Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass :
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.

Explanation

In rotational kinematics, `K.E_(T)` of the system of particles = `K.E_(T)` of the system which the particle velocities are taken w.r.t to CM + K.E of the translation of the system as a whole (i.e. of the C.M motion of the system)
`i.e. (1)/(2)m_(1)v_(1)^(2)+(1)/(2)m_(2)v_(2)^(2)+....`
`=[(1)/(2)m_(1)v._(1)^(2)+(1)/(2)m_(2)v._(2)^(2)+...]+(1)/(2)Mv^(2)`
`implies (1)/(2)m_(i)v_(i)^(2)=(1)/(2)m_(i)v_(i)^(2)+(1)/(2)mv^(2)`
`therefore k=k.+(1)/(2)MV^(2)`
where M = total mass of particles
and v = velocity of C.M motion of the system.