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Usually physics books calculate the precession rate of a gyroscope when its momentum wheel is spinning "fast enough."

I am looking for an equation to calculate the wheel speed whereat a gyroscope "falls" beyond a particular angle towards a table surface (e.g. 45 degrees around an axis parallel to the table top, closer to the table) due to gravity, because the angular momentum of the wheel is not "high enough."

In particular, I would also like be able to compute the time to fall past some angle for some wheel spin speed (though intuitively some speeds will never lend to falling beyond a particular angle).

I'm also generally interested in computing what proportion of gravity's torque gets applied along the axis of the torque versus cross to both the torque and the wheel's angular momentum, depending on the spin speed of the wheel. (Intuitively all of it is applied along the axis of gravity torque due to gravity when wheel speed is 0).

I'll define Gyroscope as the typical spinning wheel attached to a pole placed upon a support post, where the pole is parallel and above some table surface, with c.m. at the rotating wheel, and where gravity would cause precession/rotation about the axis aligned with gravity. This is the most common formulation as found in Feynman's figure 20-5 here http://www.feynmanlectures.caltech.edu/I_20.html

Thanks!

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There is a paper about gyroscope dynamics that will help you answer your question.

The authors discuss the dynamics, and they conducted an experiment to corroborate their views.

For wide coverage they used a range of wheel spin rates. That is, they also covered a couple of cases of slow spin rates. That's the aspcet that will be useful to you.

At slow spin rates nutation becomes a larger factor.

At fast spin rates nutation is quite fast, and the faster the nutation the smaller the amplitude. At fast spin rates nutation is small to begin with (pretty much unnoticably small), and its energy dissipates quickly, usually in seconds.

Conversely, the slower the spin rate, the slower (hence larger) the nutation.

Specifically to your question:
What you can calculate, given a particular setup: the amplitude of the expected nutation as a function of the spin rate at the start. The magnitude of that amplitude is, of course, an angle.

If memory serves me: the theoretical discussion in that paper will give you the clues you need. And their descriptions of the outcomes at slow spin rates will inform you what to generally expect at slow spin rates.

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