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Author Topic: Two online PDFs on angular momentum -- the key to top behavior  (Read 270 times)

Jeremy McCreary

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Two online PDFs on angular momentum -- the key to top behavior
« on: December 20, 2018, 02:56:43 AM »

In many ways, top behavior is best viewed through the lens of angular momentum (AM) -- in part, because this quantity is always conserved in the absence of external torques, even when mechanical energy is not. Problem is, many find the concept of AM a little too abstract for comfort. Here, I'd like to share 2 free online PDFs relating AM and tops. The first might make AM a little more accessible.

1. Rotational Dynamics and the Flow of Angular Momentum by physicists F. Herrmann and G. Bruno Schmid looks like a handout for undergrads. In it, we see AM portrayed in an exciting new light -- namely, as a "substance-like quantity" (SLQ) that effectively "flows" without loss from one mass to another via the torques they exert on each other.

The article lays out a profound and very useful analogy between 2 absolutely conserved SLQs -- AM and electric charge. If either of these ever seems to have gone missing, you just lost track of where it went. And since the same math turns out to govern both with rare exception, you can leverage a basic understanding of electricity into a pretty good understanding of the way AM flows through spintoys.

In this view, angular speed differences correspond to voltage differences, and AM flows to electric currents. For example, when you spin up the flywheel of a toy gyro and release the cage, the angular speed difference between the two causes AM to flow from the faster flywheel to the slower cage until their angular speeds equalize. AM ceases to flow between flywheel and cage thereafter, and the entire gyro spins down to a fall as if of one piece. But AM continues to flow from flywheel and cage to the surrounding air until each component comes to rest.

The spinning flywheel and cage separately acquire and store and lose AM like a capacitor does charge. Their axial moments of inertia are their capacitances. The flywheel bearings act as the AM conductors here, and the frictional bearing torques as the AM "currents" between flywheel and cage. Meanwhile, aerodynamic braking torques mark the ongoing leaks of AM from both flywheel and cage to the surrounding air. Once the gyro's total AM has fallen below a critical AM, for whatever reason, the whole thing topples, regardless of the relative flywheel and cage AMs at the time. Very cool.

2. The first 1.5 pages of A Historical Discussion of Angular Momentum and its Euler Equation by physicist Amelia Carolina Sparavigna offer some interesting historical notes about tops as the author works up to the modern concept of AM. (Don't let the "Euler Equation" business scare you off. It comes up after the part I'm recommending.)

A great quote from Newton identified the spinning top as a prime example of what we now call AM in action. Newton also identified aerodynamic drag as the main reason for spin decay.

Problem is, the vagaries of air and contact resistance are still hard to quantify. Top physics treatments daring to confront spin decay at all usually make up some hopelessly simplistic friction formula and ignore air resistance altogether. Models like these predict many important aspects of top behavior, but no one even tries to predict spin decay curves and spin times. Wishing spin decay away isn't an option for real top players.
« Last Edit: December 21, 2018, 03:36:08 PM by Jeremy McCreary »
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