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Hi --
I wrote up a whole bunch of notes on the simple pendulum, including
the equation of motion and the formula for the period -- not
restricted to small amplitudes.
https://av8n.com/physics/simple-pendulum.htm
The conclusions are unchanged from what I posted to this group a
couple of weeks ago. Mostly I just filled in the details. Things that
I previously asserted are now /derived/.
There's only one really tricky step in the whole calculation. The rest
is just algebra. Lots and lots of algebra. I expect few people will be
interested in the gory details, but even so, it's nice to have
something you can refer to if the need ever arises.
It's very new. There are probably lots of typos.
Please let me know of any bugs, questions, or comments.
Here's the table of contents:
1 Simple Pendulum : Period Etc.
2 Evaluating the Arithmetic-Geometric Mean
3 Connecting the Physics to the agM
4 Approximations for the Asymptotic Cases
4.1 Small Amplitudes
4.2 Large Amplitudes
5 Deriving The Equation of Motion
5.1 Basic Physics
5.2 Putting the Integral into Standard Form
6 The Sequence of Integrals
6.1 Overview
6.2 Gory Details: Cosine Factor
6.3 Gory Details: Radius Factor
6.4 Gory Details: Numerator
7 Ramifications
7.1 Numerical Robustness
7.2 Backwards Sequencing
7.3 Eponymous Constants
8 Assumptions and Details
9 Software Implementation : C++ Code
10 Notes and References
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