Simple Nature - Light and Matter

l/No matter what point you

hang the pear from, the string

lines up with the pear’s center

of mass. The center of mass

can therefore be defined as the

intersection of all the lines made

by hanging the pear in this way.

Note that the X in the figure

should not be interpreted as

implying that the center of mass

is on the surface — it is actually

inside the pear.

m/The circus performers

hang with the ropes passing

through their centers of mass.

The highjumper and the wrench are both complicated systems,

each consisting of zillions of subatomic particles. To understand

what’s going on, let’s instead look at a nice simple system, two pool

balls colliding. We assume the balls are a closed system (i.e., their

interaction with the felt surface is not important) and that their

rotation is unimportant, so that we’ll be able to treat each one as a

single particle. By symmetry, the only place their center of mass can

be is half-way in between, at anxcoordinate equal to the average

of the two balls’ positions,xcm= (x 1 +x 2 )/2.

Figure j makes it appear that the center of mass, marked with

an×, moves with constant velocity to the right, regardless of the

collision, and we can easily prove this using conservation of momen-

tum:

vcm= dxcm/dt

=

1

2

(v 1 +v 2 )

=

1

2 m

(mv 1 +mv 2 )

=

ptotal

mtotal

Since momentum is conserved, the last expression is constant, which

proves thatvcmis constant.

Rearranging this a little, we haveptotal=mtotalvcm. In other

words, the total momentum of the system is the same as if all its

mass was concentrated at the center of mass point.

Sigma notation
When there is a large, potentially unknown number of particles,
we can write sums like the ones occurring above using symbols like
“+...,” but that gets awkward. It’s more convenient to use the
Greek uppercase sigma, Σ, to indicate addition. For example, the
sum 1^2 + 2^2 + 3^2 + 4^2 = 30 could be written as

∑n

j=1

j^2 = 30,

read “the sum fromj= 1 tonofj^2 .” The variablejis a dummy

variable, just like the dxin an integral that tells you you’re integrat-

ing with respect tox, but has no significance outside the integral.

Thejbelow the sigma tells you what variable is changing from one

term of the sum to the next, butjhas no significance outside the

sum.

As an example, let’s generalize the proof ofptotal=mtotalvcmto

the case of an arbitrary numbernof identical particles moving in

one dimension, rather than just two particles. The center of mass

Section 3.1 Momentum in one dimension 143