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*From*: "Albert J. Mallinckrodt" <ajm@cpp.edu>*Date*: Tue, 8 Dec 2020 15:27:27 +0000

Yes. I’m guessing Peter had that “block“ frozen.

On Dec 8, 2020, at 6:44 AM, John Denker via Phys-l <phys-l@mail.phys-l.org> wrote:

Setting aside typos, the key idea is this:

The center of mass is given by:

∫ X dm / ∫ dm [1]

pretty much by definition, where dm is an element of mass,

and X is position.

Note X can be one dimensional in the simple introductory

situation, or higher-dimensional if you want.

Given the symmetry of the situation, you can find the CM by

inspection, based on physicist's intuition and experience,

without doing the calculus. It's in the middle.

If you want to do the calculus, it's

∫ X dX / ∫ dX

since in this situation dm is proportional to dX.

Turn the crank and find that the CM is halfway between the

limits of integration ... in agreement with the aforementioned

intuition and experience.

This comes up All The Time.

Note that the same formula [1] is also the formula for weighted

average, where dm tells you how things get weighted. In the case

where dm = dX this reduces to a simple unweighted average.

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**Follow-Ups**:**Re: [Phys-L] Ex: Re: Fluids problem***From:*Brian Whatcott <betwys1@sbcglobal.net>

**References**:**[Phys-L] Fluids problem***From:*Peter Schoch <pschoch@fandm.edu>

**Re: [Phys-L] Fluids problem***From:*Carl Mungan <mungan@usna.edu>

**Re: [Phys-L] Fluids problem***From:*John Denker <jsd@av8n.com>

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