Nobel prize winners have been fascinated by it:
You can learn about the underlying mathematics. You can solve differential equations. But do you really feel why it does not fall?
When gravity tries to topple a spinning top, the top escapes sideways, sort of, by precessing: The axis of rotation is changing slowly.
There is a force, so there is a torque. Torque is the change of angular momentum, similar to a force being the change of (linear) momentum. The torque vector is tangent to the circle the tip of the top is tracing out. So, everything is consistent. But it is only valid if the gyroscope has reached a steady state of motion. What happens if you support it and remove the support suddenly?
Richard Feynman himself figured that anybody in his right mind is correct to assume that the gyroscope will fall … a little:
The gyro actually does fall, as we would expect. But as soon as it falls, it is then turning, and if this turning were to continue, a torque would be required. In the absence of a torque in this direction, the gyro begins to “fall” in the direction opposite that of the missing force. This gives the gyro a component of motion around the vertical axis, as it would have in steady precession.
The motion of the gyroscope not subject to an external force should be easier to understand – or is it?
I’ve recently shown my artistic interpretation of a geometric way to visualize the motion of the free spinning top.
The red and violet cones symbolize the spinning top’s symmetry, at two different points of time. This red/violet cone embraces the blue one fixed in space, rolling along its surface without sliding. The axis of the red/violet cone traces out the yellow cone.
As I find the English names of these cones hard to google, I am adding a German reference. This author calls the motion of the free spinning top nutation or regular precession, explaining that the usage of these terms in the literature is ambiguous.
But wait – doesn’t precession need a torque? And there is no torque in this case? Why is it precessing then? It would have helped if I had added a few mathematical symbols and practiced calligraphy. The angular momentum is fixed in space in this case: It is the symmetry axis of the blue cone. What we see precessing is the symmetry axis of the spinning top. The explanation of precession above has swept under the rug that angular momentum, the current axis of rotation, and the symmetry axis of the gyroscope are all different. The visible symmetry axis is the red/violet one. The axis of rotation is where the blue and the red cone meet.
Nutation is usually associated with an additional wobbling superimposed on the precession. Why would you equate this regular precession with nutation? Here, I’d need to draw a realistic free spinning top supported in a point: It does not look like a cone. It is more like a heavy ring connected to the support point with thin spokes. Regular precession of such a thing could as well be called wobbling!
This big wheel would be a “rather fat” gyroscope – with an oblate ellipsoid representing its principal moments of inertia. But you could think of a spinning top floating in deep space where no support is needed. It could have any shape!
The three dancing cones would look like this: The cone associated with the spinning top does not embrace the fixed one. It is again rolling along the other one’s surface:
This is a smaller – A5-sized – drawing on watercolor paper (as opposed to gessoed canvas board before).
I’ve used only watercolor pencils and no other pens for enhancing the lines. In contrast to the Dancing Cones, you can also see the view from the side enhanced with color. The “3D” axonometric projection emerges from the view from the top (on the bottom of the drawing) and the view from the side (top right).
For this thin gyroscope, I have used yellow for the rolling cone symbolizing the spinning top, green for the “cone of nutation”, and blue for the fixed cone whose axis is the angular momentum:
But – as usual – this should not be considered a scientific visualization first!
This is an abstract pieces of art that happens to be inspired by physics … and by stained glass!
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I had written a more detailed post about the physics of the gyroscope and intution back in 2012 – for a long time this has been the most popular post on this blog.
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