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

Regarding the part of Don P's response where he writes:

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.

Indeed Don's parabolic model and mine *are* quite different. Don's model has a vertical edge at his parabola's vertex at y = 0 & x = R_1. It also curves *inward* away from the initial vertically cylindrical shape base as y increases. The slope is finite and shallowest at the top end.

My parabolic shape has a finite slope everywhere, and the parabola's vertex is out of bounds in the cut off region beyond the small radius base. That vertex corresponds to a zero radius sharp tip if the small end was not cut off. This is analogous to the ordinary frustrum of a cone where the tip of the cone is cut off to make the smaller radius end. My parabola flares outward as the wide radius end is approached so that the slope is steepest in the neck region near the small radius end and shallowest in the flared bell-end region by the wide radius end. This makes the typical horn-shaped flare at the bell end. Don's parabola has a steepest (infinite) slope at the vertex at the wide radius end and shallowest slope at the narrow radius end thus making a blunted bell end rather than a flared bell.

Using Don's notation his parabola is:
x = R_1 - (R_1 - R_2)*(y/h)^2 (where R_1 > R_2).
Note the parabola vertex is at (x_v, y_v) = (R_1, 0).

But my parabola using the same notation is:
x = (√[R_1] - (√[R_1] - √[R_2])*y/h)^2 .
This one has the parabola vertex at (x_v, y_v) = (0, h/(1 - √[R_2/R_1])) which is beyond the small radius end and thus not part of the horn.

Also this formula has the same shape with the same mean radius no matter which end has the larger radius. Exchanging the values of R_1 and R_2 merely turns the horn upside down without changing its shape or volume. That is why my formula for the average radius is symmetric in R_1 & R_2, whereas Don's is quite asymmetric under the exchange R_1 & R_2 since he explicitly has the infinite slope vertex at the R_1 end.

David Bowman