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Brian!
For linear damping the decay constant is independent of the mass. The
confusion is I'm probably using the wrong term. Should be using the
term "resistance"?, i.e. the term in front of the speed in the eq. of
motion. Equating the torques w/ the moment of inertia times the angular
acceleration: I alpha = - k*l*l theta dot - mglsin theta See. No m
in front of the damping term?
Simplifying (and linearizing) results in theta double dot + theta * g/l
+ k/m theta dot The loss part of the soln:
A = A(0) * exp(- kt/2m) This is what I want i.e. data on amplitude
change w/ time as a function of the mass. No hay nada, for bc's
search. Simanek has this in his lab manual, but retired w/o saving any
of his students' data.
bc, prays he has it "right".
p.s. In a synchronome the steady state amplitude, even when mounted
rigidly on a "brick" wall changes markedly w/ bob mass. This
theoretically won't happen if the support is absolutely rigid, as then
the amplitude, which is the measure of the total energy, equals the
input from the escapement. In the synchronome (and any other pendulum
clock?) the loss increases w/ bob mass from the support, or at least
that's the result found recently both by free decay and the amplitude.
A few other physicists have examined, of at least noticed, this also --
starting w/ Huygens.
* For a simple pendulum. I have to cheque my maths, but my impression
is, for a physical p,. it will change, because the equivalent length
changes. It will lengthen as the mass increase increasingly swamps the
moment of inertia of the pendulum's rod. This effect is rather minor
compared to the support loss.
Brian Whatcott wrote:
>At 11:09 AM 6/9/2007, you wrote:
>
>
>>PHYS-L, PHYSHARE, and TAP-L, people!
>>
>>There is a controversy among the horological community on the effect of
>>pendulum mass on the steady state amplitude of an escapement driven
>>clock, in particular the impulsing by the gravity arm of a Synchronome
>>clock. Theoretically for a simple pendulum clock with a decay constant
>>independent of the mass, there is, obviously, no effect. However, the
>>free decay (they call it "run down") is, theoretically much effected;
>>the time is proportional to the mass. I have searched somewhat
>>diligently and find no data (experimental) relating to this effect. Do
>>any of you have such?
>>
>>bc, frustrated.
>>
>>
>
>I expect I am entirely missing bc's point.
>This appears to be a question centering on mechanical Q: the ratio of
>energy stored to the energy lost per cycle.
>It seems like increasing the bob weight should increase Q,
>But this may well increase the bob weight drag,
>which should decrease Q. Hence the decay constant of a pendulum
> should not be independent of its mass.
>As to experimental data - this is the grist of engineering labs
>such as this one:
>
><http://www.physics.odu.edu/hyde/Teaching/Fall04/Lectures/ResonanceLab.html>
>
>
>
>Brian W