What determines the direction of precession of a gyroscope?

I understand how torque mathematically causes a change to the direction of angular momentum, thus precessing the gyroscope.

However, the direction, either clockwise or counterclockwise, of this precession seems to me a bit arbitrary beyond mathematical definitions. (What if torque were defined by a "left hand rule", for example?).

What's the fundamental physical reason that the precession of a gyroscope would be directed in a certain direction?

Related question.

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It is not beyond mathematical definition, there is an equation for it. –  Vladimir Kalitvianski Dec 23 '12 at 21:00
As a matter of strategy saying " I think I am asking the same question as was raised [in some earlier question]" is a way to get your question closed as a duplicate, except that I think the two are related but not-identical. –  dmckee Dec 23 '12 at 21:06
Actually in general OP is before asking urged to first investigate if a question is a duplicate. Also in general, I'm more reluctant to close a question that explicitly mentions duplicate-like Phys.SE posts, because OP obviously must not have been satisfied with the answers given there. –  Qmechanic Dec 27 '12 at 10:05

OK, the direction of precession of a gyroscope.

The first image shows a gimbal mounted gyroscope wheel. From outside to inside there is a yellow housing and a red housing.

I define three axes:

• Roll axis - the gyroscope wheel spins around the roll axis.
• Pitch axis - motion of the red housing. As you can see, the gimbal mounting ensures the pitch axis is perpendicular to the roll axis.
• Swivel axis - motion of the yellow housing.

First I will discusse a state of uniform precession:
the wheel is spinning fast.
there is some swivel.

The second image shows a single quadrant. The idea is to think of that quadrant as having a fixed position relative to the red housing, with sections of spinning wheel moving through that quadrant

The mass moving through that quadrant is moving towards the swivel axis. Think of a point particle somewhere along the wheel rim, for example the point where the green arrow starts. That point is circumnavigating the swivel axis, with a corresponding velocity. Moving closer to the swivel axis that point will tend to pull ahead of the overall swiveling motion.

The brown cilinder represents a weight that tends to pitch the gyroscope wheel.

In two of the quadrants the wheel mass moving through that quadrant is moving towards the swivel axis, in the other two away from the swivel axis.

The green arrows represent tendency for each quadrant when the wheel is spinning and swiveling. The tendencies from the four quadrants combined add up to a pitching effect.

(Incidentally, given a spin rate and precession rate one can calculate the corresponding tendency to pitch by integrating the effect around the wheel.)

The reason the brown weight is not pitching the wheel down is that the combination of the wheel spinning and swiveling gives a tendency to pitch up that keeps the brown weight from pitching down.

The key factor is motion. The tendency to pitch as represented by the green arrows arises if the wheel is spinning and swiveling. Likewise, when there is spinning and pitching then swiveling motion starts.

In a demonstration the gyroscope wheel is initially just spinning. Then a torque is added. The gyroscope wheel yields a little to the torque, the motion of yielding to the torque is a pitching motion, that gives a swiveling motion, that swiveling motion counteracts the torque, so the wheel doesn't pitch any further. In demonstrations the gyroscope wheel is usually spinning so fast that the pitching motion is imperceptably small.

General remark:
Concepts such as the spin angular momentum vector are very powerful, but they are highly abstract concepts, and inaccessible to physical understanding. For understanding cause and effect one has to track down the physics in terms of force/momentum.

This explanation is adapted from the article about gyroscope physics that is on my own website.

(Initially I had given only links, hence the conversation between me and manishearth.)

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To prevent link rot and other concerns, it is better if you write your own answer here instead of linking elsewhere. –  Manishearth Dec 26 '12 at 20:32
Do you recommend that I put a brief version of the explanation here? I feel it's superfluous to copy content. I'm aware it can be done; I saw that uploading images is possible –  Cleonis Dec 26 '12 at 22:28
Yep- a brief version, or a blockquoted (there's a button on the toolbar for that) copy-paste (maybe with some normal text above/below the blockquote to tailor the answer to the question better) –  Manishearth Dec 26 '12 at 22:53
I tried to post an excerpt from the article on my website, but as a new user I don't have permission to upload images. –  Cleonis Dec 27 '12 at 1:53
Just link to them, we can fix that for you –  Manishearth Dec 27 '12 at 3:23

Notice that angular momentum is assigned a positive or negative value based on a Foo-hand rule just like torque?

I belive that as long as you use the same hand for both direction you will get the same result for the direction of precession.

In other words, the arbitrary choice does not affect the physics, just our assignment of positive or negative values to some of the quantities.

Notice that I have not actualy said why the particulr direction occurs, but I haven't a good closed form answer, so I'll wait for someone else to cover that.

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I concur, the mathematical definitions drop out. A foo-hand rule is used to assign a direction of vector of angular momentum, and then the same foo-hand rule is applied in the rule for finding the direction of gyroscopic precession. As in all physics: as long as applied consistently throughout right-hand-rule and left-hand-rule are interexchangable; the foo-hand rule is not in any way a factor; it doesn't explain anything, it doesn't introduce arbitrariness. (Still, despite its irrelevancy I have seen well meaning physicists offering invocation of foo-hand rule magic as "explanation".) –  Cleonis Dec 27 '12 at 23:46

It basically is conservation of angular momentum that determines which way the gyroscope will rotate. The important thing to realize, is that angular momentum is a vector quantity, with certain magnitude, and direction pointing along the axis of rotation (again, apply right-hand rule).

The only thing you need to solve in principle is $$\frac{d\vec{L}}{dt}=\vec{r}\times\vec{F}$$ where $\vec{L}$ is angular momentum, and $\vec{F}$ is the force (gravity) applied at position $\vec{r}$.

The simplest model of a gyroscope that you can analyze using this equation, is a disk rotating on an axis, that is placed perpendicular to another axis (and gravity) at some distance. (Of course everything is rotating frictionless).

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I understand how the equations lead to the result, with conservation of angular momentum as a vector quantity. Maybe I'm being unreasonable, but this still feels a bit unsatisfying as a physical reason. –  high schooler Dec 24 '12 at 16:53
I think part of my confusion may arise from how we define torque and angular momentum as vectors, through the right-hand rule. If we define the torque vector as perpendicular to the rotational motion, why should it change the angular momentum as a vector? It seems to be almost like a definition made to fit with observation (that the torque is perpendicular to rotational motion and the gyroscope just happens to precess in that direction). Looking at the vector definitions of these quantities just doesn't seem to me like a complete physical explanation. –  high schooler Dec 24 '12 at 16:57
And physically, I don't see how the force/torque acting (downwards) upon the particles of the spinning wheel of the gyroscope should compel the system (the rotating wheel) to move in a certain horizontal direction; there is not one particle compelled in a certain horizontal direction with the force of gravity. –  high schooler Dec 24 '12 at 17:08
@highschooler Think about a disk rotating about an axis that is slightly shifted from the symmetry axis. What happens with the centre of mass? I think this is the way you should look at it. It also explains why precession reverses when rotation is reversed. –  Bernhard Dec 24 '12 at 17:12