Angular momentum

Angular momentum:

The quantity in rotational mechanics, analogous to linear momentum is angular momentum or moment of linear momentum. It is similar to the torque being moment of a force. If p is the instantaneous linear momentum of a particle undertaking a circular motion, its angular momentum at that instance is given by  L = r x p, where r is the position vector from the axis of rotation. In magnitude, it is the product of linear momentum and its perpendicular distance from the axis of rotation. L = P x r sinθ, where θ is the smaller angle between the directions of P and r. 

Expression for Angular Momentum in Terms of Moment of Inertia:

For a rigid body with a fixed axis of rotation, all these angular momenta are directed along the axis of rotation, and this direction can be obtained by using right hand thumb rule. As all of them have the same direction, their magnitudes can be algebraically added. Thus, magnitude of angular momentum of the body is given by 

Expression for Torque in Terms of Moment of Inertia:

Figure shows a rigid object rotating with a constant angular acceleration α about an axis perpendicular to the plane of paper. For theoretical simplification let us consider the object to be consisting of N number of particles of masses m1 , m2 , ….. mN at respective perpendicular distances r1 , r2 , …..rN from the axis of rotation. As the object