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_(t) " 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 `sum p_(t)=0`
(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'=sum r'_(t)xxp'_(t)` 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 `MRxxV` can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.
(d) Show `(dL')/(dt)=sum r'_(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.
(Hint : 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.)

Explanation

Here `vec r_(i)=vec r'_(i)+vec R+R " and " vec V_(i)=vec V'_(i)+vec V`
where `vec r'_(i) " and " vec v'_(i)` denote the radius vector and velocity of the `i^(th)` particle referred to centre of mass O as the new origin and `vec V` is the velocity of centre of mass relative to O.
(a)Momentum of `i^(th)` particle
`vec P = m_(i) vec V'_(i)`
`=m_(i) (vec V'_(i)+vec V) ("since" vec V_(i)=vec V'_(i)+vec V)`
or ` vec P=m_(i) vec V'_(i) + m_(i) vec V`
`vec P= vec P_(i) +m_(i) vec V`
(b) KInetic energy of the system of particles.
`K=(1)/(2) sum m_(i) V_(i)^(2)`
`=(1)/(2) sum m_(i) vec V_(i). vec V_(i)`
`=(1)/(2) sum m_(i) ( vec V'_(i)+vec V)(vec V'_(i)+vec V)`
`=(1)/(2) sum m_(i) (V'_(i)^(2) +V^(2)+2vec V'_(i). vec V)`
`=(1)/(2) sum m_(i) V_(i)^(2) +(1)/(2) sum m_(i) V'_(i)^(2) +sum m_(i) vec V_(i). vec V`
`=(1)/(2) MV^(2)+K'`
where `M=sum m_(i)`
= total mass of the system
`K'=(1)/(2) underset(i) sum m_(i) V'_(i)^(2)`
=kinetic energy of motion about the centre of mass
or `(1)/(2) Mv^(2)`= kinetic energy of motion of centre of mass. (Proved)
since `underset(i) sum m_(i) vec V"_(i) . vec V = sum m_(i) (dvec r_(i))/(dt). vec V`
`=(d)/(dt)(sum m_(i) vec r'_(i)). vec V=(d)/(dt)(Mvec R.vec V)`
(c ) Total angular momentum of the system of particles.
`vec L =vec r_(i)xxvec p`
`=(vec r_(i)+vec R)xx underset(i)sum m_(i) vec V_(i)`
`= (vec r'_(i)+vec R)xx underset (i)sum m_(i) (vec V'_(i) + vec V)`
` underset(i)sum (vec Rxxm_(i) vec V)+ underset(i)sum vec r'_(i)xxm_(i) vec V'_(i)+(underset(i)sum m_(i)vec r_(i))`
`xx vec V+vec Rxx underset(i)sum m_(i) vec V_(i)`
`=underset(i)sum (vec Rxxm_(i) vec V)+underset(i)sum vec r'_(i)xxm_(i) vec _(i)+(underset(i)sum m_(i) vec r'_(i))`
`xx vec V+vec Rxx(d)/(dt) (underset(i)sum m_(i) vec r'_(i))`
The last two terms vanish for both contain the factor `sum m_(i)vec r'_(i)` which is equal to
`underset(i) sum m_(i) vec r_(i) =underset(i) sum m_(i) (vec r'_(i)-vec R)=Mvec R-Mvec R=vec 0`
from the definition of centre of mass. Also
`underset(i) sum (vec Rxxm_(i)vec V)=vec Rxx Mvec V`
so that ` vec L= vec Rxx Mvec V+underset(i)sum r'_(i)xxvec P_(i)`
or `vec L =vec Rxx Mvec V+vec L'`
(d). where From previous solution
`vecL'=sumvecr_(i)'xx(dvecP_(i))/(dt)=sum(dvecr_(i)')/(dt)xxvecP_(i)`
`=sumvecr_(i)'xx(dvecP_(i))/(dt)`
`=sumvecr_(i)xxvecF_(i)^(ext)=vectau_(ext)'`
since `sum(dvecr_(i)')/(dt)xxvecP_(i)=sum(dvecr_(i)')/(dt)xxmvecv_(i)xxmvecv_(i)=0`
total torque `vectau=sumvecr_(i)xxvecF_(i)^(ext)`
`=sum(vecr_(i)+vecR)xxvecF_(i)^(ext)`
`=sum(vecr_(i)+vecR)xxvecF_(i)^(ext)`
`=sumvecr_(i)'xxvecF_(i)^(ext)+vecRxxunderset(i)SumvecF_(i)^(ext)`
`=vectau_(ext)'+vectau_(0)^(ext)`
where `vectau_(ext)` is the total torque about the centre of mass as origin and `vectau_(0)^(ext)` that about the origin O.
`vectau_(ext)'=sumvecr_(i)'xxvecF_(i)^(ext)`
`=sumvecr_(i)xx(dvecP_(i)')/(dt)=(d)/(dt)underset(i)sum(vecr_(i)xxvecP_(i))=(dvecL')/(dt)`.