Naked Science Forum
On the Lighter Side => That CAN'T be true! => Topic started by: Airthumbs on 28/06/2011 11:07:44

If all the cows in the world all lined up facing east to west would their fart power speed up the rotation of the Earth?

I really don't know, but I'll tell you this much, I wouldn't fancy being stood behind them when they try!(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fbestsmileys.com%2Fparanoid%2F2.gif&hash=b6c89a93fcdc26f07540419a36cc278d)

If all the cows in the world all lined up facing east to west would their fart power speed up the rotation of the Earth?
They would require a propoulsion system (their farts) to overcome their own individual masses and the earths mass respectively. In conclusion, to generate such energy, they would be flying our airspace first, before having an impact on the rotation of the earth.

So cows and pigs may fly first...

Fascinating thought, [;D]

The effect would be immeasureably small  but yes. Its equal and opposite reaction
edit
The result would be better if you had some system of lighting the farts  and really quite spectacular

Lighting them would be expensive but easy, all you need is a flatulent detector attached to the rear end so to speak, this could have a built in ignitor. I imagine there might be a few singed farmers though [;D]

It would make one heck of a youtube clip :)

It would make one heck of a youtube clip :)
Stop it your giving me ideas, there is a cow shed down the road!

Cows farting a big C02s problem. Its true (but I cant be bother though to find any link, but google it).

Cows farting a big C02s problem. Its true (but I cant be bother though to find any link, but google it).
Don't think it's the Co2 you have to worry about here! But if Co2 did smell like cow farts I expect breathing out in someones direction would be quite unpleasant....

Though even all the cows in the world could not turn it back  you would need something to the scale of massive energy factors. This is an educated statement, not one of empiracle fact, but I have never seen any calculations describing how much energy would be needed to stop the earths expansion. I did see this website which looked ok, but not sure how accurate it is, that the rotation of the earth depended on energies to stop it exceedingly high: http://www.madsci.org/posts/archives/199910/939644636.Ph.r.html

how much energy would be needed to stop the earths expansion.
I didn't know it was expanding.
Anyway, imatfaal is quite correct. The answer is yes. It's simply a question of the reaction produced by accelerated mass (in this case methane).

Newtons laws show that there would be no net effect. The cows' farts would push the air and the air would push the ground. After a very short while the effect would cancel out.
On the other hand, if you got all the cows to line up and walk Eastward then you would get a net push on the earth. It would be tiny, but it would be real.

Newtons laws show that there would be no net effect.
BC, think about that the next time you are in an aeroplane, because, if you are right, it ain't going anywhere.

how much energy would be needed to stop the earths expansion.
I didn't know it was expanding.
Anyway, imatfaal is quite correct. The answer is yes. It's simply a question of the reaction produced by accelerated mass (in this case methane).
lol... hahahaha
yes... rotation!!! That was melded from a different convo altogether :P

Cows farting a big C02s problem. Its true (but I cant be bother though to find any link, but google it).
Don't think it's the Co2 you have to worry about here! But if Co2 did smell like cow farts I expect breathing out in someones direction would be quite unpleasant....
http://www.independent.co.uk/environment/climatechange/cowemissionsmoredamagingtoplanetthancosub2subfromcars427843.html

Don't think it's the Co2 you have to worry about here! But if Co2 did smell like cow farts I expect breathing out in someones direction would be quite unpleasant....
Didn't you watch Mary Poppins?
"Supercalafragalisticexpihalitosis"

Thank goodness computers don't replicate images or words with smells yet!!
Ok so we have established that my original question is unlikely to effect the speed of the Earths rotation. How about things like the asteroid that contributed towards the extinction of the dinosaurs, or large volcanic eruptions. The earthquake in Japan recently had enough force to knock the earth of its axis, although a tiny amount.
What I guess I am trying to get at is that a lot of people seem to think the Earth has remained in this cyclic orbital state for millennia.
We know that several very large objects impacted the Earth over its history and I wonder to what extent these types of events could have altered the Earths climate.
It seems that the cows would probably cause some kind of methane storm and that's about the full extent of it but what about events unfolding in the past, big events, events with a huge amount of energy?

Newtons laws show that there would be no net effect.
BC, think about that the next time you are in an aeroplane, because, if you are right, it ain't going anywhere.
A lot of people might have correctly assumed that I meant no effect on the Earth's rotation.
Obviously there would be effects. For example, if you were behind the cows you would probably notice. If you were smoking at the time you might notice something else too.
However neither the flatulent cows, not my air travel make much odds to the Earth's rotation.

Newtons laws show that there would be no net effect.
BC, think about that the next time you are in an aeroplane, because, if you are right, it ain't going anywhere.
A lot of people might have correctly assumed that I meant no effect on the Earth's rotation.
Obviously there would be effects. For example, if you were behind the cows you would probably notice. If you were smoking at the time you might notice something else too.
However neither the flatulent cows, not my air travel make much odds to the Earth's rotation.
If the cows were all pointing their bums in the same direction, the net effect would be nonzero. Admittedly it would be very difficult to detect the effect, but Newton's laws would still apply.

I think science is fantastic when you can get the most intellectual people on here to state that "yes, them there cow farts all lined up would be changing the earths rotation" According to Newton that is.
I say that with the greatest of respect because most of the time the same people post stuff about quantum physics and particle accelerators that I find tough to grasp.. [;D]

Newtons laws show that there would be no net effect.
BC, think about that the next time you are in an aeroplane, because, if you are right, it ain't going anywhere.
A lot of people might have correctly assumed that I meant no effect on the Earth's rotation.
Obviously there would be effects. For example, if you were behind the cows you would probably notice. If you were smoking at the time you might notice something else too.
However neither the flatulent cows, not my air travel make much odds to the Earth's rotation.
If the cows were all pointing their bums in the same direction, the net effect would be nonzero. Admittedly it would be very difficult to detect the effect, but Newton's laws would still apply.
Still wrong.
http://en.wikipedia.org/wiki/Angular_momentum

two rockets diametrically oppposed and with opposite direction of thrust constitute an external torque; otherwise a catherine wheel wouldn't spin and a spacecraft wouldnt be able to change its inclination, heading, or yaw

So would two diametrically cows operate in the same way, given that the thrust is generated by, well you know what, lol

BC  Angular momentum only applies to a rigid body.
It's simply a case of F=ma. The accelerated mass of methane results in a force that applies a torque, communicated through the cow's hooves, to the Earth.
If angular momentum was conserved, jet engines (or rockets for that matter) would not work.

Newtons laws show that there would be no net effect.
BC, think about that the next time you are in an aeroplane, because, if you are right, it ain't going anywhere.
A lot of people might have correctly assumed that I meant no effect on the Earth's rotation.
Obviously there would be effects. For example, if you were behind the cows you would probably notice. If you were smoking at the time you might notice something else too.
However neither the flatulent cows, not my air travel make much odds to the Earth's rotation.
If the cows were all pointing their bums in the same direction, the net effect would be nonzero. Admittedly it would be very difficult to detect the effect, but Newton's laws would still apply.
Still wrong.
http://en.wikipedia.org/wiki/Angular_momentum
I'm not sure how angular momentum disproves the effect. If anything, it shows why there has to be an effect. It's a simplification, but basically the cows will cause the atmosphere to pick up a tiny bit of circulation in the direction of their derrieres. The atmosphere therefore gains some angular momentum. Since the earth is basically a closed system, it's total angular momentum can't change, so the earth's rotation would have to change slightly to make up for the atmosphere.
The real effect would be minuscule, however.
Geezer's post is another way of looking at it, where the earth is your system and the cow "rockets" are an external torque.

