Centre of Mass of a Two-Particle System

Consider two point masses $m_1$ and $m_2$, located at positions $\vec r_1$ and $\vec r_2$

respectively. The centre of mass of this two-particle system is defined as the point whose position

vector $\vec R$ is the mass-weighted average of the two particles' positions:

For particles confined to a single straight line (the $x$-axis), this reduces to the scalar form

The centre of mass is, in a precise sense, the single point that behaves as though it carried the entire

mass $m_1+m_2$ of the system -- Section 5.4 makes this idea exact by showing how this point moves under an

external force.

The centre of mass always lies on the straight line joining the two particles. This follows directly

from the defining formula: $\vec R$ is a weighted average of $\vec r_1$ and $\vec r_2$ with positive

weights $m_1/(m_1+m_2)$ and $m_2/(m_1+m_2)$ that add up to exactly $1$, and any such weighted average of

two position vectors must lie somewhere on the segment joining them, never off to one side.

It divides that line in the ratio $m_2 : m_1$, measured from $m_1$ to $m_2$. Writing $d_1$ for the

distance from $m_1$ to the centre of mass and $d_2$ for the distance from the centre of mass to $m_2$, the

defining formula (taking $m_1$ at the origin, so $x_1=0$ and $x_2 = d_1+d_2$) gives …