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 `p_(i)` 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 `i^(th)` particle relative to the centre of mass
Also, prove using the definition of the centre of mass `Sigmap_(i).=0`
(b) Show `K=K.+(1)/(2)MV^(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 `(1)/(2)MV^(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 `vecL=vecL.+vecRxxvec(MV)` where `vecL.=Sigmavec(r_(i)).xxvec(p_(i)).` is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember `vec(r._(i))=vec(r_(i))-vecR`, rest of the notation is the velcities taken relative to the centre of mass. Remember `vec(r._(i))=vec(r_(i))-vecR` rest of the notation is the standard notation used in the chapter. Note `vecL`, and `vec(MR)xxvecV` can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
(d) Show `vec(dL.)/(dt)=sumvec(r_(i).)xxvec(dp.)/(dt)` Further, show that `vec(dL.)/(dt)=tau._(ext)` where `tau._(ext)` is the sum of all external torques acting on the system about the centre of mass. (Hint : Use the definition of centre of mass and Newton.s Thrid Law. Assume the internal forces between any two particles act along the line joining the particles.)

Explanation

Here `vec(r_(i))=vec(r_(i.))+vecRandvec(V_(i))=vec(V_(i))=vecV_(i).+vecV`
where `vec(r_(i.)) and V_(I.))` are the position vector and velocity of `i^(th)` particle with respect to centre of mass O.

(a) Momentum of `i^(th)` particle
`vecp=m_(i)vecV_(i)`
`=m_(i)(vecV_(i).+vecV)`
`=m_(i)vecV_(i).+m_(i)vecV`
`vecp=vecp_(i)+m_(i)vecV`
(b) Kinetic energy of system
`K=(1)/(2)Sigmam_(i)v_(i)^(2)`
`=(1)/(2)summ_(i)vecV_(i).vecV_(i)`
`=(1)/(2)summ_(i)(vecV_(i).+vecV).(vecV_(i).+vecV)`
`=(1)/(2)summ_(i)v_(i)^(.2)+(1)/(2)summ_(i)v_(i)^(.2)+summ_(i)vecV_(i).vecV`
`=(1)/(2)MV^(2)+K.`
where `M=Sigmam_(i)=` total mass of system
`K.=(1)/(2)underset(i)summ_(i)v_(i)^(2)`
`(1)/(2)MV^(2)` = kinetic energy of centre of mass
Now, `underset(i)summ_(i)vecV_(i)..vecV=summ_(i)(vecdr_(i))/(dt).vecV`
`=(d)/(dt)(summ_(i)vec(r_(i).)).vecV`
`=(d)/(dt)(vec(MR).vecV)`
`=0`
(c ) Total momentum of system of particle
`vecL=vec(r_(i))xxvecp`
`=(vec(r_(i).)+vecR)xxunderset(i)summ_(i)vecV_(i)`
`=(vec(r_(i).)+vecR)xxunderset(i)summ_(i)(vecV_(i).+vecV)`
`=underset(i)sum(vecRxxm_(i)vecV)+underset(i)sumr_(i)+m_(i)vec(V_(i).)`
`+(underset(i)summ_(i)vec(r_(i).))xxvecV+vecRxxunderset(i)summ_(i)vecV_(i)`
`=underset(i)sum(vecRxxm_(i)vecV)+sumvec(r_(i).)+m_(i)vecV_(i)`
`+(underset(i)summ_(i)vec(r_(i).))xxvecV+vecRxx(d)/(dt)(underset(i)summ_(i)vec(r_(i).))`
There should not be last two terms
`+(underset(i)summ_(i)vec(r_(i).))xxvecV+vecRxx(d)/(dt)(underset(i)summ_(i)vec(r_(i).))`
`=MvecR-MvecR=0`
From the defination of centre of mass
`underset(i)sum(vecRxxm_(i)vecV)=vecRxxvec(MV)`
Hence `vecL=vecRxxMvecV+underset(i)sumvec(r_(i).)xxvecp_(i)`
`vecL=vecRxxMvecV+vecL.`
where `vecL.=Sigmavec(r_(i)).xxvec(p_(i))`
(d) From final answer
`vecL.=Sigmavec(r_(i)).xxvec(p_(i))`
`(vec(dL).)/(dt)=sumvec(r_(i).)xx(dvec(p_(i)))/(dt)+sum(dr_(i))/(dt)+vec(p_(i))`
`=sumvec(r_(i).)xx(dvec(p_(i)))/(dt)`
`=sumvec(r_(i).)xxF_(i)^(vec(ext))=vec(tau_(ext)^(.))`
and `sum(vec(dr)_(i).)/(dt)xxvecp_(i)=sum(vecdr_(i).)/(dt)xxvec(mv)_(i)=0`
total torque
`vectau=sumvec(r_(i))xxF_(i)^(vec(ext))`
`=sum(vec(r_(i).)+vecR)xxF_(i)^(vec(ext))`
`=sumvec(r_(i).)xxF_(i)^(vec(ext))+vecRxxsumF_(i)^(vec(ext))`
`=tau_(ext)^(vec.)+tau_(0)^(vec(ext))`
where `vec(tau_(0))` torque about point O
`tau_(ext)^(vec.)=sumvec(r_(i))xxF_(i)^(vec(ext))`
`=sumvec(r_(i))xx(vec(dp)_(i).)/(dt)`
`=(d)/(dt)sum(vec(r_(i))xxvec(p_(i)))`
`=(dvecL.)/(dt)`