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9-1 CENTER OF MASS 215

The Center of Mass

We define the center of mass(com) of a system of particles (such as a person) in
order to predict the possible motion of the system.

Here we discuss how to determine where the center of mass of a system of parti-
cles is located. We start with a system of only a few particles, and then we consider
a system of a great many particles (a solid body, such as a baseball bat). Later in
the chapter, we discuss how the center of mass of a system moves when external
forces act on the system.

Systems of Particles

Two Particles.Figure 9-2ashows two particles of masses m 1 andm 2 separated by dis-
tanced. We have arbitrarily chosen the origin of an xaxis to coincide with the particle
of mass m 1 .We definethe position of the center of mass (com) of this two-particle sys-
tem to be

(9-1)

Suppose, as an example, that m 2 0. Then there is only one particle, of mass m 1 ,
and the center of mass must lie at the position of that particle; Eq. 9-1 dutifully reduces
toxcom0. If m 1 0, there is again only one particle (of mass m 2 ), and we have, as we
expect,xcomd. If m 1 m 2 , the center of mass should be halfway between the two
particles; Eq. 9-1 reduces to again as we expect. Finally, Eq. 9-1 tells us that
if neither m 1 norm 2 is zero,xcomcan have only values that lie between zero and d; that
is, the center of mass must lie somewhere between the two particles.
We are not required to place the origin of the coordinate system on one of
the particles. Figure 9-2bshows a more generalized situation, in which the coordi-
nate system has been shifted leftward. The position of the center of mass is now
defined

as (9-2)

Note that if we put x 1 0, then x 2 becomesdand Eq. 9-2 reduces to Eq. 9-1, as
it must. Note also that in spite of the shift of the coordinate system, the center

xcom

m 1 x 1 m 2 x 2

m 1 m 2

.

xcom^12 d,

xcom

m 2

m 1 m 2

d.

Figure 9-1(a) A ball tossed into the air

follows a parabolic path. (b) The center

of mass (black dot) of a baseball bat

flipped into the air follows a parabolic

path, but all other points of the bat

follow more complicated curved paths.

(a)

(b)

Richard Megna/Fundamental Photographs

The center of mass of a system of particles is the point that moves as though

(1) all of the system’s mass were concentrated there and (2) all external forces

were applied there.

Figure 9-2(a) Two particles of masses m 1 andm 2 are separated by distance d. The dot
labeled com shows the position of the center of mass, calculated from Eq. 9-1. (b) The
same as (a) except that the origin is located farther from the particles. The position of
the center of mass is calculated from Eq. 9-2. The location of the center of mass with
respect to the particles is the same in both cases.

x

y

xcom

x 1 d

com

m 1 m 2

x 2

(b)

x

y

xcom

d

com

m 1 m 2

(a)

This is the center of mass

of the two-particle system.

Shifting the axis

does not change

the relative position

of the com.