9781118230725.pdf

Manipulating Eq. 9-23 by substituting for from Eq. 9-22 gives, for constant
massm,

Thus, the relations and are equivalent expressions of
Newton’s second law of motion for a particle.

F

:

netma

F: :

netdp

:/dt

F

:

net

dp:

dt



d

dt

(mv:)m

dv:

dt

ma:.

p:

9-3 LINEAR MOMENTUM 225

Checkpoint 3

The figure gives the magnitude pof the linear mo-

mentum versus time tfor a particle moving along

an axis. A force directed along the axis acts on the

particle. (a) Rank the four regions indicated ac-

cording to the magnitude of the force, greatest

first. (b) In which region is the particle slowing?

The Linear Momentum of a System of Particles

Let’s extend the definition of linear momentum to a system of particles. Consider
a system of nparticles, each with its own mass, velocity, and linear momentum.
The particles may interact with each other, and external forces may act on them.
The system as a whole has a total linear momentum which is defined to be the
vector sum of the individual particles’ linear momenta. Thus,

(9-24)

If we compare this equation with Eq. 9-17, we see that

(linear momentum, system of particles), (9-25)

which is another way to define the linear momentum of a system of particles:

P

:

Mv:com

m 1 :v 1 m 2 :v 2 m 3 :v 3 



mn:vn.

P

:

:p 1 p: 2 p: 3 



p:n

P

:

,

The linear momentum of a system of particles is equal to the product of the total

massMof the system and the velocity of the center of mass.

Force and Momentum.If we take the time derivative of Eq. 9-25 (the veloc-
ity can change but not the mass), we find

(9-26)

Comparing Eqs. 9-14 and 9-26 allows us to write Newton’s second law for a sys-
tem of particles in the equivalent form

(system of particles), (9-27)

where is the net external force acting on the system. This equation is the gen-
eralization of the single-particle equation to a system of many
particles. In words, the equation says that the net external force on a system
of particles changes the linear momentum of the system. Conversely, the linear
momentum can be changed only by a net external force. If there is no net exter-
nal force, cannotchange. Again, this fact gives us an extremely powerful tool
for solving problems.

P

:

P

:

F

:

net

F

:

netdp

:/dt

F

:

net

F

:

net

dP

:

dt

dP

:

dt

M

dv:com

dt

Ma:com.

p

t

1

2

3

4