Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: cavendish expt



I am now slitting my 1 / 2 " OR tape (yes I have a 60's monochrom Sony video
recorder and a few reels of tape.) However it's time for bed, and I go to RAFT
to get rid of stuff so Gate Keeper wsill be happier. (doesn't like clutter),
so you'll have to wait for a day or more for the result. (I use ~ two 1 / 2
piund lead wseights separated by five inches, the tape is 1.3 mills thick, and
21 inches long.

bc

Brian Whatcott wrote:

At 10:52 AM 12/6/02, John Denker, you wrote:

...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.

This example is as it happens, part of the Colin Chapman
legend.
With an aero engineering degree in hand, he used
to participate in the club racing scene which specified
a particularly cheap and cheerful car called an Austin 7
if I remember. This immediate post ww2 car featured a
separate chassis, which had a u section in two side frames.
This thing would wallow over bumps, until Colin closed the
fourth side of the frame tubes, to make box sections.
He knew that this multiplied their torsional stiffness
several hundred times.
From this club racing experience came the Lotus Seven,
and then before long, potent Formula One cars which
tended to be rather competitive: he knew space frames,
and the virtues of "there is nothing lighter than nothing."

A relative of my wife's, who had some insight into the
British auto components business at the time, mentioned
that he would request a stock disk brake calliper to be recast
in slimmed version: and repeated the process more
than once, driving the brake makers to distraction:
in one (apocryphal?) story, after testing a prototype
steel calliper he asked for it to be recast in
aluminum alloy. Part of his design philosophy was
(apparently) "If I can't make it break, it's too heavy."
Again, I warn the reader that these stories grow with
the telling, but he was said to know how to reinforce
a broken design more surely than he could design
in a weight loss.

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!!!!

/snip/

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.

/snip/

?

It will probably be helpful to do an experiment that fully 70%
of the readership is equipped to do, within 20 minutes
of reading my note. You might say it needs nothing more
than high-school experimental skills.

Materials:
two sheets of paper
roll of adhesive tape
two rules or heavy strips of similar size and weight.

Method:
Tape all four edges of one sheet. Tape the top surface
of this sheet to your desk edge in 'portrait' format.

Fold the other sheet in two along the long axis, and tape
all four sides together. Tape this to the edge of the desk,
long side pointing down.
These represent two laminae of equal cross section
and length, one with half the breadth, but twice the thickness
as the other.
To the lower edge of each, tape a heavy horizontal strip,
equally disposed from the vertical centerline.

The hypothesis is that John is not mistaken: the narrower sheet
will be more torsionally compliant.
The null hypothesis is that Cleyet is correct to suppose
that a wider tape suspension is more torsionally compliant in a
Cavendish pendulum.

As a measure of the stiffness in question, rotate each strip
through a small angle, and release. Observe the frequency
of oscillation.
You will likely conclude that the strip showing the higher
frequency of oscillation has the stiffer suspension.

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

Henry, Lord Cavendish was the rich son of an experimentally minded father.
We might call him an amateur scientist [though this is, I suppose,
equivalent to labeling Feynmann,a bongo player.] He left Peterhouse,
Cambridge, without completing a degree.
Better to say, a reclusive dilettante with a great interest in science,
who felt he had data to publish in physical chemistry. You are probably
familiar with the circumstance that he was never willingly in the presence
of women.

This is what he had to say [in part] about the torsion balance.

"Many years ago, the late Rev. John Mitchell of this [Royal] Society [of
London], contrived a method of determining the density of the Earth, by
rendering sensible the attraction of small quantities of matter; but, as he
was engaged in other pursuits, he did not complete the apparatus till a
short time before his death, and did not live to make any experiments with
it. After his death, the apparatus came to the Rev. Francis John Hyde
Wollaston, Jacksonian professor at Cambridge, who, not having conveniences
for making experiments with it, in the manner he could wish, was so good as
to give it to me.
...
These experiments are sufficient to show, that the attraction of the
weights on the balls is very sensible, and are also sufficiently regular to
determine the quantity of this attraction pretty nearly, as the extreme
results do not differ from each other by more than 1/10 th part. But there
is a circumstance in them, the reason of which does not readily appear,
namely, that the effect of the attraction seems to increase, for half an
hour, or an hour, after the motion of the weights; as it may be observed,
that in all three experiments, the mean position kept increasing for that
time, after moving the weight to the positive position; and kept
decreasing, after moving them from the positive to the midway position. "

It is possible, I suppose, that the copper wire which Cavendish chose, was
progressively yielding. It was circular in section, and highly stressed.

Brian Whatcott
Altus OK Eureka!