I thought the center of mass equation was derived for general forces, $$\sum{\vec{F_{ext}}}=M\vec{a_{CM}}$$
Then suddenly when the external force on the $i$ particle is of the form $m_ig_i$, where $g_i$ varies throughout the body, we have to use this equation:
$$\vec{W}=M\vec{a_{CG}}$$
where $CG$ is a different point than $CM$. So, what makes gravity special?
EDIT: I'm not asking the difference between them. They have different formulas, so obviously they have different values when $g$ is not constant.
I'm asking why doesn't the resultant gravitational force or $W$ can't be thought of as acting on the center of mass when the center of mass equation is derived for any general external force.
