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Re: [Phys-l] mass and the ring down (free decay) of a pendulum

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 Whatcott Altus OK Eureka!

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