Newtons laws show that there would be no net effect.
BC, think about that the next time you are in an aeroplane, because, if you are right, it ain't going anywhere.
A lot of people might have correctly assumed that I meant no effect on the Earth's rotation.
Obviously there would be effects. For example, if you were behind the cows you would probably notice. If you were smoking at the time you might notice something else too.
However neither the flatulent cows, not my air travel make much odds to the Earth's rotation.
If the cows were all pointing their bums in the same direction, the net effect would be nonzero. Admittedly it would be very difficult to detect the effect, but Newton's laws would still apply.
Still wrong.
http://en.wikipedia.org/wiki/Angular_momentum
I'm not sure how angular momentum disproves the effect. If anything, it shows why there has to be an effect. It's a simplification, but basically the cows will cause the atmosphere to pick up a tiny bit of circulation in the direction of their derrieres. The atmosphere therefore gains some angular momentum. Since the earth is basically a closed system, it's total angular momentum can't change, so the earth's rotation would have to change slightly to make up for the atmosphere.
The real effect would be minuscule, however.
Geezer's post is another way of looking at it, where the earth is your system and the cow "rockets" are an external torque.
So, basically you are saying the atmospheric effect is a BUM STEER?

Geezer's post is another way of looking at it, where the earth is your system and the cow "rockets" are an external torque.
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[;D]

The cows' farts would push the air and the air would push the ground. After a very short while the effect would cancel out.
Because of the friction between the earth and its atmosphere, you cannot treat the two as separate systems (for long).
Since (in anything but the very short term) there's only one system, it has conserved angular momentum and its rate of rotation is conserved too.
Also, if as Geezer thinks, angular momentum is only conserved for rigid bodies, why is the classic example in all the textbooks a spinning skater who pulls there arms in and speeds up. If they pull their arms in they cannot be rigid.
It's clear that the cows' hooves push the ground, but as I pointed out earlier, so do the farts and the two effects cancel.
They have to cancel exactly because the earth, the fart, the cow, and the atmosphere are all part of one system and the momentum of that system cannot be altered by anything internal to the system.
If the cows farted into orbit then there would be an effect.

They have to cancel exactly because the earth, the fart, the cow, and the atmosphere are all part of one system and the momentum of that system cannot be altered by anything internal to the system.
If the cows farted into orbit then there would be an effect.
They're all the same system so the total angular momentum can't change. But parts of the system can change, which is why if the angular momentum of the atmosphere decreases, that of the earth increases. This is exactly the effect caused by the cows.

Also, if as Geezer thinks, angular momentum is only conserved for rigid bodies, why is the classic example in all the textbooks a spinning skater who pulls there arms in and speeds up.
In that situation you either treat the skater as different rigid bodies at different points in time or, as a very large number of rigid bodies, each of which has its own angular momentum, and integrate their effects at any point in time.
Oh, and the angular velocity of the skater only changed because work was done within the system of the skater.

I don't think it makes sense to say that the atmosphere and the rock can be treated as separate systems when, as I have pointed out before, they are linked.
Have you noticed the thousand mile an hour wind at the equator?
No, that's because it isn't there. The atmosphere rotates along with the planet and that's because it's part of the planet.
As I have pointed out, there would be a temporary change but its wouldn't last.
If the cows walked East, then they would push the earth West, and, for as long as they kept walking, they would change the spin of the earth. But the thing is that when they stop they exert a force on the Earth and in doing so they push it East a bit and the Earth's spin goes back to what it was.
In the case of the "grand fart" the angular momentum transfer cannot be sustained because the farts don't keep going.
The bit about the skater doing work isn't relevant. You can change the angular velocity without doing net work. It would be odd, but it's perfectly possible.
I will now look forward to your description of the orbiting earth and the sun, with a vast gap between them somehow being a rigid system in order that angular momentum is conserved.
When you have done that perhaps you could explain how tidal drag which kind of requires a non rigid system (usually something sloshy) is explained in terms of the conservation of angular momentum.
Any isolated system conserves angular momentum. It does not need to be rigid. This is just as well, since no object is truly rigid anyway.

In the case of the "grand fart" the angular momentum transfer cannot be sustained because the farts don't keep going.
It does not need to "keep going". The work was done, and that's all it takes to alter the state of the system.
Take the atmosphere out of it and consider what would happen if you attached a very large rocket to the moon. It would obviously cause the moon's angular velocity to increase as long as the rocket was firing. Once it stopped firing, the moon would continue rotating with a new angular velocity.

You keep muddling up energy and momentum.
Also you missed a bit from your model
"It would obviously cause the moon's angular velocity to increase as long as the rocket was firing suff into space "
The combined angular momentum of the exhaust going clockwise is exactly the same as the anticlockwise change in the angular momentum of the moon.
The change of angular momentum of the system is still zero, and it will always be so. That's the point about conservation laws.
It's not just an approximation for rigid systems, it isn't just a figment of my imagination. It's a mathematical law, provable from first principles.
http://en.wikipedia.org/wiki/Noether%27s_theorem
It's true now, and it will still be true in the morning.

Are we talking about permanent increase in rotation or temporary increase.
It's true that immediately after the cows all fire, and before friction does too much, the earth will respond by speeding up, at least temporarily.

Unless you accept that the earth's atmosphere is part of the earth.
So, either there's no overall effect (provided that you don't think the atmosphere is part of the earth) or there's a temporary effect (which dies out rather rapidly).
Take your pick, but the idea that "It does not need to "keep going". The work was done, and that's all it takes to alter the state of the system." is wrong.
Incidentally cows generally burp more than they fart so the net effect would be the reverse of that predicted.

Unless you accept that the earth's atmosphere is part of the earth.
So, either there's no overall effect (provided that you don't think the atmosphere is part of the earth) or there's a temporary effect (which dies out rather rapidly).
Take your pick, but the idea that "It does not need to "keep going". The work was done, and that's all it takes to alter the state of the system." is wrong.
Incidentally cows generally burp more than they fart so the net effect would be the reverse of that predicted.
BC, what you are obviously failing to grasp is that the mass accelerated by the cows is a consequence of work done by chemical processes. Chemical energy was converted into kinetic energy, and added to the kinetic energy of the system as a whole.

I honestly did not think that what appeared to be such a simple question was in fact quite complicated indeed!
What if the cows all actually ran along and all farted at the same time with their hooves touching the ground? [;D]

What if the cows all actually ran along and all farted at the same time with their hooves touching the ground? [;D]
We could also ask what would happen if the skater farted at the same time as she was pulling in her arms to make herself spin faster. However, that might only complicate things [;D]

We could also ask what would happen if the skater farted at the same time as she was pulling in her arms to make herself spin faster. However, that might only complicate things [;D]
The fastest spin on ice skates was achieved by Natalia Kanounnikova, with a maximum rotational velocity of 308 RPM at Rockefeller Centre Ice Rink, New York, USA on 27 March 2006. Maybe she has a secret, Geezer?

What if the cows all actually ran along and all farted at the same time with their hooves touching the ground? [;D]
We could also ask what would happen if the skater farted at the same time as she was pulling in her arms to make herself spin faster. However, that might only complicate things [;D]
What if a cow ran by the skater and farted on her while she was spinning around in outer space?

What if the cows all actually ran along and all farted at the same time with their hooves touching the ground? [;D]
We could also ask what would happen if the skater farted at the same time as she was pulling in her arms to make herself spin faster. However, that might only complicate things [;D]
What if a cow ran by the skater and farted on her while she was spinning around in outer space?
Didn't Feynman touch on that in one of his lectures?

Unless you accept that the earth's atmosphere is part of the earth.
So, either there's no overall effect (provided that you don't think the atmosphere is part of the earth) or there's a temporary effect (which dies out rather rapidly).
Take your pick, but the idea that "It does not need to "keep going". The work was done, and that's all it takes to alter the state of the system." is wrong.
Incidentally cows generally burp more than they fart so the net effect would be the reverse of that predicted.
BC, what you are obviously failing to grasp is that the mass accelerated by the cows is a consequence of work done by chemical processes. Chemical energy was converted into kinetic energy, and added to the kinetic energy of the system as a whole.
Not only have I grasped it, I have understood the consequence of it.I also understood that the kinetic energy is rapidly converted to heat by friction within the atmosphere and between the air and the land.
There is no change in angular momentum.
What you have failed to do is accept
1) that you can change the kinetic energy and momentum independently and
2) that you are plainly wrong; nothing can change the angular momentum of an isolated system like the earth and its atmosphere.
If you are not wrong then Noether's theorem is.
Can you point out his mistake please?

