Can/Does angular momentum apply a torque?

Question

Context

I'm a psychologist and I'm interested in perception of gyroscopic forces. For that reason, I'm having to think about many of the classic issues relating to gyroscopes in reverse. That is, instead of predicting what imposed forces will do to a gyroscope assembly, I must think in terms what forces will be imposed by the assembly on a perceiver, in order to make predictions about how it will be perceived.

System

Imagine a gyroscope mounted on a rod, such that the rod could be wielded by someone. Held perfectly level, there would be no precession forces because the person's hand is applying a torque that counteracts gravity. Thus, the direction of angular momentum is not changed.

Now imagine the wielder rotates his wrist 90 degrees in a forehand direction. This applies a torque about the axis of the wrist, sweeping the rod away from our view as shown in the figure. This reorients the gyroscope by 90 degrees as well, and the angular momentum causes the end of the rod to "lift" somewhat during this rotation, because it is in the direction of precession. That is, less torque is required by the wielder to counteract gravity during this rotation when the gyroscope is spinning, than if it were not spinning.

Question

How do I characterize this apparent "lifting" force mathematically? (Is "force" even the right way to describe it?) That is, if I have a flywheel of a known mass and radius, spinning at a known rate, how do I calculate the force that it will apply to the palm of the person wielding it on a rod of known length, through a 90 degree rotation?

Answer 1

No. Torque is the time rate of change of angular momentum $$\vec{\tau} = \frac{\operatorname{d}\vec{L}}{\operatorname{d}t},$$ just like force is the time rate of change of linear momentum $$\vec{F} = \frac{\operatorname{d}\vec{p}}{\operatorname{d}t}.$$ Thus, when $\vec{L}$ is large it takes a large torque to make a significant difference in it.

Note that the angular momentum does not in any way redirect the force. $\vec{F}=m\vec{a}$ still applies. What's happening is there's gravity pulling down, and the hand pushing up. Therefore there is no net force. Because they act at different locations, though, there is a net torque that acts perpendicular to the angular momentum. Just like when $\vec{F}$ is always perpendicular to $\vec{p}$ you get circular motion, this produces circular precession.

Answer 2

How do I characterize this apparent "lifting" force mathematically? (Is "force" even the right way to describe it?)

Let me address this specific subquestion.
This is about the distinction we need to make between 'force' and 'inertia'.

First a very simple case, to establish the basics.
You have some wooden workpiece on a bench, and you want to hammer a wooden peg into a hole, the fit is tight.

If you just tap the peg the wood-wood friction is strong enough to bring your mallet to a dead stop. The tap does exert a force on the peg, but a tap is not enough.

As we know, we say that the mallet exerts a force when it strikes the peg because the mallet has inertia; a force is required to bring the moving mallet to a stop. The peg exerts that force upon the mallet. The faster you swing the mallet, the stronger the force that is required to bring it to a dead stop. At some point the wood-wood friction between the peg and the wall of the hole is not enough to stop the mallet completely, and you get to hammer the peg in.

We call it a 'force' when there is force pair. When object A exerts a force upon object B then object B exerts an equal and opposite force upon A.

Inertia itself cannot be categorized as a force because there is no 'equal and opposite pair'. The definition of inertia is that a force is required to change the velocity of an object.

So:
A complete picture features both inertia and a force pair. Of course, in everyday language we just say that the mallet strikes are forcing the peg into the hole

Spinning gyroscope wheel
Ultimately, the way a spinning gyroscope wheel responds to motion that is imposed upon it arises from inertia. (As to how that happens: I added a link to a discussion of gyroscopic precession (here on physics.stackexchange) in a comment to your question.)
With you physically holding the axle of the gyroscope wheel the wheel is indeed exerting a force upon you everytime you are in the process of changing the orientation of the spinning wheel.

Of course, there is that surprising thing that the direction of the spinning wheel response is not opposite to the direction of the force you imposed upon it. So yeah, you question: 'is "force" even the right way to describe it?' is well founded.

The difference in direction goes into internal stresses of the gyroscope wheel. A non-rigid wheel that can flex from side to side will not have the same behavior as a rigid gyroscope wheel; try to reorient it and it will deform violently.