- Centre of Mass of a Two-Particle System
Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass:
a) Show pi = p’i + miV
Where pi is the momentum of the ith particle (of mass mi) and p′ i = mi 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'_t = 0`
b) Show K = K′ +1/2MV2
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 MV2/2 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 where `L' = sumr'_t xx p'_t` is the angular momentum of the system about the centre of mass with
velocities taken relative to the centre of mass. Remember `r'_t = r_t - 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
d) Show `dL'/dt = sum r'_t xx (dp')/dt`
Further, show that
`(dL')/(dt) = t'_"ext"`
where `t'_"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 Third Law. Assume the
internal forces between any two particles act along the line joining the particles.)