Conservation of Angular Momentum vs a gyro at the North Pole

[little1406]

Consider a gyro located at the North Pole with its axis of rotation horizontal and its bearings fixed w.r.t. the Earth. When the gyro is spun up, its own angular momentum resists being rotated once per day by the Earth so the Earth must exert forces on the bearing mounts which amount to a torque that forces the gyro axis to rotate in a horizontal plane once per day. The apparent problem with this is the effect upon the Earth's angular momentum. It appears that the gyro is causing the Earth's rotation to gradually slow down....yet no external forces are acting upon the Earth-gyro system....an apparent violation of Conservation of Angular Momentum. What am I missing here?

 

[Orodruin]

No external torque on the system … and the system’s total angular momentum is conserved. There is no contradiction here.

It is no stranger than two objects colliding and changing their velocities. Linear momentum is conserved although there is no external force on the two-body system. The change of velocities is due to forces internal to the system.

 

[A.T.]

It appears that the gyro is causing the Earth's rotation to gradually slow down...

If the gyro's angular momentum vector rotates in a horizontal plane, then the applied torque vector and the equal but opposite torque vector acting on the Earth are also in that plane, and thus perpendicular to the Earth's angular momentum vector. So the gyro doesn't act to slow down the Earth's rotation.

 

[Baluncore]

Without the fixed bearings, if the gyro is constrained only to remain with its axis horizontal, then you have reinvented the gyrocompass. At a pole, its horizontal axis will appear to rotate once each sidereal day, relative to the Earth. It will remain aligned, fixed to one star.

Away from the poles, a gyrocompass will automatically align its axis with the lines of longitude, or if it is on the Equator, with the Earth's N-S axis of rotation. A gyrocompass does not work as an Earth compass, when close to the poles.

 

[little1406]

Thanks for the replies. A.T. I'm pretty sure you are wrong about the torque vector that the Earth exerts on the gyro to make it rotate once per day. It has to be vertical...i.e., orthogonal to the gyro's spin axis. Orodruin points out the main point of this apparent paradox. There are no external torques on the Earth-gyro system. Yet, AFAICT, the Earth must exert a continuous torque on the gyro support bearings to force it to rotate once per day. Baluncore, I didn't know about gyrocompasses. I was going to call it Foucoult's gyroscope.

 

[Baluncore]

The "gyroscope" is so named because it was designed to "show rotation" of the Earth.
https://en.wikipedia.org/wiki/Foucault's_gyroscope

As your ground fixed gyro rotates with the Earth, it reacts by applying a force downwards on one end bearing, and upwards on the other, (a couple). That represents a torque being applied to tilt the Earth's surface at the mounting point at the North Pole.

That couple will primarily force the North Pole to move away from the gyroscope in a consistent geographical direction. The Earth's rotation will oppose that, turning it into a reaction to the side.

 

[A.T.]

A.T. I'm pretty sure you are wrong about the torque vector that the Earth exerts on the gyro to make it rotate once per day. It has to be vertical...i.e., orthogonal to the gyro's spin axis.

There are infinitely many directions that are orthogonal to the gyro's spin axis, including horizontal ones.

Net torque is the time derivative of angular momentum. And since the angular momentum changes only within the horizontal plane, its time derivative (net torque) also lies in the horizontal plane.

 

[Orodruin]

Orodruin points out the main point of this apparent paradox. There are no external torques on the Earth-gyro system. Yet, AFAICT, the Earth must exert a continuous torque on the gyro support bearings to force it to rotate once per day.

You clearly have not understood my post.

 

[little1406]

Thanks, Baluncore, you have provided the explanation I needed…and A.T. is right about the precession torque vector lying in the horizontal plane but orthogonal to the gyro’s spin axis...my apologies. So, the conclusion I’m being dragged towards is that there is no rotation-retarding torque exerted upon the Earth. I’m going to get out my toy gyro and play with it to get a visceral understanding of all this. I don’t think it will help me understand Orodruin’s post though.

 

[berkeman]

I don’t think it will help me understand Orodruin’s post though.

Let's say that the gyroscope was oriented in the same axis as the Earth's rotation, and was initially not spinning. You use a battery to spin up the gyro with a motor's torque, which changes the Earth's angular momentum very slightly. Are any conservation laws violated?

Now mount the gyro horizontally at the north pole with your rigid mount. Do the same spin up with a battery powered motor -- are any conservation laws violated?

