Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass :
(a) Show `p=p_(i)'+m_(i)V`
where pi is the momentum of the ith particle (of mass `m_(i)`) and `p'_(i)=m_(i)v'_(i)`. Note `v'_(i)` is the velocity of the ith particle relative to the centre of mass.
Also, prove using the definition of the centre of mass `sump'_(i)=O`
(b) Show `K=K'+1//2MV^(2)`
where K is the total kinetic energy of the system of particles, K′ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and `MV^(2)//2` is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system). The result has been used in Sec. 7.14.
(c ) Show `L=L'+RxxMV`
where `L'=sumr'_(i)xxp'_(i)` is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember – `r'_(i)=r_(i)-R` , rest of the notation is the standard notation used in the chapter. Note ′ L and `MR × V` can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
(d) Show `(dL')/(dt)=sumr'_(i)xx(dp')/(dt)`
Further, show that
`(dL')/(dt)=tau'_(ext)`
where `tau'_(ext)` is the sum of all external torques acting on the system about the centre of mass.

Explanation

Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.