Does anyone know what the minimum speed of a gyro to make it so a force at 0deg will yield a movement at 90deg? or even better, what the angle is wrt speed? see http://www.copters.com/aero/gyro.html for pictures.
The post Minimum speed needed for a gyroscope to precess doesnt really answer it, but it goes in the right direction. it must be a point where the reactive [dynamic, or angular] mass exceeds the static mass. but i have found this nowhere....
There is no threshold speed for that.
Over the range of ratio of spin rate and exerted torque the transition is fully gradual.
Explanation of what I mean by 'ratio':
If the spin rate is very slow, but the exerted torque is in equal measure small then the resulting precession will be a slow motion version of the precession that you get with large spin rate.
If the spin rate is slow, but the exerted torque is one that is appropriate for fast spin rate, then the spinning object will simply flop over, totally not the motion pattern of settling into precessing motion.
We have the following property of gyroscopic precession: the faster the spin rate of the spinning object, the slower the required precession in order to oppose the exerted torque. That's what the mathematics tells us, and our direct observation bears that out.
Release a spinning top at a number of different spin rates.
High spin rate: slow precession.
The slower the spin rate upon release, the faster the required precession rate.
Those two tendencies intersect. You arrive at a range where the required precession rate is so large that there simply isn't enough time to reach it, the spinning top has already flopped over completely.
In engineering context pretty much the only relevant case is the case of large spin rate, as that is when the forces are large and things can go wrong fast.
Relevant in engineering is when the spinning top transits to precessing motion so fast that it appears instantaneously.
It cannot actually transition to precessing motion instantaneously; that would violate the laws of motion.
That 90 degrees angle is in the idealized case of arbitrarily high spin rate, and small torque.
For real life spin the angle is never quite that 90 degrees, but in the case of high spin rate it's so close that for engineering purposes the difference is negligable.
Further information:
There is a 2010 article by Svilen Kostov and Daniel Hammer. They conducted a tabletop experiment to verify the following: at the moment that the torque is applied to the spinning wheel the wheel initially yields a little to that torque. The title of their article is "It has to go down a little, in order to go around"
There is a 2012 answer by me in which I explain the mechanism of the onset of gyroscopic precession