Didn't Feynman touch on that in one of his lectures?
If he did then I was asleep when I listened to it, which begs the question, am I subconsciously learning by listening to all these lectures,documentaries and podcasts during my sleep? [:\]

Unless you accept that the earth's atmosphere is part of the earth.
So, either there's no overall effect (provided that you don't think the atmosphere is part of the earth) or there's a temporary effect (which dies out rather rapidly).
Take your pick, but the idea that "It does not need to "keep going". The work was done, and that's all it takes to alter the state of the system." is wrong.
Incidentally cows generally burp more than they fart so the net effect would be the reverse of that predicted.
BC, what you are obviously failing to grasp is that the mass accelerated by the cows is a consequence of work done by chemical processes. Chemical energy was converted into kinetic energy, and added to the kinetic energy of the system as a whole.
Not only have I grasped it, I have understood the consequence of it.I also understood that the kinetic energy is rapidly converted to heat by friction within the atmosphere and between the air and the land.
There is no change in angular momentum.
What you have failed to do is accept
1) that you can change the kinetic energy and momentum independently and
2) that you are plainly wrong; nothing can change the angular momentum of an isolated system like the earth and its atmosphere.
If you are not wrong then Noether's theorem is.
Can you point out his mistake please?
He didn't make a mistake, but you seem to be assuming his theorem requires that there will only be a temporary change
or there's a temporary effect (which dies out rather rapidly).
in the angular velocity of the Earth (that's the chunk of solid stuff you are sitting on which I believe the OP was concerned about). Just because the angular momentum of any arbitrarily defined system is conserved it does not prevent permanent changes in the kinetic energies of the components of that system.
If we were daft enough to change the period of the Earth's rotation with some really powerful rockets, the change in the Earth's period would not die out rather rapidly. Similarly, if cows were able to affect a change in the Earth's period (by converting chemical energy), there is no reason to believe that the change would die out rather rapidly.

"If we were daft enough to change the period of the Earth's rotation with some really powerful rockets"
Do you accept that we couldn't do that if the rocket exhaust was contained within the earth's atmosphere?
The rockets throw hot gas back, that gas pushes the air and that air pushes the ground so there's no net torque.
"...does not prevent permanent changes in the kinetic energies of the components of that system."
Nope, but friction and viscosity do say exactly that.
I'm not assuming there's a temporary effect.
I'm saying that, for the system as a whole (rock, cows, farts and atmosphere) i.e the earth there is never a change in angular momentum.
For some parts of that system there will be a temporary change but it will die out as the air comes to rest WRT the rock.
Do you think that the cow fart will carry on circling the world forever?
Do you realise that if it stops going round then it hasn't got angular momentum?
Imagine I get a frictionless tube all round the world and I fire a ball bearing round it.
In doing so I transfer angular momentum from the world as a whole to the ball.
From the point of view of an external observer the earth rotates a bit one way, and the ball goes the other.
From our point of view watching the stars, the world changes it's rate of spin (to a tiny extent unless it's a really big fast ball).
OK, now I add some air to the tube. It brings the ball to a halt via viscosity. The ball is slowed down by the air and by Newton's law it must exert a force on the air. That air, in turn exerts a force on the tube. The tube exerts a force on the earth and so the earth ends up spinning at exactly the same rate as it was before.
We have expended some energy, but all we did was warm the apparatus up a bit. In the long run we didn't alter the speed of the earth's rotation.
The cow farting experiment is much the same.
Nothing leaves the system, so the system conserves momentum

Sorry for the unusual formatting, but it seemed like a good idea at the time [:D]
"If we were daft enough to change the period of the Earth's rotation with some really powerful rockets"
Do you accept that we couldn't do that if the rocket exhaust was contained within the earth's atmosphere?
The rockets throw hot gas back, that gas pushes the air and that air pushes the ground so there's no net torque.
I see no reason to accept that. Your argument is based on an unproven assumption that the mass distribution within your system is has not changed.
"...does not prevent permanent changes in the kinetic energies of the components of that system."
Nope, but friction and viscosity do say exactly that.
Again, you seem to be making an assumption about the distribution of mass.
I'm not assuming there's a temporary effect.
Pardon me, but I seem to remember that's exactly what you said.
I'm saying that, for the system as a whole (rock, cows, farts and atmosphere) i.e the earth there is never a change in angular momentum.
I actually agree with you, but that in no way requires that the angular moments of the various components of the system remain constant.
For some parts of that system there will be a temporary change but it will die out as the air comes to rest WRT the rock.
Do you think that the cow fart will carry on circling the world forever?
Do you realise that if it stops going round then it hasn't got angular momentum?
See above. If you can prove that the the distribution of mass within the system remained constant, you might be able to make that case. Does friction play a part? Yes, I'm sure it does, but you have not quantified any time constants associated with that effect other than to say it "dies out".
Imagine I get a frictionless tube all round the world and I fire a ball bearing round it.
In doing so I transfer angular momentum from the world as a whole to the ball.
From the point of view of an external observer the earth rotates a bit one way, and the ball goes the other.
From our point of view watching the stars, the world changes it's rate of spin (to a tiny extent unless it's a really big fast ball).
OK, now I add some air to the tube. It brings the ball to a halt via viscosity. The ball is slowed down by the air and by Newton's law it must exert a force on the air. That air, in turn exerts a force on the tube. The tube exerts a force on the earth and so the earth ends up spinning at exactly the same rate as it was before.
We have expended some energy, but all we did was warm the apparatus up a bit. In the long run we didn't alter the speed of the earth's rotation.
The cow farting experiment is much the same.
Nothing leaves the system, so the system conserves momentum
Quite right. Yes, I'm not arguing that the system conserves angular momentum. Just because it does, it does not mean that the kinetic energies, or angular velocities, of the components of the system are conserved. Any long term changes in the mass distribution within the system will result in long term changes in the velocities of the components of the system.
Of course, the whole "cow fart" idea is a bit ridiculous, 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. If I understand BC's argument properly, he is saying that the effect would be rather short lived, and his arguments have a lot of merrit.
I'm coming at it from a different perspective. If we consider the situation of a planet with no atmosphere, I think we could make it rotate faster (or slower). To me, this suggests that an atmosphere will provide a sort of (very complicated) damping function, but ultimately, there will be a change in the angular velocity of the planet in question.

"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.

Yes, well, see, BC, I think we agree on some points here.
The angular momentum within a system is conserved  yes, I'm sure there is no arguing with that.
As you pointed out, spinning skaters can change their angular velocity by redistributing their mass.
(I think we both agree on those points).
Now, when a cow farts, there is also a redistribution of mass (and a very small amount of vectored thrust due to the accelerated mass of exhaust gas), soooo, there is a change in angular velocity, and because there was also a redistribution of mass, the angular velocities of the components of the system are under no obligation to return precisely to their former values.
In other words, while the distribution of mass remains altered, the velocities of the components can remain altered too, even though there is no change in the angular momentum of the system as a whole.
If we bring (nonanchored) spacecraft into the debate, we have a choice. We either define our system to include the spacecraft at all times (in which case angular momentum is conserved), or we allow the mass of the spacecraft to escape from the "system", in which case angular momentum is not conserved.
We have a similar option regarding the cow exhaust. If we define our system as the mass of the Earth excluding its atmosphere, the cows converted parts of Earth into gas and transferred that mass to the atmosphere. So, clearly, the cows altered the Earth's mass, and therefore, the Earth's angular momentum was not conserved.

