"I see no reason to accept that."

Imagine we put a huge spherical shell round the earth.

Do you accept that we cannot change the total angular momentum of that shell and the earth within it from inside?

That shell is called the atmosphere.

"Again, you seem to be making an assumption about the distribution of mass."

Yes, I am. I am assuming that the moment of inertia doesn't change. I will come back to that.

Suffice it to say that a building will have an effect, but it's small, and it's clearly nothing to do with kinetic energy as you were trying to suggest earlier.

"I'm not assuming there's a temporary effect.

Pardon me, but I seem to remember that's exactly what you said."

Yes, I did say that specifically in response to this.

"He didn't make a mistake, but you seem to be assuming his theorem requires that there will only be a temporary change"

The theory does say that there will be no change. If you only consider the rock, then there will be a temporary change. It seems silly to consider the two independently because, of course, they rotate together. As I said, there isn't a 1000 MPH wind at the equator.

"but it does raise an interesting question about what might happen if we really could produce sufficient thrust to alter the period of the Earth's rotation. "

Of course we could (in principle) do it, in fact, we have. We launched pioneer etc. The effect is small.

In doing so we stopped it being part of the "system" called earth and it gave us something to push against.

The angular momentum of the system composed of the Earth and Pioneer together, has exactly the same angular momentum as it always did. but since some of that is now with Pioneer, the amount left with us is different.

Now there is another effect to consider here, but it's even smaller than the direct push from the farts.

The cows do indeed, do work in forcing out the gas and that work is converted to thermal energy. It warms up the atmosphere slightly.

The atmosphere will expand (to some unimaginably tiny extent) and so, on average it will be further from the Earth's centre.

That will slightly increase the moment of inertia of the system as a whole. (If you like, the skater has put their arms out again)

That means the whole shooting match will, in fact spin slightly slower.

But, again, there's a catch.

The same would be true whichever direction the cows were facing.

If we make the (reasonable) assumption that the cows will fart anyway, and that they have done so for a long time, the effect will have already happened.

So the overall effect of the cows facing or farting Eastwards is no different from them farting around as usual.

I grant that I have made no attempt to quantify the time scale over which the effect will die out, but my best guess is that it is roughly the time taken for a disturbance in pressure at the cow's back end to reach the ground. That transfer will happen through the interaction of the gas molecules. They travel at the speed of sound. So the relaxation time for the system is (as well as I can judge it) 3 milliseconds or so.

I'd be interested if anyone has a better estimate.

You also ask if I can show that there's no change in mass distribution.

The mass of, for example, the methane and it's distribution is independent of the direction the cow faces.

Whatever effect it has, it isn't going to change when you line them up Eastwards.