Next message: Frank Reed: "Re: Timing Lunars with a Rock"
Alex Eremenko wrote:
"There is one more problem with such crude pendulum which has not been
addressed:-)
And the effect is more substantial than some other effects mentioned so far:
The period of a pendulum DOES depend on the amplitude. You can neglect this
dependence only if your pendulum is very long and the amplitude is very
small."
I don't think this is a big deal.
Theory first:
Calling the length of the pendulum L and any variation/error of estimation
of that length dL and calling the mean gravitational acceleration g and
variation of that with latitude dg and finally calling the amplitude of the swing A
(the actual distance from the bottom of the arc), the percentage change in
the period dT is given (to lowest non-vanishing order) by
dT = (1/2)*dL/L - (1/2)*dg/g + (1/16)*(A/L)^2.
Assuming my ideal pendulum length is 99.3 cm, an error of 1 cm will lead to
an error in the rate of 0.5% or in other words 3 seconds in 10 minutes. This
is the most likely source of error in a crude pendulum. The error from an
incorrect calibration for latitude (leading to a non-zero dg) could be as much
as 0.25% if the pendulum were calibrated at the pole and subsequently used at
the equator but in practice it will be an order of magnitude smaller. And
what about the swing amplitude? The pendulum should be calibrated for very small
swings which implies that A is nearly zero. Suppose this has been done, and
we mistakenly allow (and maintain, which is even harder) a swing amplitude of
20cm (that's a lot --40cm from one side to the other-- and anyone who has
seen a pendulum clock in action would be unlikely to use such a large swing but
I'm using it for the sake of argument). This will lead to an increase in the
pendulum's period of 0.25%. So swing amplitude is not a big issue until you
get to really large amplitudes, and these are relatively unsustainable.
And a little experiment:
I made myself a pendulum with a couple of washers and thread last night
measuring the length with a simple tape measure. I found that my pendulum was
about 1.5 (+/-0.3) seconds fast after five minutes. This is just about what I
would expect based on the numbers above. It's very difficult to estimate the
length of a crude pendulum like this to better than the nearest centimeter. I
tried to experiment with larger swing amplitudes, but they're difficult to
maintain. My conclusion, based on theoretical considerations and experiment, is
that an improvised pendulum, without any other time standard for comparison,
can probably be considered accurate to about +/-0.5% or +/-3 seconds in ten
minutes. Of course, it's much better than this if we have some means to
compare this pendulum against a proper time standard. Then the exact length doesn't
matter so much since we can rate the pendulum like any other timekeeper.
-FER
42.0N 87.7W, or 41.4N 72.1W.
www.HistoricalAtlas.com/lunars
