Centre of Mass

Definition: Centre of Mass

The centre of mass of a physical system of point masses is an imaginary point mass whose mass $M$ is the sum of the masses $m_1,m_2,\cdots$ of its components and whose position ${\mathbf{r}}_{\text{cm}}$ is the mass-weighted average of the positions ${\mathbf{r}}_1,{\mathbf{r}}_2,\cdots$ of its components:

$M\stackrel{\text{def}}{=}{\sum }_k{m}_k{\mathbf{r}}_{\text{cm}}\stackrel{\text{def}}{=}\frac{1}{M}{\sum }_k{m}_k{\mathbf{r}}_k$