Centre of Mass

The centre of mass ($G$) is a point about which mass is evenly balanced in every direction for a body or system of particles. It is the point through which a net force acts to produce linear acceleration without rotation.

Its position vector is given by (where $M=\sum m_i$ is total mass):
${\vec{r}}_G=\underset{\text{Discrete System}}{\underbrace{\frac{\sum m_i{\vec{r}}_i}{M}}}=\underset{\text{Continuous System}}{\underbrace{\frac{1}{M}\int \vec{r}dm}}$

Alternately, the scalar coordinate form is given by:
$\bar{x}=\frac{\int xdm}{\int dm}\bar{y}=\frac{\int ydm}{\int dm}\bar{z}=\frac{\int zdm}{\int dm}$

Composite Bodies

For a body made of several simple parts, take a mass-weighted average of each part’s own centre of mass:
$\bar{x}=\frac{\sum {\bar{x}}_i{m}_i}{\sum m_i}\bar{y}=\frac{\sum {\bar{y}}_i{m}_i}{\sum m_i}\bar{z}=\frac{\sum {\bar{z}}_i{m}_i}{\sum m_i}$

Holes / Cut-Outs

Treat removed material as negative mass in the sum above.

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