# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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 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