Conservation of angular momentum and forced precession.

[broli]

The total moment of inertia of a thin ring mass around its central axis is given by  I=mr² and I=1/2mr² around its other (diametrical) axes.



Let's take the example of a ring mass attached to an axle that is set in motion around one of this I=1/2mr² diametric axis. Using the parallel axis theorem we can find the total moment of inertia of the system to be Itotal  = Md² + 1/2Mr² as shown below.



This is all basic stuff. But what happens when we give the ring mass some spin angular momentum while we rotate it around like we did before? Well now this looks like a precessing gyroscope. However the precession here is forced due to it being rotated around. But whether it's forced or not the crucial part is that a precessing gyroscope has no angular momentum along its precession axis because its "rotation" is merely an illusion called "precession" unlike real diametrically rotating ring mass that has a MoI of 1/2Mr²

In conclusion the total moment of inertia reduces to the radius/distance squared times the center of mass of the ring because the diametric moment of inertia component is dropped due to gyroscopic precession.



That is the theory at least. The real question is does spinning mass actually affect the total moment of inertia of a system. And if it does what does it mean to have a mechanism that allows you to significantly drop the rotational mass of said system while it is rotating. Angular momentum conservation is king at the end of the day and if rotational mass suddenly drops then the only quantity that can go up proportionally is angular velocity (because of L=I⋅ω) to keep angular momentum constant. However rotational energy has a squared dependency on angular velocity E = 1/2 Iω² so where did that extra energy come from?

[Verpies]

IMO 3D rotational motion is what constitutes matter internally microscopically .  This motion has the dimensions of t³/s³ or Joule cubed.

Perhaps your macroscopic rotations subtract from this microscopic rotation.
Has anyone ever measured a gyroscope forcefully rotated in two orthogonal directions at the same time ?

[broli]

Has anyone ever measured a gyroscope forcefully rotated in two orthogonal directions at the same time ?

I haven't seen anyone measure this before and it's a rather simple experiment to do too.

Prof. Eric Laithwaite's famous video on gyroscope is perhaps the only one that comes close to it. Here he forcefully rotates a gyroscope made of single masses and you see a hint of what is going on. The masses deflect at the top and bottom as they want to make the gyroscope angular moment align with the global angular momentum.

https://youtu.be/WEdgbRGDRBI?t=3408

He explains this in a follow up video here:
https://youtu.be/smRzs07KHg8?t=973

Even if this effect is known and used I don't think many people know that a rotating gyroscope is actually precessing even if there is an absence of apparent torque which is making it precess. However the torque is very much there but it is absorbed by the axle and bearings.

But if rotating a spinning mass around its perpendicular axis of spin is exactly the same as precessing said mass then what happened to its diametric moment of inertia? And this comes then back to the original question. What happens in a transient situation were we started with a non spinning ring mass in a rotating frame and suddenly started spinning it turning it into a rotating spinning mass. It should go from maximum diametric moment of inertia to none. In other words the rotating mass suddenly drops and thus angular velocity must increase in accordance to conservation of angular momentum but this then also would lead to rotational energy increase. 

[broli]

Here is perhaps a better illustration between the two different cases.



To the naked eye they appear to be behaving exactly the same. If you didn't know the ring mass was spinning you wouldn't see a difference between both scenarios as both rings look like they are rotating about the same axis. But in reality they are completely different behaviors as one is rotating and has a real angular momentum around that rotation axis while the other is precessing and has zero angular momentum around its precession axis. This is more evident if you look at the individual velocity vectors that make up the ring mass. In the case of precession no velocity vector would give rise to angular momentum along the precession axis. This is how gyroscopes work and give the illusion of rotation around the precession axis. This is basic gyroscopic behavior and nothing new.

However the question is what would happen during a transient phase where you go from a static rotating ring to a rotating ring that starts to spin around its spin axis. You start out with a high moment of inertia and end up in situation with none. This also means angular momentum of the ring would be dropping during this transient phase. If Conservation of Angular momentum holds then there has to be a transfer of angular momentum to somewhere else to compensate. Since the gyroscope is attached to a rotating body essentially its own center of mass it becomes the only thing that can feel this induced torque. Said induced torque would need to accelerate the center of mass and increase the overall angular velocity of the system to conserve angular momentum but this would inadvertently increase the total angular energy. It almost looks like the ring mass is pushing its own center of mass in the back during this transient phase to maintain the total angular momentum.

[broli]

Here is another brain buster. I asked AI if the velocity vector is always tangential to the path of motion. I mean the definition of velocity is essentially the tangent of the path a mass takes. Obviously it said yes this is always the case and went on to cite some vector algebra.

But here is a direct contradiction to that. In the following animation I intentionally used a very low spin rate to amplify this effect.

https://youtu.be/CaHNzR9ysyU

The masses look almost like they are orbitting the (black) precession axis and have an angular momentum around this axis. But it's merely an illusion there is zero angular momentum around the precession axis even if they appear to be orbiting around it. You can actually see that their velocity vector is NOT tangent to the path of motion as you would expect. In fact it's almost perpendicular if the spin rate is low enough compared to the rate of precession.

It is very jarring and confusing to see as the path it takes clearly looks like it is orbiting the black axis but it is not and has zero angular momentum around said axis the total angular momentum (green arrow) is always completely perpendicular to the precession axis. The orbital motion is merely an illusion caused by the precession torque.

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In conclusion precessing mass has no orbital angular momentum. A simple experiment to do would be to validate that a spinning orbiting mass has a lower moment of inertia than a non spinning orbiting mass. And next up would be to see what happens to an orbiting mass that transients from non spinning to spinning. My humble guess is that it would speed up the orbiting as its orbital moment of inertia would suddenly drop and something needs to speed up to compensate for the loss of moment of inertia and thus angular momentum.