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Re: cavendish expt



Bernard Cleyet wrote:

> Aux contraire, a cylinder has the highest stiffness
> strength (tensile) for a solid member than any other simply connected
> cross-sectional shape. St. Venant ....

OK, I think I finally figured out what's going on
here. The St. Venant analysis is a horrible red
herring. As I mentioned before, and as I will
explain further in a moment, you don't need to
do the St. Venant analysis, and you shouldn't
do it, except as a reconnaissance (to find out
how irrelevant it is).

If however we are masochistic enough to do the
St. Venant analysis, I think we all agree that
the torsional stiffness of long rigid rods
when used in "screwdriver mode" is as follows:
-- stiffest: hollow tube, like a drive shaft
-- in-between: solid cylinder
-- floppiest: thin flat blade, like a putty-knife
... assuming as always constant length L in the
Z direction, uniformity of cross-section as a
function of Z, and constant total amount of
material. As a corollary, these conditions
imply constant tensile strength.

In particular, if you take a tube and slit it
lengthwise, forming a C-shaped cross section, you
unstiffen it tremendously, converting it from
something stiffer than a solid rod to something
floppier than a solid rod.

For a sold rectangular cross section of width w and
thickness t, the torsional spring constant goes like
K ~ M t^3 w / L (1)

where M is some modulus, with dimensions of stress
(same dimensions as pressure).

BUT!!! For the Cavendish experiment, equation (1)
is a red herring!!!!

==============

A screwdriver gets shorter when you twist it.
A swingset rises up when you twist it.

In the Cavendish experiment, you should load
the support-fiber almost to the breaking point.
Breaking-strengh depends on the cross-sectional
area t*w and on the intrinsic strength of the
material, which is some other number M' with
dimensions of stress.

If you do the physics (which requires nothing
more than high-school geometry and the principle
of virtual work) you find that the torsional
restoring force constant goes like
K' ~ M' t w^3 / L (2)
which is verrrry different from equation (1).

Approximating M'=M, the two equations differ by
a factor of w^2/t^2 and some minor dimensionless
constants. For videotape, w is millimeters and
t is microns, so the St. Venant contribution is
smaller by a factor of a million or so.

This agrees fully with my intuition. A few
meters of videotape hanging under its own
weight doesn't have enough St. Venant stiffness
to undo even the slightest wrinkle or twist.
It is totally unsuitable for use as a
screwdriver.

In contrast, if you attach an enormous weight
to it, it will hang straight and flat, and will
resist torsion by the elementary swing-set
mechanism. The wider the tape, the more it
will resist.

Prof. Cavendish wasn't an idiot. He had a good
reason for designing his apparatus to use a
long thin fiber.