"Now, when a cow farts, there is also a redistribution of mass (and a very small amount of vectored thrust due to the accelerated mass of exhaust gas), soooo, there is a change in angular velocity, and because there was also a redistribution of mass, the angular velocities of the components of the system are under no obligation to return precisely to their former values."
That same redistribution of mass will happen whichever way the cow is facing.
It happened yesterday, and it will happen tomorrow.
The carbon cycle (and it's friends)will ensure that, at whatever rate cow farts add methane to the air at an altitude of roughly a meter, other processes will return it (otherwise the concentration of methane in the air would increase continuously).
The "vectored thrust" you talk of is also a non starter. For a start, all thrust is a vector.
More importantly, unless you think that a cow's fart continues round the world forever, you must accept that it's brought to a halt (wrt the earth) by viscosity. The KE is degraded to heat and any momentum is transferred back to the system.
To whatever extent cow farts affect the spin of the earth, they already have done so.
I agree that, if you don't consider the atmosphere as part of the earth then you can change the angular momentum of either by transferring momentum to the other.
It's just that such a dissection isn't sensible because the two are locked together by friction/ viscous drag.
The earth rotates once a day,and so does the air.
As I have said several times now, there is not a thousand mile per hour wind at the equator.
You cannot sensibly consider the rotation of the atmosphere separately from the rotation of the earth (except for a rather short time).

It's true that the methane will eventually be removed from the atmosphere, but that's not exactly a rapid process. However, I accept that the whole debate is really silly!
It does prompt a more serious question though. In the past, large amounts of gas have been pumped into the atmosphere by volcanoes. I wonder if those emissions were great enough to alter the moment of inertia of the Earth/atmosphere system and alter its rate of rotation?

It is silly question Geezer which has had a fantastic amount of scientific input. You raise a very valid point, with regard to events such as volcanoes. How do the forces of nature effect the rotation of planets. One example for me, supposedly huge amounts of freshwater have been stored in Dams that have shown to cause a minuscule effect to the rotational speed of the Earth. So redistribution of mass can effect the rotation of the Earth. I would then ask if the tides in the oceans cause changes, as huge amounts of water are displaced everyday. I suppose they would do but as they are cyclic the tides cancel the effect. Could the sheer mass of water moved even contribute towards events such as Earthquakes even?
I would imagine that Olympus Mons has had some kind of effect on the rotation of Mars....
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The earth is smoother than a billiard ball.
The effects of mountains are tiny and those of anything else are practically non existent.
Methane in the air lasts about 10 years, but that's not the point. The cows fart constantly and the gas is removed constantly so the overall concentration (and distribution) is constant.

The earth is smoother than a billiard ball.
And if you were to redistribute the mass on a billiard ball you may also find that this effects its rotational axis.....

The earth is smoother than a billiard ball.
And if you were to redistribute the mass on a billiard ball you may also find that this effects its rotational axis.....
Indeed, for example, if I moved all the mass a metre to the left, then I would move its rotational axis a metre to the left.
However, if it started and d=ended billiard ball shaped, then I wouldn't affect its moment of inertia. (assuming uniform density)

The earth is smoother than a billiard ball.
And if you were to redistribute the mass on a billiard ball you may also find that this effects its rotational axis.....
Indeed, for example, if I moved all the mass a metre to the left, then I would move its rotational axis a metre to the left.
However, if it started and d=ended billiard ball shaped, then I wouldn't affect its moment of inertia. (assuming uniform density)
Just how big are your billiard balls? I said redistribute the mass on a billiard ball, I am pretty sure I did not say move it a meter to the left?

Sorry, it was a bit early in the morning. I read it as "the mass of...".
What I should have said was that if I redistribute the mass by rotating it through 180 degrees I might not find that the axis had changed.
The point remains that if it starts and ends billiard ball shaped, then the moment of inertia stays the same.
A better analogy might be that if I stripped the paint off a ball, then put it back, I wouldn't alter the moment of inertia very much.

And bear in mind we're talking about gas here, which will tend to redistribute fairly smoothly about the earth, rather than lumping to one side. So I'd expect the axis of rotation would remain constant.
A bit off topic, but with the big earthquakes recently, it's been found that when the mass of the earth shifts a little in such an earthquake, the rate of rotation of the earth changes because of the change of it's moment of inertia: http://www.csmonitor.com/Science/2011/0314/JapanearthquakeacceleratedEarthsrotationstudyfinds

For those that care to do the math, the mass of the Earth's atmosphere is about a million times less than the solid/liquid mass, and 90% of the mass of the atmosphere is less than 10 miles above the surface. So, even if all of the atmosphere was to suddenly disappear, it would not affect the angular velocity of the Earth very much, although it would make a measureable difference (assuming there was anyone still around to make the measurement.)

So if the cows let 'em rip with high enough velocity, could they eject the earth's atmosphere into space?

So if the cows let 'em rip with high enough velocity, could they eject the earth's atmosphere into space?
You'd probably only get enough acceleration to do that if the methane was ignited (very carefully.)

......or, we could turn it around a bit and consider how many cows it would take to release enough methane to form an explosive mixture that would consume all of the oxygen in the Earth's atmosphere in a single explosion.

The greenhouse gases in the cows' flatulence will likely cause global warming, which will start to melt the Greenland and Antarctic ice caps, which will lead to a net redistribution of mass from poles towards equator, which will slow the earth's rotation (like a skater putting her arms out). No need to even get your cows to line up!

Anyone know what the current world pop. is of cows and how much gas they produce on average each day. Of course we should include all ungulates but the task of calculating their output is inflammatory! And what is the amount of methane needed to cause the possibility of combustion at sea level in our atmosphere?
I have not heard of a farmer blowing up their cowshed due to methane buildup?

I have not heard of a farmer blowing up their cowshed due to methane buildup?
I did read of a case not so long ago where several people on a farm were asphyxiated by methane. If I remember, it was released from a slurry pond.
http://www.usatoday.com/news/nation/20070704vafarm_N.htm

There's a James Herriot story about a cow with bloat setting fire to a shed when they treated it.

I have not heard of a farmer blowing up their cowshed due to methane buildup?
I did read of a case not so long ago where several people on a farm were asphyxiated by methane. If I remember, it was released from a slurry pond.
http://www.usatoday.com/news/nation/20070704vafarm_N.htm
That is quite a sad story Geezer [:[]
I think I have read all of James Herriot's Vet books.... used to make me laugh a lot!

The conclusion seems to be that cow farts don't speed the earth up (no matter which way they are facing) for any length of time and they may slow it down (depending on whether or not the methane levels in the atmosphere attributable to cows are stable).

The conclusion seems to be that cow farts don't speed the earth up (no matter which way they are facing) for any length of time and they may slow it down (depending on whether or not the methane levels in the atmosphere attributable to cows are stable).
Wasn't the conclusion that they can speed it up for a very short length of time (that we never determined), but that this effect would be shortlived due to friction/air resistance?

I think the issue that Bored chemist had picked up on was that the consequences of the methane buildup in the atmosphere slowing down the earth, small as they are, far outweighed any effect, transitory or not, due to reactive torque.
Or in simpler language:
 A lot of cows facing in one direction and letting rip might (or might not) provide a short term rocketlike kick that would speed up the Earth's rotation.
 That effect, even if it existed, would be minute.
HOWEVER
If the polar icecaps of the Earth were to melt a little, there would be a movement of material from the poles to the tropics.
 This would slow the Earth's rotation by increasing its moment of inertia
 The effect would also be minute, but much greater than that arising from any rocketlike torque
 Such a melting could be (and probably is being) triggered by a build up of methane and CO2 in the atmosphere.
 Cow flatulence is triggered by methanogenic microorganisms  C6H12O6 (sugar/cellulose) > 3 CH4 + 3 CO2

OK, Poor phrasing on my part.
When I said "for any length of time" I meant it in the vernacular sense (i.e. "any significant length of time") rather than the literal sense.

