sci.physics.research

On Sep 19, 4:27 am, pioneer1 <1pione...@ > wrote:

> Do you think
> there is a way to compute the damping coefficient for the original
> Cavendish experiment data?

Yes. as I said on Sep 11, "Cavendish only
records three extremum points of the oscillation. This is sufficient,
however, (after a little calculus max/min analysis and some algebra)
to fit a unique offset decaying sine wave to the data"

What you want to do fit the function

x(t)=A*exp(-B*t)*sin(C*t+D)+E

to the measure extrema with coordinates (t1,x1), (t2,x2), (t3,x3)
where the first and last are assumed to be extrema of one sign and
(t2,x2) is the intemediate extremum of the other sign. The t2
measurement actually overspecifies the problem since it can be shown
that t2=(t1+t3)/2 for our fitting function (the extrema are equally
spaced for the decaying sine wave). To avoid this, we simply define
t2=(t1+t3)/2. That leaves 5 measurements and 5 fitting parameters
(A...E), subject to the constraint that the measured times correspond
to extrema. That means we need five equation. They are:

xi=x(ti), where i=1,2,3 and 0=x'(ti) where i=1,3

Prime, here, means the derivative wrt t. Setting the derivatives = 0
means t1 and t3 are extrema. Normally 5 equations and five unknowns
means a lot of work, but there are some tricks with equation ratios
and differences that makes it relatively easy. Once you find A...B
for the big balls at both extremes, you can determine G, as shown in
my final report.

Based on our exchanges, I suspect you'll find the math difficult. If
you can't manage it and if I have time in the near future I can trudge
through it and post the results. Reader contributions are welcome to,
of course. Again, the question we are trying to answer is, "How good
is Cavendish's data proper (independent of his primative analysis)?"