Re: [Phys-L] rms / conic / arithmetic / geometric averages

["Don" <dgpolvani@gmail.com>]

Many thanks to David Bowman for checking my results of 5/24/21 and finding
that my "flared horn" was actually a "blunted horn" or "bulbous horn".  Had
I taken the time to plot the parabola I chose, I would have avoided this
mistake.  At least, the stated volume and equivalent cylindrical radius
results for the "bulbous horn" were correct.  With regard to this matter
and, also, David's second comment about my spherical segment results:

1) My results for a "proper" flared (parabolic) horn (similar to a trumpet
bell) are as follows:

Parabola :	y = a*(x - c)^2
		a = h/(R_1 - R_2)^2
		c = R_1
		V = pi*h*((17*R_1^2 - 14*R_1*R_2 + 3*R_2^2)/6)
		R =   sqr((17*R_1^2 - 14*R_1*R_2 + 3*R_2^2)/6)

Where:	h = vertical height between bottom and top of the paraboloid of
revolution
		R_1 = radius (larger) of bottom base
		R_2 = radius (smaller) of top base
		V= volume of paraboloid of revolution
		R = radius of equivalent cylinder with same height and same
volume

As a high school trumpet player (long, long ago!), I am curious if anyone on
the list knows the exact mathematical shape of a trumpet bell?

2) My results for the zone and segment of two bases of a sphere, as far as I
can see, do not assume that one base is in one hemisphere and the second
base is in the other hemisphere.  David, if I interpret his comment
correctly, seems to feel that the two bases must be in different hemispheres
to result in the R shown.  My derivation does not assume this ("opposite
hemisphere") restriction.  Numerical evaluations of my result also show that
the two bases may be in the same or different hemispheres and still yield
the correct equivalent R.  Also, when the two bases are in the same
hemisphere then:

		min(R_1,R_2) <= R <= max(R_1,R_2)

But, I agree that this inequality need not be satisfied when the two bases
are in different hemispheres.  However, in that case, we still have (at
least, in the numerical case that I tried):

		R < R_sphere

Perhaps, I misunderstood David's comment about the opposite hemisphere
requirement?

All my results simply find the radius of a cylinder of the same height and
same volume as the object in question.  This "equivalent" cylindrical radius
may, or may not, agree with someone's definition of the "average" radius,
but it will always produce the same volume for the same height.

Don Polvani
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> -----Original Message-----
> From: Phys-l <phys-l-bounces@mail.phys-l.org> On Behalf Of David Bowman
> Sent: Tuesday, May 25, 2021 4:32 PM
> To: Phys-L@Phys-L.org
> Subject: Re: [Phys-L] rms / conic / arithmetic / geometric averages
>
> I just rechecked my calculation for Don P's paraboloid of revolution &
found an
> error in it.  My revised calculation now *agrees* with Don's calculation
for his
> paraboloid shape.  Sorry for adding to the confusion.
>
> David Bowman
>
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