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Re: Saturday Morning Puzzle, Finale

As promised more in reply to bw.

Fire in the stands quickly engulfed its the full length ~> 100m. So on the
other side, ~ 50m, the temperature was not significantly less than being next
to the fire for those near the center (as far as surviving). My guess is that
the "field" reduction was less rapid than 1/R, because the fire was also
likely more than 20m high.

About the air jets propelling dust and large particles (I've finally seen a
video of it), they were a result of the accordion like collapse, i.e. a
bellows; added to this (something I didn't think of) is the vacuum created by
the buildings' disappearance. Nature's abhorrence added to the wind. (Exact
physical description!)

As a result of JM's cogent argument, I've thought more on the laminae?, etc.
It's initially so counter intuitive. With a pseudo infinite lamina, the
configuration of the second mass will make no difference in the force (with in
the "near field") as the field is constant. However, to attain this condition
(pseudoinfinite) the field will be very weak. Configuring the second mass as
a second lamina will not increase the force and must not be extended to the
edges of the first where the field fringes or the force will be even smaller.

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. I have yet to do or find the integration for the field
perpendicular to the axis midway*. However, I suspect that it may also be
similar to the other orientation in which the field is the same as a sphere at
the face. however, as in the axial orientation, I suspect that the filed
falls off "faster" than 1/R^2. Likely the reason they used cylinders, as
suggested by one of the NPS faculty, is, "easier to machine."


*Problem in a Marian ed. for which I don't have the solns. book!

bc




brian whatcott wrote:

At 22:04 10/6/01 -0700, Bernard Cleyet wrote:

I must be missing some thing here. Either "I'm dead" or most of this is
common sense. -- The field E, g, e-m source, acoustical, etc. due to an
effectively infinite plane (near field) is constant (that's why g is
considered a constant "on earth."). Therefore, two of them will be
mutually attracted with considerably greater force than the equivalent
spheres and, near field, constant w/ separation (gnomonic: there is a rat
in separate). However, a sphere and a plane will have a lesser attraction
except in the case of equal radii. I'm not up on variational calc.
This is an opportunity for someone to fill that void -- perhaps Michael
Bowen?

Note: the g for a rod is 1/R direct integration or Gauss.
[1/(4piG)Integral(g dot dA) = M]

This gives the result for the field of b.w.'s laminum 2pi G rho(a).
Therefore, the force of attraction of a sphere, M, is 2pi *rho*GM.

bc

P.s. There was a fire in the stands in England. Fans on the other side of
the pitch (50 m?) were burned to death.
P.p.s Did any of you catch the article, "Compressed Air and Gravity:
Physics Finished What Terror Began?" (NYT SCIENCE Tuesday September 25,
2001?

I now sympathize with Bernard's puzzlement, and understand the general
and uncharacteristic silence which greeted parts of this puzzle.

It turns out that although there are quite practical applications
for explicit expressions for gravitational force of non spherical
objects of variable density, for instance of spiral galaxies, the
topic is 'analytically difficult' so that apart from the expression
for the thin lamina which John M gave earlier, other formulations
are few and far between.

It is not the case in general that a center of gravity can be
decided by eye either, so finding shapes that offer superior attractive
force to that between two spheres is not obvious.

In cases of this kind, I suppose one could do a brute force computation
but the topic seems to offer an opportunity for the experimental
verification of desirable shapes.

Ah, if only I were capable of formulating the algorithm which
iteratively sums the influence of each particle in [x,y,z] on
all others - which ought to be easy (ha!) then a genetic code
could optimize: what you might call a brute force squared approach.

I conclude that the lightly offered puzzle was not in fact too easy,
but really rather too hard....

Brian W

p.s. 1) There is no Latin word 'laminum' as far as I know.
2) Did embers fly across a soccer pitch to spread the conflagration?
3) I did not see this article, "Compressed Air and Gravity:
Physics Finished What Terror Began?" , but it sounds intriguing.
Where does compressed air figure in the tale?

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