasing from (practically) zero to R. One of such rings of mass dm is shown by shaded portion