Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: [Phys-L] Ex: Re: Fluids problem

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 <> wrote:

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

On Dec 8, 2020, at 6:44 AM, John Denker via Phys-l <> 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.

Forum for Physics Educators
CAUTION: This email was not sent from a Cal Poly Pomona service. Exercise caution when clicking links or opening attachments. Please forward suspicious email to<>.

Forum for Physics Educators