I was reading up on why before starting the precession the gyroscope "goes down a little", (Link at the bottom). In this paper, while looking at the graphs I observed that before reaching the steady-state precession in the transient state the gyroscope oscillates a bit (up and down oscillation), before damping out and simply precessing about the vertical axis, as you can see in the attached image from the same paper.

Image showing the damped oscillation in the transient state. Image from the paper

Q: I didn't understand why does the gyroscope oscillate (up and down) during that transient phase (damped oscillation)? And what is the restoring force acting in that oscillation?

Link for the paper: https://arxiv.org/abs/1007.5288

Directly to PDF of paper link: https://arxiv.org/ftp/arxiv/papers/1007/1007.5288.pdf

  • $\begingroup$ Does this stackexchange answer about the onset of gyroscopic precession answer your question? (It's an answer from 2012, written by me.) $\endgroup$
    – Cleonis
    Jun 10 at 16:50
  • $\begingroup$ Incidentally, my understanding is that linking to the page with the summary of the article (instead of linking to the PDF directlly) is preferred here. arxiv.org/abs/1007.5288 $\endgroup$
    – Cleonis
    Jun 10 at 16:56
  • $\begingroup$ I have edited the answer and provided the link you mentioned. Coming to the answer you linked, that answer is different from what I intend to ask, my doubt is not regarding which direction the gyroscope spins by the torque applied, but rather before achieving a steady state condition of precession, during the transient state the gyroscope is oscillating (as you can see in the graphs in that paper). I don't understand how and why the up and down oscillation is happening. Maybe I wasn't that clear in my question, I'll try to improve it a little. Thanks $\endgroup$ Jun 10 at 17:31

When a demonstration of gyroscopic precession is given the demonstrator will invariably release the gyroscope gingerly. This is not done on purpose, it just feels natural to do so.

However, by releasing gingerly nutation is suppressed. What in fact happens is that the demonstrator is providing critical damping. (Critical in the sense that nutation cycles are prevented but not the settling into steady gyroscopic precession)

Svilen Kostov and Daniel Hammer opted for a sudden release. One moment the gyroscope is still supported, the next moment they just flat out drop it.

When you do a sudden release of a (spinning) gyroscope you get that nutation: that is what a gyroscope will always do.

In actual demonstrations the students rarely if ever see the nutation because invariably the demonstrator releases the gyroscope gingerly.

(Also when the spin rate is very high the nutation is very fast and the amplitude is very small. The nutation always occurs, but when the spin rate is very high the naked eye doesn't see it.)

I have an hypothesis:
When a teacher demonstrates behavior of a gyroscope he is suppressing the nutation all the time because he feels that the nutation is some whacky disturbance.

In actual fact the nutation is the key to understanding gyroscopic precession.

[Later edit]
Kostov and Hammer write:
"In the end, Feynman dispels the “miracle” of the purely precessional motion by bringing nutation (the wobbling part) into the picture as the mechanism by which some angular momentum is transferred from the horizontal to the vertical direction."

In physics there is always a judgement call as to what to regard as the essence and what to regard as collateral effects. In, for example, a demonstration with an air track the demonstrator will announce that any effects from air resistence will be ignored.

In the case of onset of gyroscopic precession the phenomenon of nutation is inherently involved. In the it-has-to-go-down experiment the setup is designed to obtain a significant nutation; large enough for quantative analysis.

If it would have been possible to conduct the experiment under frictionless circumstances then the nutation would have persisted indefinitely.

The actual motion of the gyroscope (in the it-has-to-go-down experiment) is a superposition of two motions: nutation and gyroscopic precession. The nutation is a motion where the spin axis rapidly sweeps out a cone with a small angle. In the graph provided in the paper the up-down component of the nutation motion is most visible, but it has to be emphasized: the nutation motion has circular symmetry: the spin axis is sweeping out a cone (a cone with a small angle).

In real world circumstances there is always some friction, and so both the nutation motion and the gyroscopic precession motion decay. The nutation motion, being the faster motion, decays faster than the precessing motion.

Settling into steady gyroscopic precession

The quickest way to go from release to steady gyroscopic precession is to provide friction in the vertical direction only. That way there is no friction component that opposes precessing motion, while at the same time suppresssing nutation motion.

A complete nutation cycle goes through a full circle. With sufficient vertical friction the nutation can be suppressed to moving through only a quarter of a circle: the center of mass of the gyroscope goes down and the nutation process transforms that vertical motion into sideways motion.

(This 'one quarter of a circle' must not be taken literally. The actual motion won't be precisely that shape. What matters is the transformation process: initially moving vertically, yielding to the torque, and that motion gets transformed to horizontal motion, the steady gyroscopic precession.)

By habit physicists will tend to ignore the nutation, mis-classifying nutation as 'non-essential'.

So we end up with an odd state of affairs: physicists are applying mathematical expressions to describe gyroscopic precession, and for high spin rates those mathematical expresssions are an excellent approximation. However, generally these same physicists do not understand gyroscopic precession.

  • $\begingroup$ Thanks! I think I have a better understanding now. Also, I agree with your Hypotesis $\endgroup$ Jun 11 at 12:50
  • $\begingroup$ @IshanTandon I have added material. In particular I have added emphasis that the nutation motion has circular symmetry; the spin axis sweeps out a cone. $\endgroup$
    – Cleonis
    Jun 12 at 7:37
  • $\begingroup$ Thanks for the clear and precise answer! Nutation seems very interesting, especially the part where it sweeps a quarter circle to transfer angular momentum. I will study more about it. I don't have enough reputation to upvote your answer :(, but I hope your answer helps others understand the concept better! $\endgroup$ Jun 12 at 11:15
  • $\begingroup$ @IshanTandon The way you phrased your comment ('nutation seems very interesting') suggests you are still under the influence of the mindset that tends to see nutation as an inessential detail. In order to make progress you will have to get to the point where you recognize the central role of nutation motion. $\endgroup$
    – Cleonis
    Jun 12 at 13:48

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