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If tidal power plants are slowing down Earth's rotation then is it theoretically possible to build a power plant that would drain energy from Earth's angular momentum (thus slowing down it's rotation)?

What would such machine look like?

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There are a few important points to make before I attempt an answer. The first is just a point about terminology: you have to be careful to distinguish between energy and (angular) momentum. They are not the same thing - they have different units - and so it doesn't strictly make sense to talk about "draining energy from Earth's angular momentum", since it's impossible to convert one of those quantities into the other. This doesn't mean it's a bad question, it's just that you have to be careful to understand the difference.

The second important point is that angular momentum is conserved. If you drain it from one thing, it has to go into something else. Energy is the same, but energy can come in useful forms that we mostly call "work" and no-so-useful forms that we mostly call "heat". When we say "generating energy" we really mean converting it into a useful form.

Although you can't convert angular momentum into energy, it is possible to generate useful energy by transferring angular momentum from one body to another. The rule is that you can do this whenever the angular velocities of the two bodies are different, as Peter Shor mentioned in his answer. This is exactly the same rule as the one that allows you to extract energy by transferring charge when two voltages are different, or by transferring linear momentum when the velocities are different, or by transferring heat when the temperatures are different.

The main point is that if you want to generate energy by decreasing Earth's angular momentum, you have to transfer that anglar momentum somewhere else. One place to put it is the Moon, which is effectively what happens when you extract energy from the tides. (The moon orbits the Earth more slowly than the Earth rotates, so their angular velocities around the centre of the Earth are different.) However, it's difficult to imagine how the flow of angular momentum from the Earth to the Moon could be sped up.

However, there is another way you could do it, and that's to launch things into space. If you can get mass far enough away from Earth then it will have a lower angular velocity, and so in theory you could generate useful work by transferring matter from Earth's surface to deep space.

But the question is how you could do it. Launching stuff in rockets won't do the job, since it takes loads of energy to accelerate things to escape velocity, and there's no practical way to get it back. However, with a space elevator it would be different. Imagine a space elevator that's so long that the centrifugal force at the counterweight end is much larger than $1g$. You have to expend some energy lifting mass up to the zero-g point, but this will be more than outweighed by the energy you can gain by letting the mass power a generator as it moves from the zero-g point to the counterweight. Once it gets there its velocity will be enough to escape the Earth's gravity well and you can just let it go, sending the lift back to collect the next mass.

If you do this repeatedly you will have a net gain of energy, just from sending garbage into space, and it ultimately works because it transfers angular momentum from a region of high angular velocity (the Earth) to one of low angular velocity (objects moving in space far from Earth). However, there are enormous practical issues that would probably prevent it from ever being done. Aside from the strength of cable needed, the angular momentum will be transferred not directly from the surface of the Earth but from the whole length of the cable as the mass moves along it, which will make it swing around like crazy unless it can somehow be made to behave as if it's rigid. But nevertheless, it's possible in principle.

Come to think of it, if we're coming up with crazy schemes that would only work in principle, I guess you could build a rail track around the equator of the Earth, put a very heavy train on it, and attach the train via a cable to the moon so that it pulls the train along, powering a generator. This would be an effective way to generate work by increasing the transfer of angular momentum between the two bodies, but of course it's pure fantasy.

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+1: Very nice. I think you should make the space elevators rigid. Then it's clear how it works (sending the mass up reduces the earth's angular momentum the same way a spinning ice skater slows down when he extends his arms). So you're slightly incorrect; the angular momentum isn't transferred to the tip of the cable; it's transferred to its entire length via the Coriolis force generated when you move the load up the cable. –  Peter Shor Dec 26 '12 at 14:21
    
@PeterShor good point, I've edited the answer. I wonder if it would be possible to build an elevator that behaves rigidly enough by having a network of cables in tension, anchored to multiple points on the Earth's surface, kind of like one of these flickr.com/photos/46309585@N04/4561104341 - that would solve some of the obvious problems with having an actual rigid rod stuck into the ground... –  Nathaniel Dec 26 '12 at 14:40
    
It seems to me that this scheme might actually be practical for a fast-rotating asteroid. Have any studies been done on this? –  Peter Shor Dec 26 '12 at 17:30
    
This is the kind of answer I was looking for, thanks. –  daniel.sedlacek Aug 20 '13 at 11:55
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Yes, many things rob Earth's angular momentum and slow it down. For example, when a rocket lands on Earth, it gains/loses speed due to the Earth's rotation. Other than that, when a rocket leaves Earth, they use the fact that the Earth rotates in order to gain kinetic energy from the Earth's rotation. In other words, the rocket robs the Earth's energy.

