Center of Mass

Center of mass or barycenter (2 bodies): $r_c$ is distance of center from $m_1$ along line between bodies
$r$ is distance between bodies
$m$ is masses

$$r_c=r\frac{m_2}{m_1+m_2}$$

Center of mass or barycenter (n>2 bodies):
(Sum of product of all masses and their associated positions)

$$\overrightarrow{r_c}=\frac{\Sigma \vec{p}m}{\Sigma m}$$

Mechanical Equilibrium

$\Sigma F$ Must equal zero to maintain linear state
$\Sigma \tau$ about any point must equal zero to maintain rotational state
If you put the point you are calculating at one of the forces, there’s less algebra