A Classical Approach of Newtonian Mechanics

6 CONSERVATION OF MOMENTUM 6.3 Multi-component systems

i=1 i=1

.

.

X

i=1. (6.20)

i=1 , (6.21)

N

i=1 mi^

N

i=1 mi^

where M =

.N

mi is the total mass, and F =

.N

Fi is the net external force.

The quantity rcm is the vector displacement of the centre of mass. As before, the

centre of mass is an imaginary point whose coordinates are the mass weighted

averages of the coordinates of the objects which constitute the system. Thus,
.N
mi ri

According to Eq. (6.19), the motion of the centre of mass is equivalent to that

which would be obtained if all the mass contained in the system were collected

at the centre of mass, and this conglomerate mass were then acted upon by the

net external force. As before, the motion of the centre of mass is likely to be far

simpler than the motions of the component masses.

Suppose that there is zero net external force acting on the system, so that

F = 0. In this case, Eq. (6.19) implies that the centre of mass moves with uniform
velocity in a straight-line. In other words, the velocity of the centre of mass,

.N

mi ̇ri

is a constant of the motion. Now, the momentum of the ith object takes the form

pi = mi ̇ri. Hence, the total momentum of the system is written

N

P = mi ̇ri. (6.22)

i=1

A comparison of Eqs. (6.21) and (6.22) suggests that P is also a constant of the

motion when zero net external force acts on the system. Finally, Eq. (6.19) can

be rewritten
dP
= F. (6.23)
dt
In other words, the time derivative of the total momentum is equal to the net
external force acting on the system.

It is clear, from the above discussion, that most of the important results ob-

tained in the previous section, for the case of a two-component system moving in

1 - dimension, also apply to a multi-component system moving in 3 - dimensions.

r (^) cm =
̇r (^) cm =