Moment of inertia:

According to Newton’s first law of motion, if there is no external force is applied on a body, the body continuous in its state of rest or of uniform motion in a straight line. The property of the body by virtue of which it tends to resist the change its state of rest or of uniform motion in a straight line by itself is known as inertia. For translatory motion, the value of inertia depends only the mass of the body. The greater is the mass greater is the inertia. Kinetic energy in translational motion also depends upon the mass and the linear velocity of the body. But when a body rotates about an axis, the kinetic energy of the rotation is determined not only by its mass and angular velocity but also depends upon the position of the axis about which the body is rotated and the distribution of mass about that axis.

If a body is rotated about an axis PQ with an angular velocity , all the particles have the same angular velocity but the linear velocities are different due to the different distance from the axis of rotation.

Let us consider, the linear velocities of the particles of mass m_{1}, m_{2}, m_{3} —– , distance r_{1}, r_{2}, r_{3} —– are respectively v_{1}, v_{2}, v_{3} —— from the axis of rotation PQ.

The kinetic energy of the body is the sum of the kinetic energies of the various particles and it is given by K.E. = m_{1}v_{1}^{2} + m_{2}v_{2}^{2 }+ ——-

Since v = r, then the K.E. = + ^{ } ——-

K.E. = (mr^{2} ) = I.

Where I is the moment of inertia of the body about the axis PQ and it is equal to mr^{2}.

The moment of inertia of a body about an axis is defined as a sum of the products of the mass and the square of the distance of the different particles of a body from the axis of rotation.

The moment of inertia may also be defined as the twice the kinetic energy of a rotating body when its angular velocity is unity.

Radius of gyration:

If the total mass of the body is supposed to be concentrated at a point such that the kinetic energy of rotation of that point is the same as that of the body itself then the distance of that point from the axis of rotation is called radius of gyration of the body about that axis.

K.E. = I = (mr^{2} )= M

Therefore, Mk^{2} = mr^{2} = mn()

Where n is the number of particles each of mass m into which the given mass M is divided.

Therefore, k =

Hence the radius of gyration is the square root of the mean square distance of the particles of the body from the axis of rotation.

Physical significance of moment of inertia:

Moment of inertia plays the same role in rotatory motion as a mass does in linear motion. So, moment of inertia is an analogue of mass in linear motion.

According to Newton’s first law if there is no external force is applied on a body, the body continues in its state of rest or of uniform motion in a straight line. This property is known as inertia. A body always resists the external force tending to change its state of rest or of linear motion. Therefore, for the body of greater mass, greater force is required to produce a given linear acceleration.

Similarly, bodies process rotational inertia. When a body is free to rotate about an axis, it opposes any change in its state of rest or of rotation. Greater the moment of inertia of a body greater is the couple required to produce a given angular acceleration. The moment of inertia depends not only on the mass of a body but also on the distribution of mass about the axis of rotation.