Moment of Inertia and Torque

Inertia

The last article dealt with inertia and Newton’s first law of motion. Inertia is the tendency of the body to remain in the current motion state. We have, so far, studied the behaviour of objects in linear motion. It requires force to change the motion state of a body. Think of a situation where you have to rotate a bolt using a wrench of certain length (while holding the wrench at its far-edge). Now repeat the same experiment with a longer wrench. You would find it easy to rotate now. The problem is solved for anyone else but not a scientist. For him the job has just begun. A scientist asks ‘how’ a longer wrench made the job easy? Above that ‘why’ does the bolt refuses to rotate? The explanations are elucidated in this article.

The circular motion analogue of inertia is moment of inertia. It is the tendency of an object to not enter into circular motion or come to rest from rotation or to change the rate of rotation. As we have learnt during our Physics tuition class, inertia depends on the mass. Moment of inertia however depends on two things: one being mass and the other being the distribution of mass. If the mass is distributed away from the axis of rotation then the moment of inertia is high and is low otherwise.

A ballet dancer rotates faster when she keeps her hands close to her body and slower when the hands are spread out. The spread-out-mass increases moment of inertia there by decreasing the angular velocity. Having understood that moment of inertia is the analogue of inertia. Let’s discuss force analogue of angular motion. It is called as torque or turning force. Turning force doesn’t just depend on the force applied but also on how its applied. It is given by the product of the length of the wrench and the force applied multiplied with sin of the angle between arm of the wrench and the force applied. So if the length of the wrench is small then more force is required to obtain a given amount of turning force and the converse also holds true. Hence the explanation for long wrench making the job easy is ‘turning force’ and its dependence on the length of the wrench.

The parallels between circular motion and linear motion can also be drawn while comparing force and torque. Force can be written as mass times acceleration (F = m a) and torque can be expressed as (Torque = I x angular acceleration). ‘I’ here stands for moment of inertia and is an analogue of ‘m’ and angular acceleration and acceleration are analogues. The beauty of science is about how things fall at one place despite of apparent differences.