A Classical Approach of Newtonian Mechanics

6 CONSERVATION OF MOMENTUM 6.3 Multi-component systems

X

X X

fj i

Figure 48: A 3 - dimensional dynamical system consisting of many point mass objects.

Finally, suppose that the ith object is subject to an external force Fi.

Newton’s second law of motion applied to the ith object yields

j/=i

mi ̈ri = fij + Fi. (6.17)

j=1,N^

Note that the summation on the right-hand side of the above equation excludes

the case j = i, since the ith object cannot exert a force on itself. Let us now take

the above equation and sum it over all objects. We obtain

(^) j i
mi ̈ri =
i=1,N i,
X
j=1,N fij^ +^ i=1,NF i.^ (6.18)^
Consider the sum over all internal forces: i.e., the first term on the right-hand
side. Each element of this sum—fij, say—can be paired with another element—
fji, in this case—which is equal and opposite. In other words, the elements of the
sum all cancel out in pairs. Thus, the net value of the sum is zero. It follows that
the above equation can be written
M ̈rcm = F, (6.19)
fi j Fi
mi
ri Fj^
mj
rj