MOMENT OF INERTIA
"The moment of inertia of a particle is mathematically defined as the product of its mass and square of distance from the axis of rotation." If we assume a body to be moving along a stright line, then according to Newton's first law of motion,the body by virtue of inertia opposes any change in its motion. Greater the mass of the body , greater is its inertia or opposition to change and hence greater is the force to be applied to bring about a given change. Thus force and mass have got opposite efforts on translatory motion. Force tends to produce acceleration while mass opposes it. Hence mass of a body is taken to be a measure of its inertia for translatory motion. In rotational motion also, a body free to rotate about an axis opposes any change in its state of rotation indicating thereby that the body has got an inertia to this type of motion. This inertia of the body in rotational motion is called rotational inertia or moment of inertia. Thus the term moment of inertia of rotational motion is analogous to the term mass of translational motion.
For a rotating rigid body, moment of inertia is measure of how hard it is to change its state.The greater a body's moment of inertia, the harder it is to start the body rotating if it is at rest and the hander it is to stop its rotation if it is already rotating. That is why, moment of inertia is also called rotational inertia. The moment of inertia of a particle is mathematically defined as the product of its mass and square of distance from the axis of rotation.
Moment of Inertia of a Particle
The moment of inertia of a particle is measured by the product of its mass and square of distance from the axis of rotation.
Let a particle of mass m be at a distance r from the axis of rotation AB(Fig).Then the moment of inertia of the particle about the axis AB is given by I = r². It is neither scalar nor vector. It has different values in different directions (ie.about different axes) so it is a tensor. Nevertheless, we shall treat it as a scalar for the simple case of rotation about a fixed axis.
Moment of Inertia of a Rigid Body
A rigid body is assumed to be made of n-particles. The moment ofinertia of a body about an axis of rotation is the sum of moment ofinertia of all particles of which the body is made.
Let us consider a rigid body made up of n-particles of masses m₁, m₂, ... , mₙ separated by distances r₁, r₂,.... rₙ respectively from axis of rotation AB (Fig). If I be the moment of inertia of a body about axis AB, then we can write
1= M.I. of all the particles of body
=m₁r₁²+m₂r₂²+...+mₙrₙ²
.:. I=Σmr²
Hence algebraic sum of products of masses of each particle and square of distances from the axis of rotation is called moment of inertia of the rigid body.
Unit and Dimension
We know that
I=mr²
or [I] = [mr²]
.:. [I]=[ML²]
S.I.unit→kg m²
C.G.S.System→gm cm²
Factors affecting Moment of Inertia of a Body
1. It depends upon the mass of the body.
2. It depends on distance of the body from the axis of rotation.
3. It depends on position and orientation of axis of rotation. If axis of rotation changes, usually momentof inertia will change.
4. It depends on shape and size of a body for a given axis. If mass is casted in different shapes withthesame axis, moment of inertia will be different.
5. For a given shape, size and axis, it depends on the distribution of mass within the body with respectto rotation axis.Farther the constituent particles of a body are from the axis of rotation, larger will beits moment of inertia. That is why, in case of a hollow and solid body of same mass, radius and shapefor a given axis, moment inertia of hollow body is greater than that of the solid body.
.jpg "moment of inertia of a particle")
