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.