The Spinning Gyroscope and Intuition in Physics

If 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.

Antique spinning top

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.

Generally, 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).

10 Comments Add yours

  1. howardat58 says:

    Reading your older posts I found this one. The funny thing is that when I was a kid I had a gyroscope with eiffel tower and played with it manymany times. later when doing 3D vector mechanics I understood the gyroscope theoretical development, but I never believed in it !

    1. elkement says:

      Thanks, Howard! I guess it is because the usual explanation describes the steady-state, but neither the initial “falling a bit” nor the ongoing wobbling (nutation).
      BTW you have found one of the top posts on this blog – seems many people are searching for ‘gyroscope’… ;-)

      1. howardat58 says:

        The other thing we tried was to hold a bicycle wheel by its axle, spin it round, and then tilt the axle – big surprise, the first time anyway! Re the bit ” just as the moving bicycle is sort of stabilized by its turning wheels.” I think that this is a very minor part of the dynamic behaviour of a bike. Only if the bike had no steering would it be the main thing, but then the bike would fall over pretty soon.

        1. elkement says:

          I think that it is important that the angular momentum parallel to the symmetry axis (turning wheels) is much bigger than the additional angular momentum introduced by trying to turn the gyroscope / ‘make it fall’.

  2. M. Hatzel says:

    Physics is my intellectual blind spot, so I think I may be able to say that as a test subject I’ve probably proven your point: a light bulb flickered in my brain. I think I’ve seen this motion in figure skaters spinning. They can adjust their axis and orientation as they spin, and can appear to spin faster, but what I’m probably seeing is a change in precession as the wide and loopy spin pulls in tight again?

    What intrigues me most here is the process of rethinking what we take for granted as fundamental understanding within our fields of study. I do this with language, and the inquiry often leads me back into a new relationship with an idea, one that I (hope) is more sensitive to the reality of complex possibilities.

    1. elkement says:

      Thanks for your comment – it is interesting to hear from somebody with a non-tech perspective! There is one effect related to skaters that can be explained probably without going through the details of the gyroscope motion: When a skater pulls her arms closer the body, she spins faster. This is related to the conservation of angular momentum – the same angular momentum can be achieved with a large average distance of masses from the axis of rotation and slow rotation or small distance and fast rotation (which – I know – will trigger the question: What is angular momentum really…).
      Difficulties in grasping physics arise from the fact that the same phenomenon can really be tackled starting from different “laws” or “principles” which are actually the same under the hood (and be be derived from each other mathematically), but they seem to have different philosophical connotations. E.g. you can think about classical mechanics by considering how a force moves a body over a tiny distance, then due to the reconfiguration of the body the force changes a.s.o… In this way you “see” the motion as progressing step-by-step.
      On the other hand, you could start an explanation of many fields in physics by considering “symmetries” or “conservation laws” (such as angular momentum is constant) – this gives you the feeling that the world is in a sense more stationary or eternal. It is very interesting to see the differences between fields in physics and the historical evolution: The step-by-step / force approach is rooted in classical (ancient) mechanics, but in modern (quantum physics) you describe everything in terms of symmetries.

      1. M. Hatzel says:

        I’ve read this several times since your generous reply. I can only laugh at myself for understanding so little, but I had this lovely recollection of how I felt when first encountering literary theory. I had been so excited/scared/thrilled to discover Derrida and Heidegger. As you suggest, the translation of idea into working application is the challenge, even in ideas about culture and literature. Thanks for taking the time to explain.

      2. elkement says:

        Thanks – I am not sure if I really did explain it well in my comment. I believe everybody who has become familiar with a discipline starts using ‘shortcuts’ in plain English – not really ‘jargon’, but also everyday language terms that are used in a specific way. There is a reason that science writing is a craft in its own right, though there are some professional scientists who are great communicators at the same time – admirable.

  3. elkement says:

    Yes, I agree – dealing with fluids requires a new level of “intuition”. E.g. in “Newton’s” equation of motion for continuous matter the nature of velocity changes drastically: velocity is not any more associated with a particular point mass moving through space, but it becomes a function of the spatial coordinates (it’s a field actually).

  4. bert0001 says:

    I could follow this one without having to resort back to my old textbooks. However, the helium balloon, i would rather like to isolate as a problem in fluidomechanics (which is physics) as my first reaction was also that the air in the car will feel deceleration stronger than the helium in the balloon … but then, we can never trust our brain, only the experiment.

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