Separation of motion of a system of particles into motion of the centre of mass and motion about the centre of mass: (a)n Show vecp= vecp_(i)+m_(i)vecV where vecp_(i) is the momentum of the i^(th) particle (of mass m_(i)) and vecp_(i)=m_(i)vec v_(i), Not vec v_(i) is the velocity of the i^(th) particle relative to the centre ofmass. Also, prove usingthe definition of the centre of mass sum vec p_(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 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) (c) Show vecL= vecL+ vecR xxMV where vecL'=sum vec r_(i) xx vec p_(i) is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember vecr_(i)=vecr_(i)-vecR, reat of the notation is the standard notation used in the chapter. Note vecL' and M vecR xx vecV can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles. (d) Show (d vecL)/(dt)= sum vecr_(i)xx(d vecp')/(dt) Further show that (dvecL')/(dt)= vec tau_(ext)'where Text 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 Third Law. Assume the internal forces between any two particles aet along the line joining the particles.)