NASA uses this fact to their advantage. Their 'rob' Earth's energy to speed up their spacecraft during launch.

This relates to angular momentum because the rocket acts as a mass being added or removed from the side of a spinning 'sphere' (the Earth). If you use calculations, you would find things like conservation of energy and conservation of angular momentum.

However, we don't have power plants doing that because it is highly impractical.

It is impractical because we have an atmosphere which moves along with the Earth's rotation

Edit: Let's imagine that Earth's atmosphere doesn't exist. So we have a big rotating sphere in vacuum we could launch a spacecraft in the direction of rotation, have some turbines 'take' the spacecraft's kinetic energy, let the spacecraft land, and repeat. It's a simplified version of some system. We could even use magnets and Faraday's law (the spacecraft is a magnet) instead of some turbine.

$\Delta L = mv_{initial} r - (\Sigma m)v_{final} r $

$= m_{earth} r_{earth} v_{earth,initial} - {v_{earth,final} (m_{earth} r_{earth} + m_{rocket} r_{rocket})}$

Since $v_{initial} \approx v_{final}$, we have $\Delta L = -vrm_{rocket}$

Change of Earth's angular momentum.

Planet angular momentum to useful energy

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Note that sometimes Earth 'robs' energy from other objects, too –  raindrop Dec 21 '12 at 22:07
    
The movement of Earth's atmosphere is not caused by Earth's angular momentum. If it were, they would have come into equilibrium billions of years ago and the atmosphere would not be in motion. Instead the movement is caused by thermally driven convection due to the Sun heating air near the surface, and the air at the top losing heat due to emitting thermal radiation itself. These processes can result in a transfer of angular momentum between the Earth and its atmosphere, but of course the total is always conserved. –  Nathaniel Dec 26 '12 at 3:21
    
Also, you couldn't possibly generate energy by launching and landing spacecraft in the way you suggest, even if there were no atmosphere. It's true that kinetic energy is transferred from Earth to the spacecraft on takeoff, but this is a thermodynamically unfavourable energy transfer and work must be done to perform it. If you launch things using a space elevator it's a different story though - I'm writing an answer to explain that. –  Nathaniel Dec 26 '12 at 3:46
    
@Nathaniel I assumed that the tangential velocity of the Earth at the Earth's surface was much greater than usual wind speeds parallel to the Earth's surface, so I thought that somehow the atmosphere 'rotated along' with the Earth's rotation. I've edited the question to explain the spacecraft thing. –  raindrop Dec 26 '12 at 6:31
    
the atmosphere does rotate along with the Earth. But when you talk about "[taking] advantage of the movement of Earth’s atmosphere (caused by Earth’s angular momentum) relative to outer layers of Earth", it sounds like you're assuming it doesn't. –  Nathaniel Dec 26 '12 at 6:39
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To harness the energy of the Earth's rotation, you need something with different angular velocity (just like harnessing thermal energy requires two things at different temperature). Everything on Earth is rotating with the same speed, and so has the same angular velocity; the closest thing with a different angular velocity is the Moon.

This is in fact what is powering the tides: the Moon is gaining angular momentum and changing its orbit. However, we have no practical way to speed up the transfer of angular momentum to the Moon, so harnessing Earth's angular momentum in any way other than using the existing tides is completely impractical.

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Technically speaking, every time you exert a force on planet Earth, you are accelerating it. Take this case for example:

Earth spins counter-clockwise, or towards the East as it would be observed on Earth. Thus, should you exert a force in the same motion (walking West, applying a force on the planet to the East), you would be increasing the rate at which Earth spins (albeit this is a TINY increase). In the same way, walk to the East and you'll apply a tangential force towards the West that actually slows Earth's rotation slightly.

This behavior would obviously scale up for more massive objects exerting larger forces, resulting in the slight decrease apparently caused by the power plants you mentioned.

Back to your case for building a plant to harness this energy, though... There are currently no plans to try and harness this energy, as the facility required to do it would likely be ridiculously impractical.

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what is impractical on tidal power plants? :) –  daniel.sedlacek Dec 21 '12 at 22:22
    
When you start walking you push the Earth in one direction, but when you stop walking you push it in the opposite direction. You cannot change the total angular momentum unless you transfer it to another object away from the Earth. The tides do this by exerting a non-constant gravitational force on the moon, but most ways of generating power do not do this, so they don't affect the total angular momentum at all. –  Nathaniel Dec 26 '12 at 7:03
    
It's only counter-clockwise for those of the northern-hemisphere persuasion ;-) –  Mike Dunlavey Dec 26 '12 at 14:49
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