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.
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.
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} = |
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} = |
a _{cm} = |
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:
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.
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} |
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:
F _{ext} = Ma _{cm} |
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.
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.
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} |
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.
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.