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Physics 53
Systems of Particles
Everything should be as simple as it is, but not simpler.
— Albert Einstein
Overview
An object of ordinary size — which we call a “macroscopic” system — contains a huge
number of atoms or molecules. It is out of the question to attempt to use the laws we
have discussed for a single particle to describe separately each particle in such a system.
There are nevertheless some relatively simple aspects of the behavior of a macroscopic
system that (as we will show) follow from the basic laws for a particle. We will be able
to show that in any multiparticle system each of the mechanical quantities relevant to
the state of the system consists of two parts:
1.
One part in which the system is treated as though it were a single particle (with
the total mass of the system) located at a special point called the
center of mass
.
This part is often called the
CM motion
.
2.
Another part describing the
internal motion
of the system, as seen by an observer
located at (and moving with) the center of mass.
In this section we will give a general analysis of the variables to be used in describing
the motion of
any
system of particles, large or small. In later sections we will apply this
analysis to the case of solid objects, in the approximation that all the particles in the
object have a Fxed spatial relation to each other. In this approximation, the object is
called a “rigid body.”
Still later we will discuss ±uids, including gases, where things are more complicated
because the internal motion is dominant, and where we must take a statistical approach
in the analysis.
Center of mass
We consider a system composed of
N
point particles, each labeled by a value of the
index
i
which runs from 1 to
N
. Each particle has its own mass
m
i
and (at a particular
time) is located at its particular place
r
i
.
The
center of mass (CM)
of the system is deFned by the following position vector:
PHY 53
1
Systems
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View Full DocumentCenter of mass
r
CM
=
1
M
m
i
r
i
i
∑
where
M
is the total mass of all the particles.
This vector locates a point in space — which may or may not be the position of any of
the particles. It is the
massweighted average
position of the particles, being nearer to the
more massive particles.
As a simple example, consider a system of only two particles, of masses
m
and
2
m
, separated by a distance
. Choose the
coordinate system so that the less massive particle
is at the origin and the other is at
x
=
, as shown
in the drawing. Then we have
m
1
=
m
,
m
2
=
2
m
,
x
1
=
0
,
x
2
=
. (The
y
and
z
coordinates are zero of course.) We Fnd from the
deFnition
x
CM
=
1
3
m
m
⋅
0
+
2
m
⋅
( ) =
2
3
. The location is indicated on the drawing.
As time goes on the position vectors of the particles
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 Spring '07
 Mueller
 Physics

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