Center of Mass for Distributed Point Masses

To locate the center of mass of a collection of $n$ discrete point masses $m_1,m_2,m_3,\ldots,m_n$, the first step is to choose an origin and a coordinate system. Let $x_1,x_2,x_3,\ldots,x_n$ be the X-coordinates of the positions of these masses relative to that origin (with analogous $y_i$ and $z_i$ coordinates for the Y and Z directions). The X-coordinate of the center of mass is then the total "moment" of all the masses about the origin, $\sum_i m_ix_i$, divided by the total mass $M=\sum_i m_i$:

Exactly analogous expressions hold for the Y and Z coordinates:

Together, $(x_{CM},y_{CM},z_{CM})$ locate the center of mass of the whole distribution of point masses in Cartesian coordinates. These three equations can be combined into one compact vector equation:

where $\vec r_{CM}=x_{CM}\hat\imath+y_{CM}\hat\jmath+z_{CM}\hat k$ is the position vector of the center of mass, and $\vec r_i=x_i\hat\imath+y_i\hat\jmath+z_i\hat k$ is the position vector of the $i$-th distributed point mass, with $\hat\imath,\hat\jmath,\hat k$ the usual unit vectors along the X, Y and Z axes. This vector formula is the fundamental starting point for locating the center of mass of any collection of masses, discrete or (via integration, §5.1.5) continuous. …