Center of Mass

When calculating in single particle systems such as an atom or a defined point, the physical shape and geometry of its mass is insignificant. However in real life, no object is simply a particle and possess a great complications in its physical geometry. When dealing with such problems, we calculate the kinematics relative to its center of mass. But not only is the center of mass used for physical objects, also for multi-particle systems which can be treated as a single entity and the dynamics of each particle is calculated relative to the system's center of mass.

Now our chief concern comes down to "how do you find the center of mass?"
A nice mathematical equation is written for any n-particle system to answer this question:

Generalizing this equation for n-dimensional coordinate system, each coordinate for its respective center of mass coordinate is:

The equations should make sense intuitively; it basically takes the average of the weighted position in the system for all particles.
For example, look at the diagram below:

To find the center of mass for this system, we use a two-coordinate system that contains (x, y).
M = 2kg + 3kg + 5kg = 10kg

rcm = (-3/5, -2/5)

Sample Problem
5 particles have masses as follow: 3kg, 1kg, 4kg, 2kg and 10kg. And the respective position of each particle is
(-2, 2, 3), (-1, 3, 4), (1, 1, 1), (3, 3, 3), (0, 0, 0). Locate the center of mass. (Use fractions, if necessary)