BC: then I agree.

Well, see, it's pretty obvious really. The question was;
"Can cow farts make the Earth rotate faster?"
In fact, cow farts actually make the Earth rotate more slowly. This is because they are continuously converting "Earth mass" (grass etc.) into "atmospheric mass" in the form of methane. This results in a slight, but real, increase in the moment of inertia of the Earth/atmosphere system.
If all the cows were to suddenly stop producing methane, the moment of inertia would decrease, and the Earth would take less time to rotate.

What you are forgetting is total historical contribution for cow farts! A lot of cow fart is locked up in the form of methane in massive quantities below the ocean and vast areas of land currently in a state of permafrost and melting!
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You might say that the energy from a cow fart is dissipated, and that would be in line with the theory of infinite dilution. One molecule of methane, absorbs 20 times the amount of infra red radiation then that of Carbon Dioxide, making it a far more efficient green house gas.
So all that trapped cow fart is going to be released into the atmosphere. If all of the gas was exploded in one place on earth, the resulting explosion would definitely have an effect of the earths rotational axis, haha!
Admittedly all the cows over history passing gas have not been standing in the same direction whilst doing so. [::)]
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Just look at all that energy! What a waste... [:o]

If everyone stood up in a parked school bus (in neutral) and threw their book bags at the same instant towards the back of the bus, it would (theoretically) propel the bus forward for an instant. Until the books slammed into the back of the bus. At which point the movement of the bus would stop again.
Likewise, if everyone stood up, bent forward, and let some flatulence go...
Well, the net effect on the bus would be nil, but people might be sent screaming out of the bus!!!!!!!
If you consider the earth as a closed system like the school bus, then the net effect will be nil. However, if it was time synchronized, one might be able to imagine a slight bump, returning to zero, like the instant while the books were still flying through the air.

Great analogy; I wonder if Geezer accepts it.

Great analogy; I wonder if Geezer accepts it.
Almost, but not quite.
The bus was accelerated, therefore it was "made to move faster". Then it slowed down again and stopped.
However, the bus did not return to its original position, so we can't say the net effect was nil.

What if the bus was propelled by a certain gas produced by the animal in question? [;D]

What if the bus was propelled by a certain gas produced by the animal in question? [;D]
Well, then it would have to be green, wouldn't it?

A bit of lateral thinking  nothing to do with animal flatulence
If we arranged for every city in the Northern hemisphere to have a oneway traffic system where vehicles were constrained to work in a clockwise direction, and a similar system where every city in the Southern hemisphere forced vehicles to circulate in an anticlockwise direction, we might make a much more substantial contribution to speeding up the earth's rotation!
Even if the road rules in North America and Europe changed to drive on the left, vehicles would travel clockwise around roundabouts, and that could contribute.

vehicles would travel clockwise around roundabouts, and that could contribute.
Not in Italy or Boston. They would drive straight across the roundabouts there.

Great analogy; I wonder if Geezer accepts it.
Almost, but not quite.
The bus was accelerated, therefore it was "made to move faster". Then it slowed down again and stopped.
However, the bus did not return to its original position, so we can't say the net effect was nil.
For an ideal, "physics question" type frictionless bus, the C of G of the bus/ bags/ schoolkids system would end up in exactly the same place that it started.
So we can say the overall effect was practically nil (except that the bus would be slightly warmer and bigger afterwards).
The analogous effect of the cow farts would be that one day would be slightly shorter, but that for all subsequent days the day length would be the same. You would introduce a change in phase, but not frequency, of rotation.
Also, the vehicles on the roundabouts have the same problem. They are part of the earth/ air/ cow/ fart/ vehicle system, and can not influence the overall angular momentum of the system.
There would be short term effects, but they would stop when the cars stopped or left the roundabouts. In particular, most effects would be cancelled out when the vehicles drove home.

From Bored Chemist:
Also, the vehicles on the roundabouts have the same problem. They are part of the earth/ air/ cow/ fart/ vehicle system, and can not influence the overall angular momentum of the system.
There would be short term effects, but they would stop when the cars stopped or left the roundabouts. In particular, most effects would be cancelled out when the vehicles drove home.
NO, that is the whole point of the one way systems and roundabouts. The movement of traffic, both morning and evening, would have a continuing vorticity in the same direction that would have to be deducted from the conserved angular momentum of the earth's rotation on a continuing basis.
Angular momentum (earth + traffic) is conserved. SO if we manipulate angular momentum (traffic) on a continuing basis, we make a real and lasting change in angular momentum (earth).
The traffic flow could not alter the total angular momentum of the earth system as bored chemist rightly points out. But it could lengthen or shorten the day, because the traffic makes a contribution to the system angular momentum that is separate from and independent of the contribution from the earth's rotation.

You would introduce a change in phase, but not frequency, of rotation.
Say what? How do you change phase without altering frequency?

You would introduce a change in phase, but not frequency, of rotation.
Say what? How do you change phase without altering frequency?
I used to get up at 08:30 to go to work each day, but I changed my job.
Now I get up at 06:30.
That's a change in phase but it was always once a day so the frequency is the same.
The Sunrise does the same thing every day, it's once a day but the phase changes WRT noon.
The traffic will make a difference while it's in motion but if there's ever a moment when all the traffic stops  say the World cup final then the speed of the earth will be just the same as it was before. Same with the cow farts, once they come to rest there's no effect on the spin.
As I said, if the cows faced East and started walking they would change the rotation rate slightly. The cars on roundabouts are much the same.

I used to get up at 08:30 to go to work each day, but I changed my job.
Now I get up at 06:30.
That's a change in phase but it was always once a day so the frequency is the same.
The Sunrise does the same thing every day, it's once a day but the phase changes WRT noon.
Frequency is just the reciprocal of period (the interval between cyclic events). The interval had to change to produce a change in phase. Mathematically, it's impossible to change the period without changing the frequency.
The change in frequency may have been temporary, but it still had to change to produce a change in phase.

If you picked the right day then the change in period was 2 hours in one day about 8%.
If you loot at it over the course of a week then it's just over 1%. That year the average change was about 0.02%.
But this is a one off event so you have to average it over the whole of time. The mean change in frequency was zero.
Also, something that happens at 06:30 one day, but at 08:30 the next isn't a "cyclic event" because it doesn't repeat; It's not cyclical.

Yes, but the frequency still changed.

Yes, it changed by zero.

Yes, it changed by zero.
Ah, right! You must be using the "new math" then. Either that or you have some previously unknown proof that frequency and period are not reciprocals up your sleeve, or perhaps you believe the period didn't actually change?

OK, it was 1/86400Hz before and it was 1/86400Hz afterwards.
What does old maths give as the difference between those?
There was a change, but that was a one off event; it doesn't have a frequency (unless you count zero).

OK, it was 1/86400Hz before and it was 1/86400Hz afterwards.
What does old maths give as the difference between those?
There was a change, but that was a one off event; it doesn't have a frequency (unless you count zero).
Perhaps you are thinking that frequency can only be determined by timing entire cycles? That's not necessary. We only need to know angular velocity.
If we plot the angular velocity against time and do a Fourier transform on it, we'll see lots of interesting changes in frequency when the angular velocity is changing.

What apodisation did you plan to use for the FT?

What apodisation did you plan to use for the FT?
Enough to prove that the frequency changed.

What apodisation did you plan to use for the FT?
Enough to prove that the frequency changed.
Please show your working
My hypothetical change of job happened on April 1st this year so, in a sense, rather than being 24 hrs long, that day was just 22 hrs.
Now, you can do an FT on a pulse lasting 22 hrs and come out with a nice "spectrum". If I remember rightly it comes out as a sinc function. Sure, it will have a dominant component at a frequency of 1/22 hrs^1, but that's just a mathematical result.
It doesn't have a meaning.
There is no way that you can say that April 1 2011 has a frequency of "every 22 hrs" because it doesn't repeat itself.
It actually has a frequency of once in the whole of time.
1/ infinity is practically zero.

