The Center of Mass

of a System of Particles (AP)

Center of Mass in One Dimension

The animation at right will hopefully convince you that the center of mass of a system of n particles (of constant mass) is given by:

where x_cm is the position of the center of mass, m₁ is the mass of the first particle, x₁ is the position of the first particle, etc.

Velocity of the Center of Mass

Starting with the center of mass equation, it is easy to show that the velocity of the center of mass of a system of n particles, v_cm, is:

where v₁ is the velocity of the first particle, v₂ is the velocity of the second particle, etc., and M is the total mass of the system.

Aside from being very handy when dealing with both inelastic and elastic collisions, what is this equation trying to tell us? Rearanging the last result gives:

On the right side of this equation, m₁v₁ is the momentum of particle 1, m₂v₂ is the momentum of particle 2, etc., so the right side of the equation is the total momentum of the system of particles. Looking at the left side of the equation, Mv_cm would be the momentum of the center of mass of the system - if we assume that all of the mass of the system is located at the system's center of mass. In other words, the equations above are saying that:

1. The total momentum of the system equals the total mass of the system times the velocity of the system's center of mass (Mv_cm).
2. The system behaves as if all of its mass were located at the system's center of mass.

Acceleration of the Center of Mass

In the same way, we can derive an equation for the acceleration of the center of mass, a_cm:

where a_cm is the acceleration of the center of mass of the system, and a₁ is the acceleration of particle 1, etc. Does this equation have anything to say? Well, multiplying both sides by M gives:

Now, m₁a₁ is the net force on particle 1 (Newton's Second Law), m₂a₂ is the net force on particle 2, etc., so the right side is the net force on the system of n particles. The left side of the equation is the net force on the center of mass of the system if all of the mass of the system is located at the system's center of mass. So,

1. The net (external) force on a system of particles equals the mass of the system times the acceleration of the system's center of mass. (Newton's Second Law applied to the center of mass).
2. If the net (external) force on a system of particles is zero, the center of mass of the system will not accelerate (Newton's First Law applied to the center of mass.
3. The system behaves as if all of its mass were located at the system's center of mass.

last update February 5, 2010 by JL Stanbrough