 

[A.T.]

I don’t think it will help me understand Orodruin’s post though.

His point was merely that a gyro and Earth can exchange angular momentum, even such that the magnitudes of their angular velocities change, while the total angular momentum of Earth and gyro is constant.

But in your example they only exchange angular momentum continuously such that the directions of their angular momenta change in a cyclic way, so the transfer adds up to zero over a whole cycle. Thus neither is accumulating angular momentum.

 

[A.T.]

Let's say that the gyroscope was oriented in the same axis as the Earth's rotation, and was initially not spinning. You use a battery to spin up the gyro with a motor's torque, which changes the Earth's angular momentum very slightly. Are any conservation laws violated?

Now mount the gyro horizontally at the north pole with your rigid mount. Do the same spin up with a battery powered motor -- are any conservation laws violated?

@little1406 And then consider the combined case: You spin up the gyro while its axis is vertical (parallel to the Earth's spin axis), and then you tilt the gyro spin axis by 90° so it ends up horizontal.

What happens to the Eath's spin rate during the spin-up of the gyro, and then during tilting of the gyro?

 

[little1406]

A.T., it's easy for me to see that during the aligned-axis spin up, the gyro's acquisition of angular momentum must come from an equal and opposite change in the Earth's angular momentum. And Conservation of Angular Momentum forces me to believe that during the tilting-to-horizontal of this gyro, Earth's original angular momentum must be restored to its original value by the reaction torque that occurs when the gyro is tilted.

Full disclosure: My original question was distilled from consideration of a gyro device to extract energy from Earth's rotation. Setting aside practicality for the moment, it seems to me that, in principle, the reaction torque created as the Earth rotates the gyro axis could do work on a load connected to the gimbal. Back to my gyro oriented horizontally at the North Pole, I accept that the axis rotation imposed by Earth's rotation will cause a downward reaction force on one bearing and an upward force on the opposite bearing but, is that all there is? Is there not a torque required from the Earth to accomplish this rotation? If not, then all my troubles disappear and I just have to readjust my intuition about gyro behavior.

(Thanks again to everyone chiming in on this)

 

[A.T.]

device to extract energy from Earth's rotation

We already have that:
https://en.m.wikipedia.org/wiki/Tidal_power

... it seems to me that, in principle, the reaction torque created as the Earth rotates the gyro axis could do work on a load connected to the gimbal.

How, if the gyro axis is fixed relative to the Earth?

Back to my gyro oriented horizontally at the North Pole, I accept that the axis rotation imposed by Earth's rotation will cause a downward reaction force on one bearing and an upward force on the opposite bearing but, is that all there is? Is there not a torque required from the Earth to accomplish this rotation?

Those two opposite forces transmit the torque required from the Earth to accomplish this rotation of the gyro's angular momentum vector.

As already said, net torque is the time derivate of the angular momentum vector.

 

[little1406]

A.T. the fixed-to-the-Earth gyro was just a simplification to try to understand the forces/torques involved. The real energy-producing gyro would be in a complicated gimbal connected to a gearbox which allowed the gyro axis to rotate probably at 1/2 revolution per day to extract maximum power. If it's fixed it does no work as you point out. And if it's perfectly free it does no work either. We must have torque x angular velocity to obtain power. But I'm beginning to understand that this scheme cannot operate continuously as I initially imagined. Instead, the best it could do is to rotate 90 degrees until the spin axis was vertical (we're still on the North Pole) and then it would cease interaction with the Earth's rotation.

I have read about tidal power. The Wikipedia article explains that the loss of angular momentum by the Earth is offset by a gain in angular momentum of our Moon....the Moon's orbit is actually boosted by the action of the tides.

 

[A.T.]

I have read about tidal power. The Wikipedia article explains that the loss of angular momentum by the Earth is offset by a gain in angular momentum of our Moon.

It also slows down Earth's rotation:
https://en.wikipedia.org/wiki/Tidal_locking#Locking_of_the_larger_body

 

[berkeman]

But I'm beginning to understand that this scheme cannot operate continuously as I initially imagined. Instead, the best it could do is to rotate 90 degrees until the spin axis was vertical (we're still on the North Pole) and then it would cease interaction with the Earth's rotation.

That's good, because we do not allow discussions of PMMs or Over-Unity Mechanisms here at PF.