Please show your working
I've got a better idea. You show how frequency is not the reciprocal of period.

BTW BC, your "change in job" analogy is
a load of twaddle a bit suspect.
The Earth's frequency is a function of its angular velocity. The time at which you start work has nothing to do with your angular velocity, so there was no phase shift in that case.

The Earth's frequency is a function of its angular velocity. The time at which you start work has nothing to do with your angular velocity, so there was no phase shift in that case.
The Earth's frequency... is that the Schumann thingy me bob? Angular velocity, sounds like the kind of math you would need to determine the outcome of my original post which has somewhat drifted a wee bit, not that I mind as to be honest this thread was destined for the manure heap from it's conception. 98, 99 100.... [:P]

Something that happens once, and only once doesn't have a frequency. It doesn't matter how long it takes, Only things that repeat have a meaningful frequency.
My granny may have lived for a hundred years, but she didn't live ten times per millennium.
And I'm sorry you don't understand that my working day and the earth's rotation have a simple fixed phase relation, except once when I changed it.
I used to get up at 08:30, that's about 60 degrees of the earth's rotation before the Sun is overhead. Now I get up at 06:30; that's about 90 degrees before noon. (I'm ignoring the half hours to keep the arithmetic easy.)
The change of 30 degrees is a real phase shift.
(obviously I'm simplifying it by also ignoring weekends, BST, and such)

That's why we need to analyse the case that's pertinent to the original question of an object with mass being accelerated by a force. In that situation, the object has a cycle defined by 360 degrees of rotation. There can be no discontinuity in its rotation.
One or more cycles will differ from 24 hours. Therefore, there was a change in the period during some number of cycles. And you can't change the period without changing the frequency.

"There can be no discontinuity in its rotation."
The rotation sped up and slowed down, it may have been continuous, but it wasn't steady either.
Anyway, as I have said before "international cow fart day" happened once. It doesn't have a frequency.
The frequency of rotation was always 1 in 24 hrs. What the cows did was slightly affect the phase.

The frequency of rotation was always 1 in 24 hrs. What the cows did was slightly affect the phase.
Assuming we are talking about the Earth, the only way the phase could change without affecting the 24 hour cycle would be if the phase shifted in one direction, then shifted back an equal amount in the other direction, and both shifts happened within a 24 hour cycle. Otherwise, the rotation of at least one cycle would have to deviate from 24 hours. Are you saying that did not happen? I think you will have to accept that it actually did.
Knowing that, within some number of cycles that includes the ones that deviated from 24 hours, if we sum all their cycle times and divide the result by the number of cycles to get the period, the result is not going to be 24 hours. We might pick a very large number of cycles to minimize the apparent deviation from 24 hours, but unless we pick an infinite number of cycles, there will always be a deviation.
The only other way to avoid the change in period is to ignore any cycles that deviate from 24 hours.
Your "working day" model is interesting, but it has a teensy flaw. Your "clock" frequency is the same before and after the phase shift. But you make no mention of what happened to your clock during the phase shift.
Let's assume your clock is the alarm clock that wakes you up to go to work. In that case, the phase shift occurred while you were adjusting the time that the alarm goes off. The rate at which you adjusted the wakeup time determines the spectrum of frequencies produced during the phase shift.
For example, if you adjusted the wakeup time in zero time, you would generate an infinite series of frequencies. That might make it make it a bit difficult to claim there was no change in frequency.

Something that happens once, and only once doesn't have a frequency. It doesn't matter how long it takes, Only things that repeat have a meaningful frequency.
My granny may have lived for a hundred years, but she didn't live ten times per millennium.
And I'm sorry you don't understand that my working day and the earth's rotation have a simple fixed phase relation, except once when I changed it.
I used to get up at 08:30, that's about 60 degrees of the earth's rotation before the Sun is overhead. Now I get up at 06:30; that's about 90 degrees before noon. (I'm ignoring the half hours to keep the arithmetic easy.)
The change of 30 degrees is a real phase shift.
(obviously I'm simplifying it by also ignoring weekends, BST, and such)
One of the important things about my redesign of the traffic scheme is that it is an ongoing effect  it will last as long as our civilization or fossil fuel reserves do. It is not a constant effect, so we would have to talk about an increase in average frequency, but it would constitute a real frequency shift rather than just a phase shift.
Similarly with the cows  if they were lined up for an event, then sure, the net effect is only a phase shift. But if they were to stay lined up, and someone were to conduct an ongoing symphony, then there would be a frequency shift.

but it would constitute a real frequency shift rather than just a phase shift.
Not to be picky, but any phase shift produces a real change in frequency (it might be small, but it's still real.)
Please see my previous post  I was adding some stuff to it while you were posting  possibly a "nono", but I didn't think anyone else was actually awake [;D]

" but unless we pick an infinite number of cycles"
At last! You have got it! A single oneoff event in the whole of time doesn't produce a frequency shift.
Incidentally, any other number of cycles would be totally arbitrary and hard to justify.
We can stop now.

A single oneoff event in the whole of time doesn't produce a frequency shift.
It's not a oneoff event. As you pointed out yourself, it's a phase shift, and we can't change phase without changing frequency.
It might be a oneoff event if the Earth instantaneously rotated through some number of degrees, but as that is clearly impossible, we don't need to worry about it.
We can stop now.

Just to throw a monkey into the butter:
Since the frequency spectrum is the Fourier transform of the signal, terms like "frequency change" are somewhat problematic. If I do a Fourier transform of your signal, I get a frequency spectrum for the entire signal, including the phase change. This frequency doesn't change over time.
By the way, I worked out the Fourier transform for a sinusoidal signal with an instantaneous phase change and it gives you the same frequency components associated with a single sinusoid.
If the phase change is gradual, it will generate more frequency components.

If you do want to talk about frequency changing over time, I suggest using a spectrogram, in which case even an instantaneous phase change causes a frequency shift in the spectrogram.
Can we stop now?

Since the frequency spectrum is the Fourier transform of the signal, terms like "frequency change" are somewhat problematic.
It's only problematic if you assume "frequency" implies sinusoidal.
I think we can stop now.

True. It's probably best to agree on a definition of frequency before arguing over specific examples.
You have shamed me into stopping.
For now.

It's clear that Geezer and I have differing opinions of the meaning of frequency.
I think that something needs to repeat before it has a meaningful frequency.
Sure, you can FT a single spike or top hat, but what you get depends on the apodisation. If you don't arbitrarily crop the time domain then you get a zero frequency. If you accept that the input function is infinitely wide you don't get a spectrum.
Of course, if you could get the cows to line up and fart regularly, say every Tuesday, that would be different.

It's clear that Geezer and I have differing opinions of the meaning of frequency.
I think that something needs to repeat before it has a meaningful frequency.
Sure, you can FT a single spike or top hat, but what you get depends on the apodisation. If you don't arbitrarily crop the time domain then you get a zero frequency. If you accept that the input function is infinitely wide you don't get a spectrum.
Of course, if you could get the cows to line up and fart regularly, say every Tuesday, that would be different.
Aha! Now we are getting somewhere.
I can think of three cases. There may be more.
a) A purely sinusoidal "frequency". You can't change its phase at all.
b) A nonsinusoidal "frequency". You can change its phase, but not without changing its "frequency".
c) A nonsinusoidal function composed of more than one sinusoidal "frequency". You can change the phase of the function by altering the amplitudes of the sinusoidal frequencies.

Geezer, what's the definition of frequency that gives you the above three cases? I'm betting that if you give a definition, we can work out pretty quickly why BC's definition doesn't agree with it (and for that matter, why the Fourier transform definition doesn't agree with either).

The one in Wikipedia seems reasonable enough.
"Frequency is the number of occurrences of a repeating event per unit time."
I think a repeating event could be lots of different things, but two ought to be sufficient. The unit of time could be anything you like.

