Question

Establish the relation between torque and angular acceleration. Hence define the moment of inertia.

Answer 1

Hint:The relation between torque and angular acceleration can be established using Newton’s second law, which is force exerted is directly proportional to the acceleration produced, force is the product of mass and acceleration.

Complete step by step answer:
We are given to establish a relation between torque and angular acceleration.
According to Newton’s second law, Force acting on a body is mass times acceleration. Acceleration is the rate of change of velocity per unit time.
Consider a force F acting on an object.
Let r be the distance from the axis of rotation to the point of application of force.
Let us apply Newton’s second law for the tangential component of the force $F_T$ which produces tangential acceleration $a_{\tan }$.
From Newton’s second law,
$F_{\tan }=m\times a_{\tan }\to (1)$
We know that tangential acceleration is the product of angular acceleration and the distance from the axis of rotation to the point of application force, $a_{\tan }=r\alpha$
Where α is the angular acceleration
Substituting the expression for tangential acceleration in equation:
$F_{\tan }=m{a}_{\tan }=mr\alpha$
If we multiply both sides by r, the equation becomes
$F_{\tan }\times r=mr\alpha \times r$
The left side of the equation is torque, torque is the product of force acting on the body and the distance.
$F_{\tan }r=torque=m{r}^2\alpha$
Let $m{r}^2=I$
Where I is the moment of inertia and Moment of inertia of a body is the quantity expressing a body’s tendency to resist angular acceleration.
$\therefore Torque=I\alpha$

Note: Work is the product of force and displacement in a linear motion whereas torque is the product of tangential force and angular distance to rotate the object about an axis in circular motion. The units are the same for both but work and torque are different.