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

This topic interested me. I looked at three cases. John Denker's frustum of a right circular cone, David Bowman's "flared horn" (parabolic case), and a spherical "zone and segment of two bases" (i.e. segment of sphere after two parallel slices). In each case, R_1 is the lower radius, R_2 is the upper radius, h is the vertical distance between bases, and R is the radius of the equivalent (by volume, V)) right circular cylinder with height h.

The technique I used was simple. Set the formula for the object's volume equal to the volume of the equivalent cylinder and solve for the equivalent cylinder's radius R.

1) Frustum of a right circular cone

V = (pi/3)*h*(R_1^2 + R_1*R_2 + R_2^2)
V_cyl = pi*h*R^2
R = sqr((R_1^2 + R_1*R_2 + R_2^2)/3)

This agrees with John Denker's result for his N = 1 (conic) case

2) Flared (Parabolic) Horn

I integrated the volume of a parabola rotated about the y-axis with R_1 the bottom (larger)radius of revolution, R_2 the top (smaller) radius of revolution, and h = vertical distance between the bottom and top:

Parabola: x = a*y^2 + c
a = (R_2 - R_1)/h^2
c = R_1
V= pi*h*((3*R_2^2 + 4*R_1*R_2 + 8*R_1^2)/15)
V_cyl = pi*h*R^2
R = sqr((3*R_2^2 + 4*R_1*R_2 + 8*R_1^2)/15)

For a similar case, David Bowman got:

R = sqr((R_2^2 + R_2^3/2 * R_1^1/2 + R_2* R_1 + R_2^1/2 * R_1^3/2 + R_1^2)/5)

This does not agree with my expression. However, both expressions yield the correct result in the limiting case when a = 0 (i.e. R_2 = R_1). So, the "parabola" is simply a straight vertical line rotated about the y=axis and the volume is that of a right circular cylinder with radius R = R_1 = R_2.

I assume this means that David's parabola choice is somehow different from mine and both results are correct.

3) Zone and Segment of Sphere

V = (pi*h/6)*(3*R_1^2 + 3*R_2^2 + h^2)
V_cyl = pi*h*R^2
R = sqr((3*R_1^2 + 3*R_2^2 + h^2)/6)

This case explicitly involves the height h in the expression for R. It does check in the hemispherical case (R_1 = R_sph, R_2 = 0, h = R_sph ) and full sphere case (R_2 = R_1 = 0, h = 2*R_sph).

Don Polvani

-----Original Message-----
From: Phys-l <phys-l-bounces@mail.phys-l.org> On Behalf Of David Bowman
Sent: Saturday, May 22, 2021 10:38 PM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] rms / conic / arithmetic / geometric averages

For some reason JD's post reminds me of an xkcd comic of a while back.
https://xkcd.com/2435/

Also I tried looking at a generalization of his 'average' radius truncated cone
average problem wherein the trucated cone's sides are not straight lines but are
more generally power law curves so that for the generic truncated "cone" it looks
like the flared bell of a horn, and the exponent, p, of the power law is a
parameter of the horn's shape. So p=1 is an ordinary truncated cone, p=2
corresponds to a horn with a parabolic/quadratic bell flare, and p=3 is, likewise, a
cubically flared horn bell, etc. It ends up in such a situation the appropriate
formula for the 'mean' radius is
R_avg = √[((r_2^(2 + 1/p) - r_1^(2 + 1/p))/( r_2^(1/p) - r_1^(1/p)))/(2*p+1)]
& where r_1 & r_2 are the radii at each of the ends of the horn. For such a
shape the only 2 values of p correspond to instances of JD's formula for his
generic average. Those 2 values are p = 1 which gives the N=1 ordinary conic
case (i.e. a cheerleader's horn), and p=1/2 which gives the N=0 RMS case. (But
because the p=1/2 value is less than 1 the 'horn' is not really flared like horn, but
is more shaped like a truncated bowl since the curve is reversed.) Other values
of p give an average (above) that don't correspond to JD's N formula because
even when the above quotient of differences of powers results in a finite
telescoping sum there are many cross terms of products of various powers
where the sum of powers on each such cross term is 2. But JD's formula just
has 1 kind of cross term with a single power for each radial factor with a
coefficient of N. The quotient of differences results in a finite sum when p is
either a natural number or a positive half-integer. For other values of p the
quotient of differences does not divide out to a finite sum. As an illustration of
what I mean consider the p = 2 case. This generates the 5 term finite
telescoping sum in the overall square root:
R_avg = √[(r_2^2 + r_2^(3/2)*r_1^(1/2) + r_2*r_1 + r_2^(1/2)*r_1^(3/2 ) +
r_1^2))/5] which does not correspond to JD's N=3 case. Similarly, the p = 5/2
generates a finite 6 term telescoping sum:
R_avg = √[(r_2^2 + r_2^(8/5)*r_1^(2/5) + r_2^(6/5)*r_1^(4/5) + r_2^(4/5)*r_1^(6/5)
+ r_2^(2/5)*r_1^(8/5 ) + r_1^2))/5], and the p = 3 case generates a finite 7 term
telescoping sum:
R_avg = √[(r_2^2 + r_2^(5/3)*r_1^(1/3) + r_2^(4/3)*r_1^(2/3) + r_2*r_1 +
r_2^(2/3)*r_1^(4/3) + r_2^(1/3)*r_1^(5/3 ) + r_1^2))/7], etc.

BTW, in case anyone is interested, the geometric mean of the harmonic mean
and the arithmetic mean of any 2 numbers is their geometric mean, i.e.
GM(HM(x,y),AM(x,y)) = GM(x,y).

David Bowman

-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@mail.phys-l.org] On Behalf Of John Denker
via Phys-l
Sent: Saturday, May 22, 2021 5:18 PM
To: Forum for Physics Educators <Phys-L@Phys-L.org>
Cc: John Denker <jsd@av8n.com>
Subject: [Phys-L] rms / conic / arithmetic / geometric averages

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Hi --

I recently noticed something cute about averages. Maybe everybody but me
already knew this, but I was surprised.

Context: Suppose you have a round pot or bucket with uniformly tapering sides.
In other words, it is a truncated cone. You know the height, and you know the ID
at the bottom and at the top.

We all know the formula for the volume of a cylinder, and this is "almost" a
cylinder, so maybe we can use the same formula, using some sort of *average*
radius.

What sort of average should we use? RMS? Arithmetic? Geometric????

For a full cone, we know the volume is 1/3 of the volume of a cylinder with the
same diameter and radius. So RMS is no good, because that would predict 1/2
of the volume rather than 1/3.
And arithmetic is no good, because it would predict 1/4 of the volume (half of the
diameter) instead of 1/3 of the volume.
Geometric is even worse, because it would predict 0 instead of 1/3. We want
something that is somewhere between RMS and arithmetic.

The cute thing is that all of the above form a family. You can interpolate between
them. The general formula is:
average = √[(Bot2+N*Bot*Top+Top2) / (2+N)]

for all N from 0 to ∞. In particular:

N=0 --> RMS
N=1 --> conic
N=2 --> arithmetic
N=∞ --> geometric

The relationships are graphed here:
https://www.av8n.com/physics/img48/rms-conic-arith-geom-avg.png

The N=1 average is in fact exactly correct for a truncated cone, as you can
easily verify in lots of ways. (Grind out the integrals.
Or just subtract one full cone from another.)

Related: Here is an online calculator to figure out the volume of a truncated
cone:
https://www.av8n.com/hobbies/volume-calculator.html
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