# Moment of Inertia

According to Newton’s first law of motion, everybody continues in its state of rest or of uniform linear motion, unless an external force acts on it to change that state. This inability of a body to change by itself its state of rest or of linear uniform motion is called inertia. Similarly, a body rotating about a given axis tends to maintain its state of uniform rotation, unless an external torque is applied on it to change that state. This property of a body by virtue of which it opposes the torque tending to change its state of rest or of uniform rotation about an axis is called rotational inertia or moment of inertia.

#### Definition of Moment of Inertia

The moment of inertia of a rigid body about a fixed axis is defined as the sum of the products of the masses of the particles constituting the body and the squares of their respective distances from the axis of rotation.

#### Physical significance of moment of inertia:

The mass of a body resists a change in its state of linear motion, it is a measure of its inertia in linear motion. Similarly, the moment of inertia of a body about an axis of rotation resists a change in its rotational motion. The greater the moment of inertia of a body, the greater is the torque required to change its state of rotation. Thus, moment of inertia of a body can be regarded as the measure of rotational inertia of the body. The moment of inertia of a body plays the same role in the rotational motion as the mass plays in linear motion. That is why moment of inertia is called the rotational analogue of mass in linear motion.

#### Factors on which the moment of inertia depends:

The moment of inertia of a body is the measure of the manner in which its different parts are distributed at different distances from the axis of rotation. Unlike mass, it is not a fixed quantity as it depends on the position and orientation of the axis of rotation with respect to the body as a whole.

The moment of inertia of a body depends on

(i)    Mass of the body

(ii)   Size and shape of the body.

(iii) Distribution of mass about the axis of rotation.

(iv) Position and orientation of the axis of rotation w.r.t. the body.

#### Practical applications of moment of inertia

(i) A heavy wheel, called flywheel, is attached to the shaft of steam engine, automobile engine, etc. Because of its large moment of inertia, the flywheel opposes the sudden increase or decrease of the speed of the vehicle. It allows a gradual change in the speed and prevents jerky motions and hence ensures smooth ride for the passengers.

(ii) In a bicycle, bullock-cart, etc., the moment of inertia is increased by concentrating most of the mass at the rim of the wheel and connecting the rim to the axle through the spokes. Even after we stop paddling, the wheels of a bicycle continue to rotate for some time due to their large moment of inertia.