In the C.M. frame, the angular momentum of system of particles is independent of the choice of the point, relative to which it is determined.
We denote the angular momentum of the system of particles, relative to the point O, by →M. Since the internal and proper angular momentum ~→M, in the C.M. frame, does not depend on the choice of the point O′, this point may be taken coincident with the point O of the K-frame, at a given moment of time. Then at that moment, the radius vectors of all the particles, in both reference frames, are equal (→ri′=→ri) and the velocities are related by the equation,
→vi=~→vi+→vc
where →vc is the velocity of C.M. frame, relative to the K-frame. Consequently, we may write,
→M=∑mi(→ri×→vi)=∑mi(→ri′×~→vi)+∑mi(→ri×→vc)
or, →M=~→M+m(→rc×→vc), as ∑mi→ri=m→rc, where m=∑mi
or, →M=~→M+(→rc×m→vc)=~→M+(→rc×→p)