Kinetics: Consider a rigid body moving with general motion in space. Axes x-y-z are attached to the body with origin at the mass center G.
Let a particle of infinitesimal mass dm is located by the position vector r. The angular momentum of this particle about the mass center G is given as
where is the absolute angular momentum of the body about the mass center.
If the angular velocity of the body is , the velocity v of the particle dm is
where is the velocity of the center G whose position vector with respect to G is r. Thus,
The first integral is zero, since for the origin at mass center Thus,
If the axes were attached to a fixed point instead of the mass center, v would have been equal to .
The angular momentum about the fixed point O is
This expression is same as . Hence for mass center as well as for fixed point in the body, we shall write a common formula
Putting , substituting in the above equation and carrying out the vector algebra,