Center of Mass of Two Point Masses

Applying the general center-of-mass formula to the simplest possible case — just two point masses $m_1$ and $m_2$ on the X-axis, at positions $x_1$ and $x_2$ — gives three equivalent, but differently convenient, expressions depending on where the origin is placed.

1. Origin placed arbitrarily, both masses on the positive X-axis. The general formula gives directly

   $x_{CM}=\frac{m_1x_1+m_2x_2}{m_1+m_2}.$

2. Origin placed at one of the masses. Choosing the origin to coincide with $m_1$ (so $x_1=0$) simplifies this to

   $x_{CM}=\frac{m_2x_2}{m_1+m_2}.$

   This case makes a general fact plain: the center of mass always lies closer to whichever of the two masses is larger, since $x_{CM}$ is only a fraction $\dfrac{m_2}{m_1+m_2}$ of the full separation $x_2$ from the heavier mass $m_1$.
3. Origin placed at the center of mass itself. If the origin coincides with the center of mass, $x_{CM}=0$ by construction, and $m_1$ turns out to sit on the negative side of this origin (at $-x_1$) while $m_2$ sits on the positive side (at $+x_2$). Substituting into the general formula and simplifying yields $m_1x_1=m_2x_2$ …