LINEAR MOMENTUM: CONSERVATION OF MOMENTUM
Center
of Mass
Up to this point in our
study of classical mechanics, we have studied primarily the motion of a single
particle or body. To further our comprehension of mechanics we must begin to
examine the interactions of many particles at once. To begin this study, we
define and examine a new concept, the center of mass, which will allow us to
make mechanical calculations for a system of particles.
The
Center of Mass of Two Particles
We start by defining and
explaining the concept of the center of mass for the simplest possible system
of particles, one containing only two particles. From our work in this section
we will generalize for systems containing many particles.
Before quantifying our idea
of a center of mass, we must explain it conceptually. The concept of the center
of mass allows us to describe the movement of a system of particles by the
movement of a single point. We will use the center of mass to calculate
the kinematics anddynamics of
the system as a whole, regardless of the motion of the individual particles.
Center
of Mass for Two Particles in One Dimension
If a particle with
mass m 1 has a position of x 1 and
a particle with mass m 2 has a position
of x 2 , then the position of the center of
mass of the two particles is given by:
|
x cm =
|
|
Thus
the position of the center of mass is a point in space that is not necessarily
part of either particle. This phenomenon makes intuitive sense: connect the two
objects with a light but rigid pole. If you hold the pole at the position of
the center of mass of the objects, they will balance. That balancing point will
often not exist within either object.
Center
of Mass for Two Particles beyond One Dimension
Now that we have the
position, we extend the concept of the center of mass to velocity and
acceleration, and thus give ourselves the tools to describe the motion of a system
of particles. Taking a simple time derivative of our expression for x cmwe
see that:
|
v cm =
|
|
Thus
we have a very similar expression for the velocity of the center of mass.
Differentiating again, we can generate an expression for acceleration:
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a cm =
|
|
With
this set of three equations we have generated the necessary elements of the
kinematics of a system of particles.
From our last equation,
however, we can also extend to the dynamics of the center of mass. Consider two
mutually interacting particles in a system with no external forces. Let the
force exerted on m 2 by m 1 be F 21 ,
and the force exerted on m 1 by m2 by F 12 .
By applying Newton's Second Law we can state that F 12 = m 1 a 1 and F 21 =m 2 a 2 .
We can now substitute this into our expression for the acceleration of the
center of mass:
a cm =
However,
by Newton's Third Law F 12 and F 21 are
reactive forces, and F 12 = - F21 .
Thus a cm = 0 . Thus, if a system of
particles experiences no net external force, the center of mass of the system
will move at a constant velocity.
But what if there is a net
force? Can we predict how the system will move? Consider again our example of a
two body system, with m 1 experiencing an
external force ofF 1 and m 2 experiencing
a force of F 2 . We also must continue to take
into account the forces between the two particles, F 21 and F 12 .
By Newton's Second Law:
|
F 1 + F 12
|
=
|
m 1 a 1
|
|
|
F 2 + F 21
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=
|
m 2 a 2
|
|
Substituting
this expression into our center of mass acceleration equation we get:
F 1 + F 2 + F 12 + F 21 = m 1 a 1 + m 2 a 2
Again,
however, F 12 = - F 21 ,
and we can sum the external forces, producing:
Let
M be the total mass of the system. Thus M = m 1 + m 2 and:
|
|
|
This
equation bears a striking resemblance to Newton's Second Law. In this case,
however, we are not speaking of the acceleration of individual particles, but
that of the entire system. The overall acceleration of a system of particles,
no matter how the individual particles move, can be calculated by this
equation. Consider now a single particle of mass M placed at
the center of mass of the system. Exposed to the same forces, the single
particle will accelerate in the same way as the system would. This leads us to
an important statement:
The
overall motion of a system of particles can be found by applying Newton's Laws
as if the total mass of the system were concentrated at the center of mass, and
the external force were applied at this point.
Systems
of More than Two Particles
We have derived a method of
making mechanical calculations for a system of particles. For simplicity's
sake, however, we only derived this for a two- particle system. A derivation
for an n particle system would be quite complex. A simple extension of our two
particle equations to an n particle system will suffice.
Center
of Mass of Many Particles
Previously, M was
defined as M = m 1 + m 2 .
However, to continue the study of center of mass we must make this definition
more general. If there are n particles in a system, M = m 1 + m 2 + m 3 + ... + m n .
In other words, M gives the total mass of the system. Equipped
with this definition, we can simply state the equations for the position,
velocity, and acceleration of the center of mass of a many particle system,
similar to the two-particle case. Thus for a system of n particles:
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x cm
|
=
|
|
|
|
v cm
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=
|
|
|
|
a cm
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=
|
|
|
|
|
=
|
Ma cm
|
|
These equations require little explanation, as they are identical in form to our two particle equations. All these equations for center of mass dynamics may seem confusing, however, so we will discuss a short example to clarify.
Consider a missile composed
of four parts, traveling in a parabolic path through the air. At a certain
point, an explosive mechanism on the missile breaks it into its four parts, all
of which shoot off in various directions, as shown below.
Figure %: A missile breaking into pieces
What
can be said about the motion of the system of the four parts? We know that all
forces applied to the missile parts upon the explosion were internal forces,
and were thus cancelled out by some other reactive force: Newton's Third Law.
The only external force that acts upon the system is gravity, and it acts in
the same way it did before the explosion. Thus, though the missile pieces fly
off in unpredictable directions, we can confidently predict that the center of
mass of the four pieces will continue in the same parabolic path it had
traveled in before the collision.
Such an example displays
the power of the notion of a center of mass. With this concept we can predict
emergent behavior of a set of particles traveling in unpredictable ways.
We have now shown a way to
calculate the motion of the system of particles as a whole. But to truly
explain the motion we must generate a law for how each of the individual
particles react. We do so by introducing the concept of linear momentum in
the next section.
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