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Well, my math was alright, but my English and terminology was completely screwed up:

First, the c.g. is equal to the sum of the arms, divided by the total weight. The arm is the distance to the center of mass of each component, times its distance from our origin (in this case, the bottom of the can).

The c.g. is the total moments of the components, divided by their weight. The moment of any component is the distance of its center of mass from the appropriate datum (here, the bottom of the can) times its mass.

I think everything else was essentially correct.

And I saw that Rev pulled out the calculus, did the derivative, noted that only the numerator of the derivative needed to have its roots found, and showed that that yields the same answer.

Yay! Math works!

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