# Why is center of gravity a different quantity than center of mass? [duplicate]

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.

## marked as duplicate by Mitchell, Qmechanic♦Dec 25 '17 at 12:41

• the center of mass equation is derived for any general external force: Not true. The center of mass is simply a (mass) weighted average position, it makes no reference to any force, generic or not. – stafusa Dec 25 '17 at 13:56