What do you understand by torque and moment of inertia ? Obtain the relation between torque and moment of inertia for a rigid body.

Explanation

Torque : Torque is a measure of how much a force acting on an object causes that object to rotate.
`tau=rxxF`
Moment of Inertia : The moment of inertia of a particle about an axis is given by the product of the mass of the particle and the square of the distance of the particle from the axis of rotation. `I=sigma mr^(2)`
Derivation : Suppose a body acted upon by a torque `tau` is rotating about an axis passing through a fixed point O. Let it has a constant angular acceleration `alpha` The angular acceleration of all particles of the body will be the same `(alpha)` but their linear acceleration will be different.
`a_(1)=alphar_(1)`
If `F_(1)=m_(1)a_(1)=m_(1)r_(1)alpha`
The moment of this force about the axis :
`tau_(1)=F_(1)xxr_(1)=m_(1)r_(1)alphar_(1)=m_(1)r_(1)^(2)alpha`
Similarly if the masses of other particle be `m_(2),m_(3)` . . . and their distance from the axis be `r_(2),r_(3)` . . .
Then the torques will be `m_(2)r_(2)^(2)alpha,m_(3)r_(" 3")^(2)alpha` . ..
The sum of torque
`tau=m_(1)r_(1)^(2)alpha+m_(2)r_(2)^(2)alpha+m_(3)r_(3)^(2)alpha+ . . . .`
`tau=(sigmam r^(2))alpha`
But `sigmamr^(2)=I`
`tau=I.alpha`