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312 CHAPTER 10 Systems of Particles

Likewise
dp 2
F F (10.9)
dt 2,int 2,ext

If we add the left sides of these equations and the right sides, the contributions from
the internal forces cancel (that is, F F 0), since they are action–reaction
1,int 2,int
pairs. What remains is
dp dp
1 2
F 1,ext F 2,ext (10.10)
dt dt
The sum of the rates of change of the momenta is the same as the rate of change
of the sum of the momenta; hence,
d(p p )
1
2
F 1,ext F 2,ext (10.11)
dt
The sum P p p is the total momentum, and the sum F F is the
1 2 1,ext 2,ext
For any number of total external force on the particle system. Thus, Eq. (10.11) states that the rate of
particles, the mutual
forces of each pair are change of the total momentum of the two-particle system equals the total external
m 1 equal and opposite. force.
For a system containing more than two particles, we can obtain similar results. If
the system is isolated so that there are no external forces, then the mutual interparti-
m 3
cle forces acting between pairs of particles merely transfer momentum from one par-
ticle of the pair to the other, just as in the case of two particles. Since all the internal
forces necessarily arise from such forces between pairs of particles, these internal forces
cannot change the total momentum. For example, Fig. 10.5 shows three isolated par-
m 2
ticles exerting forces on one another. Consider particle 1; the mutual forces between
FIGURE 10.5 Three particles exerting particles 1 and 2 exchange momentum between these two, while the mutual forces
forces on each other. As in the case of two between particles 1 and 3 exhange momentum between those two. But none of these
particles, the mutual forces between pairs of
momentum transfers will change the total momentum. The same holds for particles
particles merely exchange momentum
2 and 3. Consequently, the total momentum is constant. More generally, for an isolated
between them.
system of n particles, the total momentum P p p p obeys the con-
1 2 n
servation law
momentum conservation for a P [constant] (no external forces) (10.12)
system of particles

If, besides the internal forces, there are external forces, then the latter will change
the momentum.The rate of change can be calculated in essentially the same way as for
the two-particle system, and again, the rate of change of the total momentum is equal
to the total external force. We can write this as

Second Law for a system of particles d P F (10.13)
dt ext

where F F F F is the total external force acting on the system.
ext 1,ext 2,ext n,ext
Equations (10.12) and (10.13) have exactly the same mathematical form as Eqs.
(10.2) and (10.3), and they may be regarded as the generalizations for a system of par-
ticles of Newton’s First and Second Laws. As we will see in Section 10.3, Eq. (10.13)
is an equation of motion for the system of particles—it determines the overall trans-
lational motion of the system.