ROTATION MOTION

Moment of inertia  
Definition:

Moment of inertia/Rotation inertia of a body about an axis -is a measure of the difficult in starting or stopping or changing the motion of the body about that axis.

Moment of inertia of a rigid body
Consider figure number one, below

Rigid body about an axis of rotation

When a rigid body rotates, it posses purely kinetic energy[K.E]

Where M=mass of rigid body and V=linear velocity of rigid body
Total kinetic energy of the body= sum of kinetic energy of each particle that a body composed

In this case we assume the body have n-particles, but

then we have

is a product of a mass of particle and square of the perpendicular distance from the axis of rotation, this is what called moment of inertia. 

TORQUE AND MOMENT OF INERTIA

From linear motion, torque T_L=force (F) x perp0endicular distance (d)

T_L=Fd

Rotation torque T_R=Fd but F=Mass (m) x acceleration (a) and d=r=perpendicular distance from the axis of rotation then

T_R=ma x r but a=αr where α=angular acceleration

=mαr x r

=mαr² but mr²=I then

T_R=Iα

Conclusion Torque in rotation motion is equivalent to force in linear motion