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    Re: Timing Lunars with a Rock
    From: Frank Reed CT
    Date: 2005 Jul 19, 18:11 EDT

    Alex Eremenko wrote:
    "There is one more  problem with such crude pendulum which has not been
    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
    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.
    42.0N 87.7W, or 41.4N  72.1W.

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