Re: [Phys-l] Another practical problem (challenge problem)

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding BC's pendulum modifications:

> ...
> Brian!
>
> Not quite my problem.   I'm not using a clock w/ a "driving"
> escapement.*  I'm trying to explain the ("instantaneous")** change
> in amplitude when a portion of the bob falls from a freely decaying
> pendulum.  I do find changes in the Q, because I fit to an
> exponential decaying amplitude to find the amplitude.  [decay
> constant  ~ 0.003)  This "study" was prompted by my friend's
> failure to detect a change in amplitude (measured by the flag
> interrupt time at BDC of a photogate) when "sliding off" three
> quarters (0.02kg) from a 77 kg tower clock's bob. *** [Easily
> explained by the minuscule mass (0.02kg) compared to the bob's, and
> the detector's insensitivity.]  I find significant changes in
> amplitude using a rotary motion detector as the suspension and
> electromagnetically dropping a 0.07kg mass from a 0.30kg pendulum
> (rod 0.025kg).
> ---------------------
> pics here:
>
> http://www.cleyet.org/Horological/weight%20pan%20effect/
>
> I've just increased the mass dropped for further trials from 0.07
> to 0.120kg.
>  ---------------------
> Your discussion is well known to horologists (centuries), and is
> complicated by support loss.  A few years ago an horologist (AH)
> did a study exactly as you describe w/ constant size using W,
> brass, steel and Al. (IIRC)  The expected Q increase w/mass did not
> occur, because of the increased support loss even when the support
> (many kg cast iron) was bolted to a concrete column.  In the
> clock's wood case the clock failed to operate w/ the W bob!   He
> measured the motion of the support, inter alia, using a light
> lever.  
>
> I reported this back in '07:
>
> https://carnot.physics.buffalo.edu/archives/2007/06_2007/msg00024.html
>
> n.b. my confusion of force and E.
>
>
> * Not incidentally, the better clocks vary the drive to maintain
> constant amplitude as part of the method of mitigating circular
> error.  e.g. the Synchronome and Littlemore.  I don't remember what
> Woodward did w/his clocks.
>
> ** Now I understand the cause of the confusion.  I shoulda writ
> originally "instantaneous change" not implied a change in the
> change (decay) of the amplitude when part of the bob drops.
>
> *** there's another problem here.  If the photogate did detect a
> change in amplitude, it would not be correct unless the mass'
> original position was above the C of M of the bob.  Otherwise,
> there would be an unintended torque change and the position of BDC
> would change. I position my added mass on the electromagnet so that
> BDC doesn't change.
>
> bc
>
> p.s.  Indeed, the decay constant does increase for the 0.12kg drop
> from  0.00239 to 0.00278   Using the usual assumptions (i.e. linear
> dissipation and decay constant <<
> 1)  Q = angular frequency / 2(decay constant), my result is: before
> drop, Q= 1076 and after, Q = 944

This discussion has motivated me to propose (a la JD) a challenge problem to the group related to the period of a simple pendulum.

We know that an ideal simple pendulum of length R is equivalent to a point mass sliding frictionlessly on the bottom of a circular/spherical bowl when the radius of curvature of the bowl is R.  (If you don't see the equivalence immediately, just stop & think about it for a while.)  We also know that the period of the motion for the pendulum/mass-on-bowl depends on the amplitude of the motion for sufficiently large amplitudes when the small angle approximation is violated during the swing making causing the restoring force/torque along the path to not be proportional to the displacement along the path.  So the oscillatory motion only approximates simple harmonic motion in the limit of small amplitude oscillations.  Here we assume the pendulum and mass-on-bowl are moving in a vertical plane in a uniform gravitational field.

Problem:
Suppose you are allowed to change the shape of the bowl from a circle/sphere to some other concave shape that causes the resulting period of the sliding mass to actually execute true simple harmonic motion for any amplitude (within reason) when the motion is still in a vertical plane.  Question:  What is the shape that the bowl must have for this to work out?  Extra credit: After you have found this curve/shape, is there some way, in hindsight, whereby you could have figured out the shape without doing all that work?  Stipulation: The amplitude of the motion is to be measured by the path length traveled by the mass/bob for whatever path shape is to be used.

David Bowman