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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.

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    $\begingroup$ Possible duplicate of What's the difference between centre of mass & centre of gravity for massive bodies? $\endgroup$
    – Mitchell
    Commented Dec 25, 2017 at 12:37
  • $\begingroup$ @Mitchell I'm asking why the weight of a body can't be thought of as acting on the center of mass? I thought the center of mass equation is derived for general external forces. Which part of the derivation is not valid when the external force is gravity? $\endgroup$
    – user78237
    Commented Dec 25, 2017 at 12:48
  • $\begingroup$ At least give that question a read. $\endgroup$
    – Mitchell
    Commented Dec 25, 2017 at 12:50
  • $\begingroup$ @Mitchell I did. The accepted answer just gives two different formulas. Why does gravity pull an object at the point calculated by the second formula while of all other types of forces act on the point given by first formula? $\endgroup$
    – user78237
    Commented Dec 25, 2017 at 13:06
  • $\begingroup$ 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. $\endgroup$
    – stafusa
    Commented Dec 25, 2017 at 13:56

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