A certain system of particles possesses a total momentum `p` and an angular momentum `M` relative to a point O. Find its angular momentum `M^'` relative to a point `O^'` whose position with respect to the point O is determined by the radius vector `r_0`. Find out when the angular momentum of the system of particles does not depend on the choice of the point O.

Explanation

Let position vectors of the particles of the system be `vecr_i` and `overset(rarr')r_i` with respect to the points O and `O^'` respectively. Then we have,
`vecr_i=overset(rarr')r_i+vecr_0` (1)
where `vecr_0` is the radius vector of `O^'` with respect to O.
Now, the angular momentum of the system relative to the point O can be written as follows:
`vecM=sum(vecr_ixxvecp_i)=sum(overset(rarr')r_ixxvecp_i)+sum(vecr_0xxp_i)` [using (1)]
or, `vecM=overset(rarr')M+(vecr_0xxvecp)`, where, `vecp=sumvecp_i`
From (2), if the total linear momentum of the system, `vecp=0`, then its angular momentum does not depend on the choice of the point O.
Note that in the C.M. frame, the system of particles, as a whole is at rest.