Here's a question my first year physics students raised in revising for their upcoming exam.

You are all probably familiar with the experiment were you seat a person on a stool which is free to rotate, with their feet off the ground. They hold a bicycle wheel mounted on an extended axle, with one hand on either side of the wheel. Let's say the axle of the wheel is initially horizontal, and the person holds the two ends of the axle, one in each hand, with one hand on each side of the wheel. The wheel is given a spin. The person then moves one hand up and the other hand down, tilting the wheel slightly. The result of this is that the person and stool start to rotate about a vertical axis.

This is typically used as an example of conservation of angular momentum of the person-wheel-stool system. Now, here's the problem.

Looking at the person-stool system alone (i.e. ignoring the wheel), the person started with zero angular momentum, and ended up rotating around a vertical axis with some positive angular momentum. Angular momentum is not, of course, conserved here since the person-stool system is not isolated.

Since the final angular momentum vector of the person-stool system is directed, say, upwards, we conclude that there must have been a new torque upwards as well. However, if we look at the torque which the person applied to the wheel in tilting it, that torque was directed parallel to the floor (i.e. horizontally, not vertically). Hence, it seems that the reaction forces of the wheel on the person cannot account for the person and stool starting to rotate.

So, where does the required torque come from to cause the person-stool system to start rotating?

So, where does the required torque come from to cause the person-stool system to start rotating?

I remember I was the poor student who had to sit on the chair during the experiment in front of the entire auditorium ;).

I think the torque comes from the wheel; which is something you can feel if you undergo the experiment: the force that the wheel exerts on your hand e.g. towards the left from the perspective of the person sitting on the chair, is quite strong. The torque associated with that force is pointing upwards (or downwards if the force is directed to the right, causes the opposite direction of rotation), and I would say that the force of the wheel on the person is what causes the person to spin (e.g. direct the angular momentum towards the torque of the wheel-force).

In the composite system, this force "disappears" (cancels out) because of Newton's third law, perhaps that you overlooked this ? This sounds like a typical "ok, there must be a force and a torque somewhere, where do we need to introduce it a posteriori to explain what happens" problem of Newtonian mechanics.

Yes, in this case the torque is quite strong because there is friction associated with the turning as well as inertial mass. Of course u would still turn even if u were using a frictionless wheel!

Crisp,

Thanks, but I'm looking for a slightly more complete explanation.

I haven't overlooked the composite system. In the usual first-year presentation, the composite system of the wheel plus the student-stool system is examined, and conservation of angular momentum is invoked to explain the observed rotation of the stool when the wheel is tilted. But it must be possible to explain the motion of the stool alone, and that definitely requires a vertical torque from somewhere. The question is: where?

James R,

I seem to recall that gyroscopic forces operate through a 90 degree shift which induces the precessional velocity. If you restrain precessional torque the reflected counterforce returns and cancels gyroscopic force. The gyroscope doesn't act like a gyroscope anymore but will fall to the table if you stop its precessional rotation.

It seems to me the body in the chair is free to move and is merely riding along with the precessional torque.

Precessional torque occurs as gravity lowers the angle (reorients) of the gyroscope as it loses rotational energy. In this case you are forcing the reorientation just as gravity would for a gyroscope suspended by one end.

Is it any easier or harder to tilt the wheel if it is done closer to the stool's axis?

It's harder, Pete. You need a bigger force to create the same torque.

If you are handed the spinning wheel with the axle in the vertical position, what happens? Does the stool spin or do you just sit there?

You just sit there, unless you start tilting the wheel.

I have always disliked that spinning wheel and stool thing.

I think you are pushing against the precession with a net horizontal force to the left or right as you tilt the wheel. This horizontal force causes a moment about the stool axis and gives the vertical torque.

Say if the wheel is spinning away from you and the axle is parallel to the floor. You twist the wheel counterclockwise. The wheel is spinning clockwise as seen from above and you and the stool must now be spinning counterclockwise. A horizontal force was acted on your arms to your left. If the wheel was rotating towards you initially and you twisted it counterclockwise again, then there must be a horizontal force acting to the right to make you spin clockwise.

There seems like there would also be a torque applied parallel to the floor when twisting like you said, but you are obviously not free to turn about that axis so you don't realize it.

I think it has to be explained through overcoming the precession of the wheel by pushing to the left or right horizontally as you twist the wheel.
Like I said, I never liked the experiment so I am trying to figure it out myself.

http://science.howstuffworks.com/gyroscope2.htm

I think there are 2 torques that are produced from the single external torque as the wheel spins. Like here

http://static.howstuffworks.com/mpeg/gyro.mpg

the wheel not only resists gravity torque about the horizontal axis but it also begins to spin about the vertical axis of the string. If you follow the forces around the wheel here as shown, there is the position when they have rotated 90 degrees from the start location. At this point they are now producing a torque about a vertical axis. As they rotate another 90 degrees, they are back to resisting torque about the horizontal axis.

http://science.howstuffworks.com/gyroscope2.htm

Here is another example of the torque being produced about the orthogonal axis of external torque applied

http://www.gyroscopes.org/behaviour.asp

Thanks, haynewp.

I think I've more or less solved the problem. The solution is the the torque is the rate of change of angular momentum. While the average torque for a finite tilting of the wheel might seem to result in no reaction torque on the stool in the right direction to cause it to rotate, when you look at the instantaneous torque, referenced to the rate of change of the angular momentum, there is indeed a torque in the right direction during the time the axis of the wheel is on the move.

It's an interesting thing, I think, particularly since none of the first year texts (at least the ones I've read) examine this aspect of the problem. I'd still like to come up with a concise explanation, though.