Centre of mass

Given points $A_1, A_2, \ldots, A_n$ with masses $m_1, m_2, \ldots, m_n$, the centre of mass is the weighted average of their positions. In coordinates, if $A_i = (x_i, y_i)$, the centre of mass $G$ is at $\left(\frac{\sum m_i x_i}{\sum m_i}, \frac{\sum m_i y_i}{\sum m_i}\right)$. If all masses are equal, this is just the average of the coordinates.

Important properties

• The centre of mass lies inside the convex hull of the points if all masses are positive.

• If the system is balanced at the centre of mass, it will not tip over.

• The centre of mass is independent of the order of points.

• For two points $A$ and $B$ with masses $m$ and $n$, the centre of mass divides $AB$ in the ratio $n:m$.