angular momentumgyroscopesnewtonian-mechanicsrotational-dynamicstorque
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?
The reason why a gyroscope does behave in this strange way is that if you try to rotate it's axis in some direction, the "endpoints" of this axis have to be pushed perpendicular to what our first intuition would say.
In order to verify why the axis starts rotating in this strange way, let's make some simplifications: the gyroscope consists of two identical rotating particles and the particles rotate much fasten than we rotate the axis.
To visualize how the particles move, I've made the following drawing: the circle depicts a sphere on which the particles revolve, the center of the particles is always the center of the sphere and the axis of rotation precesses counterclockwise. The rotation axis is initially in up-down direction. Blue and red correspond to different particles, continuous and non-continuous lines correspond to the particle being on our side or the back side of the sphere. Blue starts at B and red at I.
On the figure, you can notice that because of the curvatures of the trajectories, the particles on our side must be pushed up while those of the back side must be pushed down (when the axis hasn't yet rotated much). This force has to be compensated - in order to keep the axis rotating counterclockwise, the axis has to be pushed away from the downside and pulled from the upside.
Going back to the gyroscope, if the axis is tilted in the beginning, gravitation "tries" to make it fall down, but instead this axis will rotate in the direction perpendicular to it. Given that information, you can probably figure out yourself in which direction the gyroscope will preceed. I think it should do this in the same direction as it rotates, if I visualized this process correctly with a pencil.
The use of the words clockwise and anticlockwise also require a statement about the viewing direction.
Imagine that you want to find the torque due to a force $\vec A$ about the origin as shown in the diagrams below.
In both cases the torque $\vec \tau$ is given by $$\vec d \times \vec A = d\hat x \times A \hat y = dA\hat z$$ ie in the positive z-direction although in the left hand diagram it is an "anticlockwise" torque and in the right hand diagram it is a "clockwise" torque.
Best Answer
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:
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.)