In "The Science of Clocks and Watches" Bateman gives a formula for the Q
of a freely decaying pendulum. He assumes the dissipation is linear (a
function of the speed of the pendulum) by using the damping constant and
relating it to the logarithmic decrement. His specific case is the
number of periods (cycles) for the amplitude to decrease to half value.
This is the standard method used by professional and amateur
horologists. Since this consumes more than an hour for a "good" clock,
I have suggested that the maker of a device that continuously measures
the relative amplitude to use Bateman's original formula, videlicet:
Q = Pi * n / ln (A(1)/A(n) (1)
The problem is I can't explain to him why the natural log must be used
w/o resorting to the solution of the SHM differential equation, etc.
Can anyone explain the derivation of (1), quoting:
"... something I can put in instructions that 60 year old clock
hobbyists will understand." "... who wants to understand, but has
never taken algebra or physics. " And finally:
"Why is the natural log used?"
Is it possible?
bc has enuff trouble following the derivation, and has, admittedly a
very old, PhD.