Re: Saturday Morning Puzzle, Finale

[brian whatcott <inet@INTELLISYS.NET>]

At 22:38 10/13/01 -0700, Bernard Cleyet wrote:
> I suspect there is a law that relates the fact that the sphere is best (for
> fields) with the fact that is is the solid with the least area for a given
> volume.
>
> WRT the original question, I reported that the ca. 80's experimenters used
> cylinders.
....
> *Problem in a Marian ed. for which I don't have the solns. book!
>
> bc

I expect you're right about the superiority of spheres.
I couldn't quite convince myself though, so I wrote a few lines of
code in qbasic
(which should please Ludwik, at least?)

I opposed a cylinder to a sphere. I summed the force contributions of
100 rings of mass of varying radius across the flat face of the cylinder, then
in each of 100 layers to the back face.

This miserably defective code seems to show that when the sphere and cylinder
are separated by one unit, a cylinder of radius 0.943 to that of the sphere
(and whose depth is naturally 1.5 sphere radii) provides 23% more force than
a second sphere of similar mass.

I wonder where I went wrong?
****************************************************************
gravity.bas

5 rem  calculating a relative measure of force between a sphere and a cyl.
6 rem volume of sphere 4/3 pi r cubed, vol of cylinder pi z squared n
7 rem  total force
10 ftot = 0
20 pi = 3.14159
22 rem the universal Gravitational constant is set to one here.
30 g = 1
40 INPUT "radius of sphere r ", r
50 INPUT "distance between objects m ", m
60 INPUT "depth of test cylinder n (equals sphere's mass) ", n
65 rem calculate the radius of a cylinder of equal mass to the sphere.
70  z = (4 * r ^ 3 / (3 * n)) ^ .5
80 PRINT "z, the radius of the cylinder is ", z
85 rem masssp is the mass of the sphere of radius r
90 masssp = 4 / 3 * pi * r ^ 3
92 rem examine laminae at increasing distance
100 FOR n1 = 0 TO n STEP n / 100
102 rem examine rings of increasing radii
110 FOR z1 = z / 100 TO z STEP z / 100
112 rem slant distance of a particle from sphere
120 denom = z1 ^ 2 + (n1 + m + r) ^ 2
130 force = g * (2 * pi * z1) * (z / 100) * (n / 100) * masssp / denom
140 ftot = ftot + force
150 NEXT z1
160 NEXT n1
170 PRINT ftot
180 GOTO 10

***************************************************


brian whatcott <inet@intellisys.net>  Altus OK
                Eureka!