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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ Possible duplicate of What's the difference between centre of mass & centre of gravity for massive bodies? $\endgroup$ – Mitchell Dec 25 '17 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? $\endgroup$ – Dove Dec 25 '17 at 12:48
  • $\begingroup$ At least give that question a read. $\endgroup$ – Mitchell Dec 25 '17 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? $\endgroup$ – Dove Dec 25 '17 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 Dec 25 '17 at 13:56