Gyroscopic bicycle wheel - forces on the wheel only

This is a standard problem in university physics, which I studied at university in depth. With higher concepts like angular momentum, it is fairly straightforward to explain the motion.

However, one of my students (who is below uni level) recently asked me a basic question about the system, and I was embarassed to realise that I didn't have a good answer for him.

We know that if you analyse any part of the system separately, the forces on the centre of mass must be in the same direction as the acceleration of the centre of mass. Let's just take the wheel itself (not the rod). There is a centripetal force, supplied by the rod, and the weight of the wheel. But in order for the wheel to only precess, and not fall, there must be an upwards force balancing out the weight of the wheel. What provides that force?

There is only one possible answer, given that the wheel is only in contact with the rod, and that is that the rod somehow provides an upwards force as well as a centripetal force. Is this correct?

This is what I told my student, and he then asked how that can work. It is not a well formed question, but I nevertheless am in search of a better explanation for this specific part of the problem.

First I want to say some things about using the concept of angular momentum vector to account for the motion of a gyroscope wheel, then I will get into your question. (To skip this section: scroll down to the bold face heading 'gyroscopic precession')

The word 'explanation' is quite the wonderful metaphor. If a piece of paper is crumpled into a wad most of the paper is hidden from view.

So you open up the wad of paper, unfolding it back to a plane: explanation.

The operations with the angular momentum vector and the vector cross product do the job of accounting for the motion, but they don't help towards visceral understanding. You get the mathematical answer, but you don't get access to the inside.

I submit the following: there is a high bar to claiming that one understands something. If you understand some physical phenomenon you understand why the mathematics that describes it is the way it is. On the other hand, if your capability is limited to just reproducing the mathematics, then your level of capability is merely that of parroting jargon.

Gyroscopic precession

As to what keeps the precessing wheel up:

The precessing motion gives a tendency to pitch up.

For the mechanism of what is happening: see my 2012 discussion of gyroscopic precession.

The process of settling into a pattern of gyroscopic precession is a self-adjusting process.

When released the gyro wheel initially pitches down a little. The downward pitching motion converts to swiveling motion, the swiveling motion gives a tendency to pitch up.

There is a precession rate such that the tendency of gravity to pitch the gyro wheel down is fully counteracted by the tendency of the precessing motion to pitch the gyro wheel up. As a gyro wheel goes into precessing motion the precession rate settles onto that dynamic equilibrium state.

In classroom demonstrations the usual way to release the gyro wheel is to release gingerly. This cautious release suppresses nutation. The thing is: the phenomenon of nutation is crucially important. However, demonstrators tend to assume that nutation is a spurious side-effect. Invariably demonstrators do not pay any attention to nutation.

Experiment

The fact that initially the gyro wheel pitches down a little has been experimentally confirmed in a tabletop experiment.

Svilen Kostov, Daniel Hammer, 2010, 'It Has to Go Down A Little, In Order to Go Around'- Following Feynman on the Gyroscope