9781118230725.pdf

Newton’s Second Law for a System of Particles

Now that we know how to locate the center of mass of a system of particles, we

discuss how external forces can move a center of mass. Let us start with a simple

system of two billiard balls.

If you roll a cue ball at a second billiard ball that is at rest, you expect that the

two-ball system will continue to have some forward motion after impact. You

would be surprised, for example, if both balls came back toward you or if both

moved to the right or to the left. You already have an intuitive sense that some-

thingcontinues to move forward.

What continues to move forward, its steady motion completely unaf-

fected by the collision, is the center of mass of the two-ball system. If you fo-

cus on this point — which is always halfway between these bodies because

they have identical masses — you can easily convince yourself by trial at a bil-

liard table that this is so. No matter whether the collision is glancing, head-on,

or somewhere in between, the center of mass continues to move forward, as if

the collision had never occurred. Let us look into this center-of-mass motion

in more detail.

Motion of a System’s com.To do so, we replace the pair of billiard balls with

a system of nparticles of (possibly) different masses. We are interested not in the

individual motions of these particles but onlyin the motion of the center of mass

of the system. Although the center of mass is just a point, it moves like a particle

whose mass is equal to the total mass of the system; we can assign a position, a ve-

locity, and an acceleration to it. We state (and shall prove next) that the vector

equation that governs the motion of the center of mass of such a system of parti-

cles is

(system of particles). (9-14)

This equation is Newton’s second law for the motion of the center of mass of

a system of particles. Note that its form is the same as the form of the equation

F

:

netMa

:

com

220 CHAPTER 9 CENTER OF MASS AND LINEAR MOMENTUM

9-2NEWTON’S SECOND LAW FOR A SYSTEM OF PARTICLES

After reading this module, you should be able to...

9.04Apply Newton’s second law to a system of particles by re-
lating the net force (of the forces acting on the particles) to
the acceleration of the system’s center of mass.
9.05Apply the constant-acceleration equations to the motion
of the individual particles in a system and to the motion of
the system’s center of mass.
9.06Given the mass and velocity of the particles in a system,
calculate the velocity of the system’s center of mass.
9.07Given the mass and acceleration of the particles in a
system, calculate the acceleration of the system’s center
of mass.

9.08Given the position of a system’s center of mass as a func-

tion of time, determine the velocity of the center of mass.

9.09Given the velocity of a system’s center of mass as a

function of time, determine the acceleration of the center

of mass.

9.10Calculate the change in the velocity of a com by integrat-

ing the com’s acceleration function with respect to time.

9.11Calculate a com’s displacement by integrating the

com’s velocity function with respect to time.

9.12When the particles in a two-particle system move with-

out the system’s com moving, relate the displacements of

the particles and the velocities of the particles.

●The motion of the center of mass of any system of particles
is governed by Newton’s second law for a system of parti-
cles, which is

F.

:

netMa

:

com

Here is the net force of all the externalforces acting on

the system, Mis the total mass of the system, and is the

acceleration of the system’s center of mass.

:acom

F

:

net

Learning Objectives

Key Idea