Intuition and the Magic of the Gyroscope – Reloaded

I am baffled by the fact that my article The Spinning Gyroscope and Intuition in Physics is the top article on this blog so far.

So I believe I owe you, dear readers, an update.

In the previous article I have summarized the textbook explanation, some more intuitive comments in Feynman’s Physics Lectures, and a new paper by Eugene Butikov.

But there is an explanation of the gyroscope’s motion that might become my new favorite:

Gyroscope Physics by Cleon Teunissen

It is not an accident that Cleon is also the main author of the Wikipedia article on the Coriolis flow meter, as his ingenious take on explaining the gyroscope’s precession is closely related to his explanation of the flow meter.

The Coriolis force is a so-called pseudo-force you “feel” in a rotating frame of reference: Imagine yourself walking across a rotating disk, or rolling a ball soaked in white color rolling across a black disk and watch its trace. In the center of the rotating disk, there is no centrifugal force. But you would still feel being dragged to the right if the disk is rotating counter-clockwise (viewed from the top).

This force dragging you to the right or making the path of the ball bend even in the center – this is the Coriolis force. It also makes tubes bend in the following way when a liquid flows through them and allows for determining the flow velocity from the extent of bending:

Vibration pattern of the tubes during mass flow. Credits: Cleon Teunissen.

Vibration pattern of the tubes during mass flow. Credits: Cleon Teunissen, used with permission – Coriolis flow meter

The “loop” formed by the tubes rotates about an axis which is parallel to the direction of the flow whose speed should be measured (though not the full 360°). Now the Coriolis force always drags a moving particle “to the right” – same with the volume elements in the liquid. Note that the force is always directed perpendicular to the axis of rotation and perpendicular to the velocity of the flowing volume element (mind the example of the rolling ball).

In the flow meter, the liquid moves away from the axis in one arm, but to the axis in the other arm. Thus the forces acting on each arm are antiparallel to each other and a torque is exerted on the “loop” that consists of the two arms. The loop is flexible and thus bent by the torque in the way shown in the figure above.

Now imagine a gyroscope:

Gyroscope with Coriolis force per quadrant indicated. Credits: Cleon Teunissen.

Gyroscope with Coriolis force per quadrant indicated. Credits: Cleon Teunissen – Gyroscope Physics

Gravity (acting on the weight mounted on the gyroscope’s axis) tries to make the gyroscope pitch. Cleon now shows why precession results in an “upward pitch” that compensates for that downward pitch and thus finally keeps the gyroscope stable.

The clue is to consider the 4 quadrants the gyroscope wheel consist of separately – in a way similar to evaluating the Coriolis force acting on each of the arms of the flow meter:

The tangential velocity associated with the rotation about the symmetry axis of the gyroscope (“roll”, spinning of the wheel) is equivalent to the velocity of the flowing liquid. In each quadrant, mass is moving – “flowing” – away from or to the “swivel axis” – the axis of precession (indicated in black, parallel to gravity).

The per-quadrant Coriolis force is again perpendicular to the swivel axis and perpendicular to the “flow velocity”. Imaging yourself sitting on the blue wheel and looking into the direction of the tangential velocity: Again you are “dragged to the right” if precession in counter-clockwise. As right is defined in relation to the tangential velocity, the direction of the force is reversed in the two lower quadrants.

A torque tries to pitch the wheel “upward”.

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If you want to play with gyroscopes yourself: I have stumped upon a nice shop selling gyroscopes – incl. a steampunk version, miniature Stirling engines, combustion engines (… and strange materials such as ferrofluids).

The Spinning Gyroscope and Intuition in Physics

Antique spinning topIf we would set this spinning top into motion, it would not fall, even if its axis would not be oriented perpendicular to the floor. Instead, its axis would change its orientation slowly. The spinning motion seems to stabilize the gyroscope, just as the moving bicycle is sort of stabilized by its turning wheels. This sounds simple and familiar, but can this really be grasped by intuition immediately?

I do not think so – otherwise it would not have taken us 2000 years to get over Aristotle’s assumptions on motion and rest. And simple experiments demonstrated in science shows would not baffle us – such as the motion of a helium balloon in an accelerating car.