Ok, what about the cow example? I'll make up some numbers, so don't complain if they're quite a bit off. :)
Let's say the earth rotates 1 time/24 hours. One afternoon, and only once, the cows all fart and that day is shortened by 6 hours. When the sun rises again and all following day, the day is once again 24 hours long. So you have days of 24 hours, one day of 18 hours, followed by days of 24 hours.
What would your definition tell us about the frequency of the earth's rotation in this case? If you need more information, feel free to specify how the earth speeds up for that one day, for example.

Ok, what about the cow example? I'll make up some numbers, so don't complain if they're quite a bit off. :)
Let's say the earth rotates 1 time/24 hours. One afternoon, and only once, the cows all fart and that day is shortened by 6 hours. When the sun rises again and all following day, the day is once again 24 hours long. So you have days of 24 hours, one day of 18 hours, followed by days of 24 hours.
What would your definition tell us about the frequency of the earth's rotation in this case? If you need more information, feel free to specify how the earth speeds up for that one day, for example.
That would not work. We'd end up shortening a lot of days if the cows knocked six hours off one of them. I would think the retarding torque produced by atmospheric friction as a result of the cow exhaust would be rather small, and it would probably decay exponentially. In the interest of keeping it as simple as possible, let's make that assumption. I'm also assuming the moment of inertia of the cow/earth system is constant, even although it would not be.
I'd suggest we assume that, after a (an?) heroic effort, the cows manage to whack a whole second off one day. Although I think it's pretty unlikely, that way we might assume the Earth's angular velocity returned to its previous value in less than 360 degrees.
Again, in the interests of not making it too complicated, let's assume there was constant angular acceleration for one hour sufficient to knock one second off a day, followed by constant angular deceleration for 11 hours, after which the angular velocity returned to 360 degrees/day. (Assuming a "day" is determined by an atomic clock.)
Applying the Wiki definition, and assuming the unit of time is one day, we have a frequency of about 1.0000115 cycles/day.
Personally, I'd prefer to define the Earth's frequency using something with a lot more resolution like, for instance, arcseconds per second (should be around 15 Hz), as that would make the frequency change much more apparent, but it really won't make any difference as long as we understand the angular acceleration.
(BTW, if you do use arcseconds per second, the average frequency during our "short day" is 15.000173 Hz)

"Frequency is the number of occurrences of a repeating event per unit time."
So we need an event to repeat. I'm guessing here, but is your repeating event one sunrise or sunset (or a 2*pi radian rotation of the earth)? If so, you can get frequency by calculating (# of radians rotated through)/(# of seconds).
If you're doing it this way, you have to specify two things:
1) How many radians or seconds are you averaging over?
2) What is the starting time of your averaging?
For your case, I believe it was 2*pi radians and you started right when the cows farted?

For your case, I believe it was 2*pi radians and you started right when the cows farted?
That would work.

For your case, I believe it was 2*pi radians and you started right when the cows farted?
That would work.
Ok, I understand your definition and agree that with that definition, you get a changing frequency.
I think that BC's definition of a repeating event was that the event (one rotation of the earth through 360 degrees) had to be identical in duration to the other events to be considered repeating, though he can correct me if I'm wrong. So in that case, you can't even think of frequency defined over any interval that includes the short day.
Your definition is that it the event is rotation through 360 degrees, but in terms of repetition, you count it as repeating so long as the earth keeps rotating, even if the time it takes for each rotation varies.
The Wiki definition of frequency is pretty vague, so I think both of those interpretations make sense within its bounds.
For whatever it's worth, my own preferred definition of frequency would be in terms of the Fourier transform, which has aspects of both of these definitions.

FFS!
International cow fart day happens just once.
Not once a day.
Not twice, separated by some time period (of any duration you might want to choose).
Not alternate Tuesdays.
Not every full moon.
Once. Only once just on one occasion.
Once.
It doesn't repeat.
It has no repetition.
Since it happens once, and never again, it does not happen more than once.
Am I beginning to get the message through here?
Since, as wiki says, ""Frequency is the number of occurrences of a repeating event per unit time."" and we are not talking about a repeating event, it doesn't have a frequency.
Ok, now to look at a nice easy case of the maths.
Imagine it's not cows and farts but some huge gun and it launches a big cannon ball but the ball is hooked to a long chain which is nailed to the earth.
The gun fires and pushes the earth.
After a while the chain pulls tight and it then tugs on the earth. The earth is initially slowed down, but the tug on the chain pulls it back to it's original speed.
It's a really big fast ball, and a really strong long chain.
So big that when the gun fires, it stops the earth dead. (OK you would need two on opposite sides of the earth to get a torque). 6 hrs later the chain goes tight an pulls the world back to spinning again once every 24 hrs, but 6 hours late a rather big phase shift of 6/24 *360 ie 90 degrees.
Not only does this cause absolute mayhem as the tides slosh round and the air gets whipped up to superhurricanes it does something really odd.
From a nice safe distance there's a bloke on another planet watching the earth. He sees me waving to him
As the earth turns I move across the surface of the earth (from his point of view) He looks at where I am and, by some odd coincidence he sees that my position traces out an exact sine wave (while I'm in sight) with respect to his local time.
On "big gun day" I happen to be in his field of view.
He sees me initially tracing out a sine wave. He sees me stop (I'm holding on really tight), then he sees me restart.
"Odd!" he thinks.
Then he goes back to looking at my traversing of the world and he spots that I'm now tracing out a cosine wave.
The cosine wave and the sine wave are both easy to do a FT analysis on. Each has exactly one frequency component, and it's the same.
Now, I may be mistaken, but I think that you can do the same thing with a smaller gun, if it only held the world back for 3 hrs the phase shift would be 45 degrees
The FT would give equal sine and cosine components but with amplitudes (If I have the maths right) of cos 45 and sine 45.
If you only have a cow fart worth of phase shift, you only introduce a very small amount of cos theta into the equation but what you don't do is add any other frequencies.

Oh, I see where you might be going wrong BC.
It doesn't repeat.
It has no repetition.
Since it happens once, and never again, it does not happen more than once.
The farting does not need to be repeated. We're measuring the effect the farts have on the Earth's rotation in terms of angular displacement in time, not the frequency of farts, and the Earth's angular rotation certainly does repeat.
Please consider this:
Prior to IFD (International Fart Day), the Earth was rotating with uniform angular velocity. If we plot the displacement of a point on the surface near the equator against time, the plot should be perfectly sinusoidal (I hope you would agree with that.)
Likewise, on the days after IFD, the plot will be perfectly sinusoidal with the same frequency and period as the days prior to IFD. (No disagreement so far, I hope.)
During IFD the angular velocity was not uniform. It increased a bit, then it slowed back down so that the daily cycle time was reduced and that resulted in a phase shift of the Earth's rotational cycle relative to our atomic clock "day" (pretty hard to argue with that).
As the angular velocity was not uniform during IFD, the plot of the displacement of the point cannot be perfectly sinusoidal during IFD. In other words, it's distorted. (I'm sure you would agree with that too.)
The only tricky bit is understanding what the distortion from the pure sine means.
I'm sure that the time between two repeating events on the nonsine wave will be less than the time between two repeating events on the pure sine wave, so that alone qualifies as change in frequency. If you look at it in FT terms, I'm pretty confident that will also reveal changes in frequencies.
I'd rather not get into a debate around your cannon ball and chain model until we solve the fart question, but you could post it as a new question.

