Center of Mass

Consider a system of particles 1,2,...,n. Their total mass is:

M = Si=1,...,n mi

Their total momentum is:

= Si=1,...,n i = Si=1,...,n mi i

Define the center-of-mass coordinate vector:

cm = (1/M) Si=1,...,n mi i

The x-coordinate of the center of mass is, for example:

\[ \rm \mathbf{x_{cm} = \frac{m_{1}x_{1} + m_{2}x_{2} + ... + m_{n}x_{n}}{M}} \]

Why is this useful?

We can find the equation of motion for the center of mass by taking two time derivatives and multiplying with M => Newton's Second Law for a system of particles:

Note that only the net external force enters here. The internal forces between the particles pairwise add up to 0 because of Newton's Third Law.

Consequences:

  1. No matter how complicated the motion of the individual particles, the motion of the center of mass is usually very simple.
  2. If we calculate in a coordinate system that is moving with the center of mass, then the total momentum is 0 in this coordinate system, and all momenta of all particles must add up to 0.