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 and
dynamics 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_{cm} we 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:
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 m_{2} 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} =  F_{21}. 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.