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Re: A ball in a rotating dish

At 13:37 8/2/99 -0400, Robert Cohen wrote:
On Mon, 2 Aug 1999, Ludwik Kowalski wrote:

... Was
the generality of the formula lost when the sin(TET) was
canceled on both sides of the force=force equation?

for TET=40 degr --> f=1.272 rot/second
for TET=20 degr --> f=1.149
for TET=10 degr --> f=1.121
for TET= 1 degr --> f=1.114
for TET=0.1 degr --> f=1.112

As expected, the frequency (or period=1/f) does not
depend on the amplitude (TET) when TET is small.
Nothing different from what one finds from the well
known pendulum formula.

Speculations from my previous message should be
ignored. But what is wrong with the derivation?

My speculation was backwards. Rather than there being a minimum radius,
there is a minimum rotation rate.


| Robert Cohen

I see that Robert came through at the same time as me.
This was a splendid piece of subtle misdirection, in which
all the derivations were 100% correct, and still it left us checking and
rechecking....

If you read this:
"We increase the rotating frequency of the [sphere/dish] and check
the deflection angle of the ball theta"

...and later you see the correctly derived expression:
cos (theta) = g/(R*omega^2)

What is more natural than to see a table of theta (deflection angle)
for given rotation rates?

And then, what on earth can be wrong with the expression when
a slow rotation rate gives nonsense results?

The answer is:
A wonderfully contrived transposition of the dependent and
independent variables in our minds.
It turns out that the simple pendulum and the conical pendulum
and the bank angle of a race track all share the same formalism:

F characteristic = 1/(2*pi sqrt(L/g)) approximately
F characteristic = 1/[2*pi sqrt (L*cos (theta)/g)] exactly

L length of pendulum or radius, theta is angular deflection.

When presented in this exact form, there is no difficulty
in following Galileo and noting the frequency which RESULTS from
a given length and or deflection has a lower limit, whereas the
independent variable theta may be indefinitely reduced.
Sketching the data brings out the idea quickly.

Revised table:
Theta Frequency
80 deg 2.7 Hz
70 1.9 Hz
60 1.6 Hz
50 1.4 Hz
40 1.3 Hz
30 1.20 Hz
20 1.15 Hz
10 1.12 Hz
5 1.117 Hz
1 1.114 Hz









brian whatcott <inet@intellisys.net>
Altus OK