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Let's begin by insisting that angular momentum is conserved everywhere and at all times. Since it is a vector quantity, direction matters.

Consider a gyroscope on the end of shaft, the other end of which sits on a free pivot point. After being spun up the gyroscope is dropped so that it begins to precess around a vertical axis centered at the pivot.

The precession is caused by a torque from gravity acting on the spinning mass of the gyroscope. The energy involved in precession has its origin in the kinetic energy in the spinning gyroscope. So far so good.

For the angular momentum of the precession (not of the gyroscope wheel itself) to be preserved, there must be an equal and opposite pointing angular momentum somewhere in the system.

Where is it?

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  • $\begingroup$ Should have said "conserved" instead of "preserved". $\endgroup$ – puzzled Sep 22 '19 at 20:56
  • $\begingroup$ Thank you. So the kinetic energy for the precession comes from the gyroscope falling a little bit at the beginning. If I have a little motor in the gyroscope causing it to continue spinnng, the precession never stops even though its vertical position might not change, that is, even though it may not continue to fall. Also, the precession doesn't continue once the gyroscope is turned off. Wouldn't that initial transfer of potential to kinetic energy cause precession to continue indefinitely in the absence of any force to stop it? Yet it stops when the gyro stops. $\endgroup$ – puzzled Sep 23 '19 at 3:28
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The energy involved in precession has its origin in the kinetic energy in the spinning gyroscope. So far so good.

No.

You spin the gyroscope (upright) so that it has an angular velocity $\omega$. The kinetic energy associated with this motion is:

$$ \frac{1}{2}I_{\mathrm{axle}}\omega^2,$$ where $I_{\mathrm{axle}}$ is the moment of intertia of the gyro about its axle.

The gyroscope then falls a bit under gravity and then it starts precessing. The angular velocity of the gyroscope about its axle is still $\omega$ -- assuming a point contact with the ground and no air resistance, there was no torque in that direction that could have reduced it.

The gyroscope then starts precessing about the $z$ axis (gravity) with angular velocity $\Omega$. This introduces a kinetic energy $$ \frac{1}{2}I_z \Omega^2,$$ where $I_z$ is the momentum of intertia of the gyro assembly about the $z$ axis, not about its axle.

This kinetic energy, not originally present in the system, comes form the gravitational potential energy $mgL(1-\cos\theta)$ lost when the gyro drops by angle $\theta$ before precessing.

For the angular momentum of the precession (not of the gyroscope wheel itself) to be preserved, there must be an equal and opposite pointing angular momentum somewhere in the system.

Angular momentum (about an axis and/or a point) is only conserved in a closed system, i.e. a system which no external torques act upon.

For the $\omega$-associated angular momentum, that's still conserved as said before, there are no torques tangential to the gyro that can slow down its rotation.

For the total angular momentum, about (say) the point of contact, gravity provides the external torque.

Hence you expect a change in angular momentum:

$$ \Delta \mathbf{L} = \Gamma \Delta t = (\mathbf{r}\times m\mathbf{g})\Delta t,$$

which is exactly what causes precession.

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  • $\begingroup$ Thank you. So if we now consider the earth and gyroscope together that would be (essentially) a closed system, correct? So when the precession begins, is there not an equal and opposite angular momentum involving (imperceptibly of course) the earth? If not, where is the opposing momentum? $\endgroup$ – puzzled Sep 24 '19 at 17:02
  • $\begingroup$ Yes the gyro also exerts a torque on the earth, which would gain an equal and opposite angular momentum $\endgroup$ – SuperCiocia Sep 24 '19 at 17:29
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"Conserved" doesn't mean "unchanging". It means "only changed by inputs and outflows".

In the case you mentioned, gyroscopic precession due to gravity, gravity is exerting a torque on the assembly. That's what causes the changing angular momentum as the assembly rotates.

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  • $\begingroup$ I understand that gravity causes the change in momentum. But doesn't that change have to be matched by an opposite angular momentum somewhere? If I grabbed hold of the gyroscope and forcibly spun it around in a forced precession, me and the earrth would be experiencing an equal and opposite change in angular momentum, no? So the earth (gravity) is twisting ever so slightly in an opposite direction? How does that happen? $\endgroup$ – puzzled Sep 22 '19 at 22:15
  • $\begingroup$ The mass of the gyroscope pulls up on earth. The pivot pushes down. That’s a torque. It’s equal but opposite to the torque on the gyro because each force is part of an N3L pair. $\endgroup$ – Bob Jacobsen Sep 22 '19 at 22:18

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