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This is a common phenomenon that I observe whilst preparing a meal. Assume that we have a plate of diameter $r$ and we drop it when there is an angle $\theta$ between it and the table and we also apply a force $F$ at the side of the plate.

Suppose that the gravitational acceleration is $g$ how can we determine this plates "wobbling" frequency, how this frequency depends on time and when would this motion stop?

cross section of the plate

https://drive.google.com/open?id=0B2pRI_pd_h4-WW5ZaHJHay0zQW8 video of the action called as "wobbling"

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  • $\begingroup$ Welcome to Physics SE! Please note that we don't answer homework or worked example type questions. Please see this Meta post on asking homework/exercise questions and this Meta post for "check my work" problems. $\endgroup$
    – Yashas
    Mar 1, 2017 at 15:00
  • $\begingroup$ This is not actually a homework question. I have gave letters just for the ease of discussion. $\endgroup$
    – user147133
    Mar 1, 2017 at 15:01
  • $\begingroup$ Richard Feynman wrestled with a similar problem, although his plate was wobbling in the air while spinning. If you can get this article from the American Journal of Physics 75, 665 (2007), it undoubtedly will help you understand the complexity of this problem: aapt.scitation.org/doi/abs/10.1119/1.2402156?journalCode=ajp $\endgroup$
    – Ernie
    Mar 1, 2017 at 15:16
  • $\begingroup$ Unfortunately I cannot. Which link I click on the site you suggested it comes back to the actual website you gave. $\endgroup$
    – user147133
    Mar 1, 2017 at 15:51
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    $\begingroup$ Possible duplicate of What is the physics of a spinning coin? $\endgroup$
    – John Rennie
    Mar 2, 2017 at 10:17

1 Answer 1

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The frequency of the wobbling plate (or the spinning coin) is an interesting problem. We can make certain idealized assumptions to compute the frequency as a function of angle of tilt; as the plate loses energy through friction, the angle of tilt will decrease and this will affect the frequency.

The problem is known as "Euler's disk", and a detailed analysis of the motion is given on this page - the conclusion is that the rate of rotation as a function of angle $\theta$ is given by

$$\omega = \sqrt{\frac{4g}{r\sin\theta}}$$

Which makes sense given our observation that the rate of wobble goes up as the plate settles down.

The damping is a difficult question - according to wikipedia the problem was addressed in a Nature article by Moffatt, who showed that the spinning time is indeed finite, and scales with the third power of the initial angle. It's not clear to me whether that is the definitive answer - there is some disagreement on whether the dissipation is due to viscous effects or rolling friction.

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