9781118230725.pdf

of mass is still the same distance from each particle. The com is a property of the

physical particles, not the coordinate system we happen to use.

We can rewrite Eq. 9-2 as

(9-3)

in which Mis the total mass of the system. (Here,Mm 1 m 2 .)

Many Particles.We can extend this equation to a more general situation in

whichnparticles are strung out along the xaxis. Then the total mass is Mm 1 

m 2 mn, and the location of the center of mass is

(9-4)

The subscript iis an index that takes on all integer values from 1 to n.

Three Dimensions.If the particles are distributed in three dimensions, the cen-

ter of mass must be identified by three coordinates. By extension of Eq. 9-4, they are

(9-5)

We can also define the center of mass with the language of vectors. First

recall that the position of a particle at coordinates xi,yi, and ziis given by a posi-

tion vector (it points from the origin to the particle):

(9-6)

Here the index identifies the particle, and iˆ,ˆj, and k are unit vectors pointing,ˆ

respectively, in the positive direction of the x,y, and zaxes. Similarly, the position

of the center of mass of a system of particles is given by a position vector:

(9-7)

If you are a fan of concise notation, the three scalar equations of Eq. 9-5 can now

be replaced by a single vector equation,

(9-8)

where again Mis the total mass of the system. You can check that this equation

is correct by substituting Eqs. 9-6 and 9-7 into it, and then separating out the x,

y, and zcomponents. The scalar relations of Eq. 9-5 result.

Solid Bodies

An ordinary object, such as a baseball bat, contains so many particles (atoms)

that we can best treat it as a continuous distribution of matter. The “particles”

then become differential mass elements dm, the sums of Eq. 9-5 become inte-

grals, and the coordinates of the center of mass are defined as

(9-9)

whereMis now the mass of the object. The integrals effectively allow us to use Eq.

9-5 for a huge number of particles, an effort that otherwise would take many years.

Evaluating these integrals for most common objects (such as a television set or

a moose) would be difficult, so here we consider only uniformobjects. Such objects

have uniform density,or mass per unit volume; that is, the density r(Greek letter

xcom

1

M

xdm,^ ycom

1

M

ydm,^ zcom

1

M

zdm,

r:com

1

M 

n

i 1

mir:i,

r:comxcomˆiycomˆjzcomk.ˆ

r:ixiiˆyijˆzikˆ.

xcom

1

M 

n

i 1

mixi, ycom

1

M 

n

i 1

miyi, zcom

1

M 

n

i 1

mizi.



1

M 

n

i 1

mixi.

xcom

m 1 x 1 m 2 x 2 m 3 x 3 



mnxn

M

xcom

m 1 x 1 m 2 x 2

M

,

216 CHAPTER 9 CENTER OF MASS AND LINEAR MOMENTUM