The standard text-book explanation goes like this: There is gravity, as we assume that the spinning top is not supported in its center of gravity. Thus there is a torque. The gyroscope is whirling, thus it has angular momentum. A torque corresponds to a change in angular momentum, analogous to a force resulting in a change of (linear) momentum. The torque vector is perpendicular to gravity and to the axis of the gyroscope. Thus the change in angular momentum is always perpendicular to the current angular momentum vector and the tip of the spinning top moves in a circle. The angular momentum vectors changes all the time – not in length, but in direction – which is called precession.

As Richard Feynman pointed out in his Physics Lectures, this explanation constitutes rather mathematical step-by-step instructions than a real explanation. We do not see immediately why the spinning top precesses instead of falling to the ground.

Our skepticism is justified: The text-book explanation does not fully expound the dynamics of the systems and explain what really happens – in the very moment the spinning top starts to move. It rather refers to a self-consistent solution: If the gyroscope would already precess in a circle, that circular movement is consistent with the torque. As everybody in his right mind (R. Feynman) would assume, it actually might fall a bit if it is released.

Spinning Top - PrecessionGenerally, the tip of the gyroscope keeps tracing out a wavy or loopy path, which is called nutation.

If the spinning top nutates / starts falling, it looses potential energy. This has to compensated by an increase in rotational energy, the velocity of the tip of the gyroscope is not a constant. (Note that the total angular momentum of the gyroscope is composed of contributions from the fast spinning motion and the slow precession). The tip of the gyroscope moves on a curved trajectory bending upwards, which finally leads to overshooting the average height.

Friction can make the wobbling decay and finally turn the trajectory into the simple-text-book-path. This simulation allows for turning on friction (which is also equivalent to Feynman’s explanation).

An excellent explanation can be found in this remarkable paper (related to the simulation): The gyroscope is set into rotational motion while still supported. When “gravity is suddenly turned on” by removing the support, the additional vertical component of the angular momentum – due to to precession – is suddenly turned on. The point is that the initial angular momentum is parallel to the symmetry axis of the gyroscope, and the axis starts from velocity zero. The total angular momentum – still parallel to the symmetry axis – is the sum of the one related to precession and the one related to the gyroscope’s fast movement. So the latter is not parallel to the axis any more: The tip of the axis starts tracing out the loopy path (nutation) when it precesses. Only if we tune the angular frequency carefully before we release the spinning top, the text-book solution can be obtained. In this case precession is really maintained by the torque.

So do we understand the gyroscope intuitively now? A deep understanding of angular momentum and torque is a pre-requisite in my point of view. On principle, all of classical mechanics can be derived from Newton’s laws, so the notions of force and momentum should be sufficient. Nevertheless, without introducing angular momentum, there is no way to explain the motion of the gyroscope briefly.

Why do we need “torque” in general? Such concepts are shortcuts that allow for a concise description, but they also reveal the underlying symmetry or essential aspects of a problem. You could describe the dynamics of a rigid body by considering the motion of all little pieces the body is composed of. But since it is rigid, actually two points would be sufficient. You can select any two points or basically any set of independent coordinates – 6 independent numbers.

The preferred choice is: 3 numbers – such as Cartesian co-ordinates, x,y,z – describing the motion of the center of gravity and 3 numbers describing the rotation of the body. You need two numbers to denote the direction of the axis about which to rotate (similar to two longitude and latitude to describe a point on a sphere), and one number to denote the angle – how much you rotate. You could also describe any rotation in terms of the components discussed for the gyroscope: precession, nutation and internal rotation.

Then Newton’s equation of motion for the rigid body can be re-written as a law of motion for the center of gravity (Force equal change of momentum of the center of gravity) and a law for two new properties of the system: the torque equals the change of the angular momentum. Actually, this equation defines what these properties really are. Checking the definitions that have evolved from the law of motion we conclude that the angular momentum is linear momentum times the lever arm, and the torque is force times lever arm. But these definitions as such would not make sense if they would not have been generated by the reformulation of Newton’s law.

I think we sometimes adopt or memorize definitions carelessly and consider this learning because these definitions are required by standards / semi-legal requirements and used within a specific community of experts. But there is no shortcut and no replacement of understanding by rote learning.

I believe you need to keep the whole entangled web of relations between fundamental laws in mind, but it is hard to restrict the scope. We could now advance from gyroscope and angular momentum to the deeper connections between symmetries and conservation laws. In order not get stuck in these philosophical musings all the time – and do something useful (e.g. as an engineer), you need to be able to switch to shut-up-and-calculate-mode (‘Shut up and calculate’ is often attributed to Richard Feynman, but I could not find an authoritative confirmation).