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*From*: Brian Whatcott <betwys1@sbcglobal.net>*Date*: Tue, 8 Dec 2020 15:37:39 +0000 (UTC)

Key concept: a cylinder of water shape-shifts to an infinitesimally thin lamina of water when pumped at lowest energy cost. On Tuesday, December 8, 2020, 09:27:40 AM CST, Albert J. Mallinckrodt <ajm@cpp.edu> wrote:

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] [ext] Re: Ex: Re: Fluids problem***From:*Ken Caviness <caviness@southern.edu>

**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>

**Re: [Phys-L] Ex: Re: Fluids problem***From:*"Albert J. Mallinckrodt" <ajm@cpp.edu>

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