I had an idea for energy generation: put a gyroscope at the north or south pole connected to a dynamo or similar. The idea was that when the gyroscope resists the Earth's rotation, it would drive the dynamo to create electricity. (The gyroscope's axis of rotation would be perpendicular to the Earth's, of course.) The source of the energy would be the Earth's angular momentum.

Trying to work out the energy generation, however, it became apparent that it in fact could not generate any energy. Due to conservation of angular momentum, the angular momentum of the Earth before starting the machine is the same as after stopping the machine, so no energy was taken from the Earth.

So, what exactly does happen? It is apparent that the machine can at least temporarily steal some energy from the Earth, so how does it give it back, so to speak?

EDIT: The gyroscope is not on a gimbal. It's axis can only rotate in the direction of the Earth's rotation, and only by turning the dynamo.

  • $\begingroup$ Please explain the reasoning behind your second paragraph: "Due to conservation of angular momentum, the angular momentum of the Earth before starting the machine is the same as after stopping the machine, so no energy was taken from the Earth." $\endgroup$
    – S. McGrew
    Apr 7 '19 at 15:25
  • $\begingroup$ @S.McGrew right, so before the machine starts, it has 0 angular momentum. The same thing is true after it stops. Due to conservation of angular momentum, this means the Earth's angular momentum has not changed. $\endgroup$
    – PyRulez
    Apr 7 '19 at 15:28
  • $\begingroup$ Due to confusion of the words 'parallel' and 'perpendicular' your latest edit to the question has left it in a self-contradictory state. The current final paragraph contains the statement: "It's axis can rotate only in the direction of the Earth's rotation." $\endgroup$
    – Cleonis
    Apr 7 '19 at 15:32
  • $\begingroup$ @Cleonis that part is correct. The axis is perpendicular to the Earth's axis of rotation, but rotates in that direction. In other words, the gyroscope spins perpendicular to the Earth, but the axis of rotation is itself rotating in same direction as the Earth. $\endgroup$
    – PyRulez
    Apr 7 '19 at 15:39

If I understand you correctly:
You describe a gyroscope wheel with the spin axis perpendicular to the Earth's axis, mounted in such a way that there is no freedom to pitch, but the swivel axis is connected to electric generators.

To establish names for directions of rotation I add an image of a gimbal mounted gyroscope

gimbal mounted gyroscope

  • Spinning of the blue gyroscope wheel: spin axis

  • Motion of the red frame relative to the yellow frame: pitching

  • Motion of the yellow frame: swiveling

As I understand your description:
A setup where there is no possibility of pitching motion. Compared to the setup in the image: regard the red frame and the yellow frame as a single solid unit, a perfectly rigid structure.

For comparison let me first describe what would happen if the setup does allow pitching motion but with brakes that you can quickly release.

You spin up the gyroscope wheel, initially using the brakes to prevent pitching motion. As the gyroscope wheel spins up the combination of the spinning and the co-rotating with the Earth along the swivel axis gives a tendency to pitch. Since the Earth's rotation is so slow that pitching tendency is very small, but with a sufficiently high spin rate you can eventually achieve a significant tendency to pitch.

When you release the brake of the pitching motion you can attempt to harvest energy from the pitching motion. Any harvesting that you achieve stops when the spin axis has become aligned with the Earth's axis.

Now the case with no freedom for pitching motion:
As you spin up nothing will happen.
Prior to spinning up the swiveling motion of the gyroscope was co-rotating with the Earth's roatation, and all the way while spinning up the swiveling motion will remain co-rotating with the Earth's rotation. Because of the tendency to pitch stress in the supporting structure will mount, proportional to the spin rate. But this setup is described as perfectly rigid, so it can entirely prevent pitching motion.

[LATER EDIT, 3 hours after initial posting]

I think I need to clarify my answer.
The key factor in gyroscopic effects is motion.

When you have a spinning motion, and you add another rotational motion, along a different axis, then in response the wheel will move along a third axis.

The simplest case is when the secondary rotational motion is perpendicular to the spinning motion. Then the response is along an axis perpendicular to both of the first two axes.

But then that third rotational motion has its own effect, and it's easy to misinterpret that.

So take the case depicted in the image. You spin up the wheel, and then you put your fingers on the outer frame to try and swivel the gyroscope wheel. In response the wheel pitches, and that pitching motion results in a tendency that opposes your attempt to make the wheel swivel.

So it feels as if your attempt to make the wheel swivel is opposed by the spinning wheel just like that. If the spin rate is very high a very slow pitching rate is sufficient to give a palpable opposition to the attempt to sustain swivel. It's easy to think of the pitching motion as insignificant, but it very much isn't.

It is difficult to suppress pitching motion altogether, the usual mountings aren't set up for that. Those are factors that contribute to misinterpretation of how a spinning gyroscope wheel responds to being nudged.

  • $\begingroup$ Pitching motion is not required, just pitching force. It's similar to how a wall can produce normal force without moving. $\endgroup$
    – PyRulez
    Apr 8 '19 at 2:51
  • $\begingroup$ @PyRulez Ah, clearly your understanding of gyroscopic effect and my understanding of it are incompatible with each other. The only way to settle this would be to physically construct a setup as described, and try it. Maybe someday. $\endgroup$
    – Cleonis
    Apr 9 '19 at 21:42

Let's say that the gyroscope is started spinning at the pole, with its axis perpendicular to the axis of the Earth, and the gyro is on a gimbal mount that allows the gyro to tilt freely. As the Earth turns, the gyro experiences a torque that rotates the gyro's axis toward alignment with the axis of the Earth. The closer the alignment, the less torque. So what happens is that the gyro aligns its axis parallel to the Earth's axis, and then just stays aligned.

There is a lot to be found on the internet about this idea, including this patent, this, and this.

  • $\begingroup$ It's not on a gimble, see the edit. $\endgroup$
    – PyRulez
    Apr 7 '19 at 15:06
  • $\begingroup$ As you presumably know, the process that you describe is the operating principle of a gyrocompass. That raises an interesting question. Can a design based on the operating principle of a gyrocompass be used to harvest rotational kinetic energy from the Earth? If not, why not? That seems like a question worth asking here on physics.stackexchange. $\endgroup$
    – Cleonis
    Apr 7 '19 at 15:27

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