What is Moment of Inertia ?

Moment of inertia (I) in rotatory motion is similar to effect mass of the body in transitory motion.This physical quantity opposes the rotatory motion of the body and to overcome this we need to apply torque.

I is independent of the magnitude of the angular velocity. It is a characteristic of the rigid body and the axis about which it rotates. In rotation (about a fixed axis), the moment of inertia plays a similar role as mass does in linear motion.

As 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, as the moment of inertia about a given axis of rotation resists a change in its rotational motion, it can be regarded as a measure of rotational inertia of the body; it is a measure of the way in which different parts of the body are distributed at different distances from the axis.

Unlike the mass of a body, the moment of inertia is not a fixed quantity but depends on the orientation and position of the axis of rotation with respect to the body as a whole. As a measure of the way in which the mass of a rotating rigid body is distributed with respect to the axis of rotation.

The radius of gyration of a body about an axis may be defined as the distance from the axis of a mass point whose mass is equal to the mass of the whole body and whose moment of inertia is equal to the moment of inertia of the body about the axis.

Thus, the moment of inertia of a rigid body depends on the mass of the body, its shape and size; distribution of mass about the axis of rotation, and the position and orientation of the axis of rotation.(24)

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What is Torque ?

It is a physical quantity similar to force in Linear motion which represents the its effect on the rotatory motion of a body. If the body is fixed at a point or along a line, it has only rotational motion. Force is needed to change the translational state of a body, i.e. to produce linear acceleration.

In the case of rotatory motion it is not only the force that determines the motion and also its position at which it is applied.

Example :

Let opening or closing of a door is the case . A door is a rigid body which can rotate about a fixed vertical axis passing through the hinges. Unless a force is applied the door does not rotate. But any force does not do the job. A force applied to the hinge line cannot produce any rotation at all, whereas a force of given magnitude applied at right angles to the door at its outer edge is most effective in producing rotation. It is not the force alone, but how and where the force is applied is important in rotational motion.

Torque is also called as moment of force. The meaning of moment is always multiply with displacement. That means torque is the vector product of force and displacement.

Expression :

If a force acts on a single particle at a point P whose position with respect to the origin O is given by the position vector r , the moment of the force acting on the particle with respect to the origin O is defined as the vector product

τ = r × F

The moment of force (or torque) is a vector quantity. The symbol τ stands for the Greek letter tau. The magnitude of τ is τ = rF sinθ

where r is the magnitude of the position vector r, i.e. the length OP, F is the magnitude of force F and θ is the angle between r and F as shown.

If τ = 0 if r = 0, F = 0 or θ = 00 or 1800 . Thus, the moment of a force vanishes if either the magnitude of the force is zero, or if the line of action of the force passes through the origin.
Since r × F is a vector product, properties of a vector product of two vectors apply to it. If the direction of F is reversed, the direction of the moment of force is reversed. If directions of both r and F are reversed, the direction of the moment of force remains the same.
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