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Re: cavendish expt



BW beat me to it!

Here's what I began yesterday:
[I've added in square brackets.]

John!

interleaved:

"John S. Denker" wrote:

I wrote:

"C) Conclude that a round solid fiber gives
minimal stiffness for a given strength."

Bernard Cleyet wrote:

If "a given strength", is the same as a given cross sectional area,
then not true. Aux contraire, a cylinder has the highest stiffness
strength (tensile) for a solid member than any other simply connected
cross sectional shape. St. Venant showed this some time ago (ca.
1855). continuing:

Given strength means given tensile strength in the
vertical tape, as should be clear from my original
posting. Also, it seems self-evident that tensile
strength in such situations is proportional to the
cross sectional area; I hope we can agree on that.

Just making certain. As I wrote above, one may infer equal areas result in
equal tensile strength



I don't know what means "highest stiffness strength
(tensile)". First of all, stiffness is not equivalent
to strength, as I noted in my original post. Secondly,
the thing I asserted was lowest was not tensile but
torsional.

my post script of yesterday [Wed] @ ~ 1800 explained: I repeat it
"correction: highest stiffness - tensile strength ratio"



So what I said was true: a solid cylindrical fiber
has the minimal torsional stiffness for a given
tensional strength (assuming constant length in
the Z direction).

no a ribbon has much smaller torsional rigidity. That is why they are used
for galvanometer and torsion pendulum suspensions, drive belts, etc.




"B) Again split the tape, but rather than
moving them apart, move them together so they
share the same attachment point. This will
give less stiffness than the original full-width
tape."

I stand by that, too.

by the approximate formula the rigidity will be the same [thickness <<
width]. Using a less approximate formula the reduction depends on the
width thickness ratio. For example the OR [open reel magnetic recording]
tape is one mill by 1/4 inch. Two 1/8 inch lengths will have a reduction
of ~ 0.005%. While a square member cut in half will suffer a decrease of
60% in rigidity. [ I've since found what I think is a more credible
formula. It gives a similar result.]



why do you think the split tape will have significantly less stiffness
than the original?

Uhh, because of elementary physics?

1) I can't imagine a case where splitting something
gives it more stiffness.

I wrote significantly less, not increase. [see above for approximate
amount]



2) After splitting it, moving the two halves together
reduces the stiffness.

Do the calculation. You don't even need to bother
with a full St. Venant analysis; just imagine a
swing-set supported by two very thin vertical ropes.
The resistance to torsion goes like the square of
the distance between the ropes, again assuming
constant length in the Z direction.

p.s. a quick calc. for the torsion constants (per unit length and
same shear modulus) of a cylinder (r= 0.0178 inch) and a ribbon two
mills thick and 1/2 inch wide give a factor of 120 X (the cylinder
being the more rigid)

That's preposterous. Does that perhaps come from
misapplying the formula for flexional stiffness
across the width of the tape, rather than the
formula for torsional stiffness?

[send me your formulae and I'll send mine]





Let's get real:
-- Birds have hollow bones. Do we really think
they would do so if solid bones were stiffer?
-- The driveshaft on your car is hollow. Flagpoles
are hollow. Do we really think that people would
build such things if solid rods were stiffer?
-- Why would people bother to fabricate I-beams
if solid rods were stiffer?
-- Et cetera...............

Quite true a cylinder with the same cross sectional area as a [solid] rod
has greater rigidity. That's because the neutral area is used as active
(torsional and flexure) area. When I wrote the cylinder had the greatest
rigidity I meant rod. However, I did write

"Aux contraire, a cylinder has the highest stiffness strength
(tensile) for a SOLID [emphasis added] member than any other simply
connected
cross sectional shape. St. Venant showed this some time ago (ca.
1855)."

[e.g. the rigidity of an elliptical cross section rod is proportional to:
(a^3 * b^3) / (a^2 + b^2), where a and b are the semi axises. By
inspection the maximum is with a = b, a circle. I found this formula in
two of my texts and several institutional net sites (one derives it). Note
the approximation with a << b: a^3 * b. From this one may guess the
formula for a tape.]


bc