What is the source of additional energy in a precessing gyroscope?

[kmm]

I'm trying to think about gyroscope precession in terms of energy, and I'm a little confused. If you hold the spinning gyroscope at some tilt so that it doesn't precess, it will have energy associated with it's rotation. When you let it go, it will now have additional energy due to the precession. What accounts for this additional energy? I understand that the gyroscope being tilted will experience a torque due to gravity and this causes a change in angular momentum so that it precesses. Although, if the gyroscope precesses about the z-axis, it will have experienced a change in angular momentum along this axis even though there was no torque in this direction. Unless, we're saying gravity indirectly causes a torque in this direction, so we would say that gravity accounts for the increase in energy. However, in addition to that, it's angular velocity doesn't continue to increase therefore it's energy doesn't continue to increase. But if gravity is the source of energy why did it's energy increase initially, then stop increasing?

 

[Doug Huffman]

[ ... ] But if gravity is the source of energy why did it's energy increase initially, then stop increasing?

If only we could make gravity a source of energy. In a gyroscope, energy and momentum are conserved - PERIOD. If you find some energy sneaking into your freebody diagram then you're doing it wrong.

 

[kmm]

If only we could make gravity a source of energy. In a gyroscope, energy and momentum are conserved - PERIOD. If you find some energy sneaking into your freebody diagram then you're doing it wrong.

OK, but when I let go of the gyroscope it now has energy it didn't have before, correct? There is also angular momentum due to precession that it didn't have before. But yes, the magnitude of the angular momentum due to it's spin is conserved. Also, gravity is a source of energy due to motion which is what I'm referring to here.

 

[mfb]

OK, but when I let go of the gyroscope it now has energy it didn't have before, correct?

You just moved energy around from one component to another. Depending on what you do, the center of mass of the gyroscope goes down a bit, or the rotation frequency of it goes down a bit, or both.
Total energy stays the same.

 

[A.T.]

OK, but when I let go of the gyroscope it now has energy it didn't have before, correct?

Nope.

There is also angular momentum due to precession that it didn't have before.

If external torques acting acting, angular momentum is not conserved.

Also, gravity is a source of energy due to motion which is what I'm referring to here.

If the center of mass goes down, potential energy is converted into kinetic energy.

 

[kmm]

Nope.

So initially it had energy due to spin. But when I let it go, it doesn't now have energy due to spin+precession?

If external torques acting acting, angular momentum is not conserved.

I agree.

Why?

Well I was ignoring friction, but the torque due to gravity is perpendicular to the angular momentum so only changes it's direction but not it's magnitude.

If the center of mass goes down, potential energy is converted into kinetic energy.

But initially, it's center of mass doesn't go down does it? This would require a change in angular momentum downward, but the change in angular momentum due to gravity is parallel to the surface of the ground.

 

[mfb]

So initially it had energy due to spin. But when I let it go, it doesn't now have energy due to spin+precession?

If I get money from the bank, I start with "money at the bank" and now have "money at the bank and money in my wallet", that does not mean I have more money than before. The amount of energy in the individual energy components is important.

Well I was ignoring friction

Fine.

But initially, it's center of mass doesn't go down does it? This would require a change in angular momentum downward, but the change in angular momentum due to gravity is parallel to the surface of the ground.

You just never consider angular momentum around the axis where the initial rotation happens because its component is small.

 

[kmm]

If I get money from the bank, I start with "money at the bank" and now have "money at the bank and money in my wallet", that does not mean I have more money than before. The amount of energy in the individual energy components is important."

You just never consider angular momentum around the axis where the initial rotation happens because its component is small.

I agree that energy is ultimately coming from somewhere and we aren't making energy from nothing, I'm trying to figure out where. So according to your analogy, applying it to what I said before regarding spin and precession energy, I have energy due to angular momentum taken from it's spin (the bank) and converted it to angular momentum in precession(the wallet). The way I see it though, the magnitude of angular momentum is conserved (ignoring friction). So I understand if we're saying the energy in the precession comes from some gravitational potential energy. But I still don't see how the center of mass would drop initially. Even if it did, I also don't see why it would stop dropping (also ignoring friction).

 

[Doug Huffman]

[ ... ]precession comes from some gravitational potential energy. [ ... ]

Bzzzt!

 

[A.T.]

but the torque due to gravity is perpendicular to the angular momentum so only changes it's direction but not it's magnitude.

The total magnitude, not the magnitude along the gyro axis.

 

[jbriggs444]

The way I see it though, the magnitude of angular momentum is conserved (ignoring friction).

In the first year physics approximation, one pretends that the angular momentum is all in a plane of symmetry (or around an axis normal to that plane) and that "precession" is what happens as the direction of the angular momentum and the plane of symmetry changes over time. The assumption is that the angular momentum of a gyroscope aligns with the axis of symmetry. For a gyroscope that is spinning rapidly and precessing slowly, that assumption is approximately true. But only approximately.

The angular momentum of a gyroscope that is precessing does not align perfectly with the axis of symmetry. A torque applied at right angles to the axis of symmetry need not be at right angles to the current angular momentum pseudo-vector -- though for a rapidly spinning gyroscope it will be approximately so.

 

[kmm]

The total magnitude, not the magnitude along the gyro axis.

So this would mean that some energy of the spin about the gyro axis is converted to energy of precession, correct? I'm going to have to think more about how the magnitude of angular momentum along the gyro axis changes though.

In the first year physics approximation, one pretends that the angular momentum is all in a plane of symmetry (or around an axis normal to that plane) and that "precession" is what happens as the direction of the angular momentum and the plane of symmetry changes over time. The assumption is that the angular momentum of a gyroscope aligns with the axis of symmetry. For a gyroscope that is spinning rapidly and precessing slowly, that assumption is approximately true. But only approximately.

The angular momentum of a gyroscope that is precessing does not align perfectly with the axis of symmetry. A torque applied at right angles to the axis of symmetry need not be at right angles to the current angular momentum pseudo-vector -- though for a rapidly spinning gyroscope it will be approximately so.

Would this be why what A.T. said in post #10 is true?

 

[Dale]

The Lagrangian for a gyroscope is pretty complicated and has a lot of cross terms. See equation 53 here.
http://ocw.mit.edu/courses/aeronaut...fall-2009/lecture-notes/MIT16_07F09_Lec30.pdf

 

[kmm]

The Lagrangian for a gyroscope is pretty complicated and has a lot of cross terms. See equation 53 here.
http://ocw.mit.edu/courses/aeronaut...fall-2009/lecture-notes/MIT16_07F09_Lec30.pdf

Thanks, I'm going to look over this. It looks like this is the type of analysis I needed.

 

[kmm]

The Lagrangian for a gyroscope is pretty complicated and has a lot of cross terms. See equation 53 here.
http://ocw.mit.edu/courses/aeronaut...fall-2009/lecture-notes/MIT16_07F09_Lec30.pdf

This was helpful. Seeing the total energy given by the gravitational potential energy and the total kinetic energy due to the different motions helped me better understand what's going on in a gyroscope in terms of energy.