In checking the total E derivation (example in The Berkeley Physics Course V 1), I missed the use of the approximation!!
So both methods use the approximation. Sorry for all the bother.
Now here's another problem:
What happens to the amplitude when g in T~= 2Pi (l/g)^0.5 is changed?
From the total E derivation (after the approximation):
Angular speed (at BDC) = A(g/L)^0.5 or
(speed/A) / L^0.5 = g^0.5
so if g is reduced the ratio becomes smaller. Which changes or both?
This is not just a theory problem, but practical.
V. good pendula detect the tidal change of g due to the moon. One clock, the Littlemore**, thought to be the best made, did a poor job of detection compared to a Shortt clock***. Bryan**** thinks it's due to its method of maintaining constant amplitude (to prevent circular error). His description of his moon tidal model and a discrepant amplitude measurement is here.****
To model tidal effect, Bryan added a repulsive force on the bob using an electromagnet and a PM on the bottom of the bob. Unfortunately, this has the complications of, at least, non-homogenous field and eddy current loss. I think the orientation of the magnet WRT the field also changes.
I propose repeating the experiment using a free pendulum***** w/a non conducting bob and rod, a ceramic magnet, and Helmholtz coils. Before I go to all this trouble, I'd appreciate v. much comment from you-all, especially on the effect of g change on the amplitude.