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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 of F_{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}

=

m_{2}a_{2}

Substituting this expression into our center of mass acceleration equation we
get:

Let M be the total mass of the system. Thus M = m_{1} + m_{2} and:

F_{ext} = Ma_{cm}

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:

x_{cm}

=

m_{n}x_{n}

v_{cm}

=

m_{n}v_{n}

a_{cm}

=

m_{n}a_{n}

F_{ext}

=

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.

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.