The cosine wave and the sine wave are both easy to do a FT analysis on. Each has exactly one frequency component, and it's the same.
BC, you're also wrong about how to do a Fourier transform of this signal.
1) Your signal isn't a sine + a cosine. It's a sine over part of the domain, a cosine over another part, and a continuous transition over the third.
2) The FT of a sine or a cosine over a part of the domain is not the same as the FT of a sine or cosine over the whole domain.
3) The FT of the transition region gives you a frequency spread.
4) The FT is linear, so FT(sine bit + cosine bit + transition bit) gets you FT(sine bit)+FT(cosine bit)+FT(transition bit)
5) Because the FT of each bit has multiple frequency components, the entire thing does as well. I believe the FT(sine bit)+FT(cosine bit)'s multiple components actually cancel each other out if the transition is instantaneous. But if the transition isn't instantaneous, then I'm extremely confident that you get a spread of frequencies.
By the way, the FT of a sine or a cosine has two frequency components, not one.
By the way, I've done a great deal of work in the area of timefrequency analysis (http://en.wikipedia.org/wiki/Time%E2%80%93frequency_analysis), so I can assure you I know precisely what I'm talking about on this one.

It's really just another example of FMF (Fart Modulated Frequency).

JP,
I agree that " if the transition isn't instantaneous, then I'm extremely confident that you get a spread of frequencies".
I'm just saying that, compared to the whole of time, the change is instant.
Taking any other time scale would, as I have said earlier, be an arbitrary choice (and give an equally arbitrary outcome).
If it happened twice then you could (just) use the time between those two events but this cow farting was a one off (I think I may have mentioned that sorry if I didn't make it clear)
It's the same point I made earlier about apodisation; if you fail to divide by the infinity you don't get zero.(mathematicians of a nervous disposition will want to pretend that I talked about things tending to zero as the reciprocal tends to infinity).
Incidentally, My experience with FT isn't in time/ frequency domain analyses, its in the 2D ones used in optics and the 3D ones used in crystalography, though we do use 1D FTs in spectroscopy.
Geezer,
you say "During IFD the angular velocity was not uniform. It increased a bit, then it slowed back down so that the daily cycle time was reduced and that resulted in a phase shift of the Earth's rotational cycle relative to our atomic clock "day" (pretty hard to argue with that)."
Yes, the earth had an off day in terms of timekeeping.
But only one bad day in the whole of forever. On average, it didn't happen.
"I'd rather not get into a debate around your cannon ball and chain model until we solve the fart question, but you could post it as a new question."
It's much the same system, something gets launched and imparts a torque to the earth, it gets stopped and imparts another, opposite torque. In one case it's a small amount of gas brought to a halt by atmospheric friction, in the other it's a bloody great iron ball brought to a halt by a chain.
The difference is one of magnitude only. I wanted something big enough to halt the world for 6 hours and I didn't want to strain the cow fart analogy too far.

Did someone say that the Earth is a closed system? Does that exclude, Light, gravity, cosmic rays, dark matter, asteroids, meteors, comets and UFOs? Not to mention interplanetary dust. In the cosmic context I do not think earth is a closed system, if it was we could not exist as the dinosaurs would still be running about.. [:o]

Geezer,
you say "During IFD the angular velocity was not uniform. It increased a bit, then it slowed back down so that the daily cycle time was reduced and that resulted in a phase shift of the Earth's rotational cycle relative to our atomic clock "day" (pretty hard to argue with that)."
Yes, the earth had an off day in terms of timekeeping.
But only one bad day in the whole of forever. On average, it didn't happen.
No argument. It depends on your interpretation of "make the Earth rotate faster". The "faster" bit did happen, just not for very long.
My greater objection was to the point that phase and frequency are independent. That goes against forty years of peering at oscilliscopes.
Understood about the ball and chain thing, but I'm not sure it's any more realistic than the cows. I think I'd prefer some some gigantic rockets. It should not be too difficult to determine how much thrust they would have to develop in order to accelerate the Earth's rotation by a measureable amount.
I'm clueless about how to determine how long the deceleration would take. I suspect the function would be exponential, but how would we calculate the atmospheric friction torque, even to a crude approximation?

Wait a minute! BC, you said "On average, it didn't happen".
After the short day, the average absolutely will reveal a difference, unless you are going to count cycles that have not happened yet, so you can't say "it didn't happen", particularly when a permanent change in phase marks the time when it really did happen.

Did someone say that the Earth is a closed system? Does that exclude, Light, gravity, cosmic rays, dark matter, asteroids, meteors, comets and UFOs? Not to mention interplanetary dust. In the cosmic context I do not think earth is a closed system, if it was we could not exist as the dinosaurs would still be running about.. [:o]
We are ignoring quite a lot of things to simplify the model.
Geezer, I'm averaging over an infinite past history. (which is one such simplifying assumption).

Geezer, I'm averaging over an infinite past history. (which is one such simplifying assumption).
Are you really sure you want to do that? If the Earth doubled its angular velocity tomorrow, would you say it wasn't happening because it had never happened before?

If it did it briefly enough I might.

If it only happens once, you can chalk it up to observational error and ignore it, right? :)

If it only happens once, you can chalk it up to observational error and ignore it, right? :)
That's what I usually do.
"What the bleep was that?"
"Beats me. Anyway, it doesn't fit the model, so it's obviously a fluke. Ignore it."
Off topic, that's a very common occurrence when testing just about anything with a computer in it. The test engineers see some weird behaviour, but because they can't reproduce the problem, the development engineers tell them they either screwed up or were hallucinating, and close out the problem report.
The weird behaviour usually reappears about two hours before the product is supposed to be released to the market.

It's like the old joke about a mathematician, physicist and engineer trying to prove that all odd numbers are primes:
The mathematician says "1 is prime, 3 is prime, 5 is prime, so by induction all odds are primes."
The physicist says "1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 isn't prime (but that's experimental error), 11 is prime, so all odds are prime."
The engineer says "1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, so all odds are prime."

Ok, I just came up with this argument that seems to prove by contradiction that "fart day" generates extra frequencies (via the Fourier transform). Let me know what you think.
First, you need to know that the Fourier transform has an inverse. From a signal over time, you can uniquely get the frequency spectrum of that signal, and from a frequency spectrum you can uniquely recover the signal over time.
Second, if the earth rotated unimpeded by farting cows, you could model it by a periodic sinusoid, s(t). Maybe this sinusoid starts and stops and maybe it goes off to infinity. It doesn't matter. It generates a frequency spectrum, say S(f). You can go back and forth from S(f) to s(t) by Fourier transforms and inverse Fourier transforms. I could just as easily have told you that the frequency spectrum is S(f) and you could have recovered the signal over time, s(t). There is no loss of information in the Fourier transform.
Let's assume BC is right and that the frequency spectrum with the cows farting is the same as without. If that's the case, then it's also given by S(f). By the properties of the Fourier transform, a frequency spectrum S(f) means that the earth's rotation is given by s(t), which we know is true from above.
But this is identical to the signal without the cows farting, and we know the signals cannot be identical (there's a phase shift). So the frequency spectra cannot be identical.
QED?

Ok, I just came up with this argument that seems to prove by contradiction that "fart day" generates extra frequencies (via the Fourier transform). Let me know what you think.
First, you need to know that the Fourier transform has an inverse. From a signal over time, you can uniquely get the frequency spectrum of that signal, and from a frequency spectrum you can uniquely recover the signal over time.
Second, if the earth rotated unimpeded by farting cows, you could model it by a periodic sinusoid, s(t). Maybe this sinusoid starts and stops and maybe it goes off to infinity. It doesn't matter. It generates a frequency spectrum, say S(f). You can go back and forth from S(f) to s(t) by Fourier transforms and inverse Fourier transforms. I could just as easily have told you that the frequency spectrum is S(f) and you could have recovered the signal over time, s(t). There is no loss of information in the Fourier transform.
Let's assume BC is right and that the frequency spectrum with the cows farting is the same as without. If that's the case, then it's also given by S(f). By the properties of the Fourier transform, a frequency spectrum S(f) means that the earth's rotation is given by s(t), which we know is true from above.
But this is identical to the signal without the cows farting, and we know the signals cannot be identical (there's a phase shift). So the frequency spectra cannot be identical.
QED?
I think that the problem with this analysis has precisely to do with the phase shift. A frequency spectrum S(f) does not uniquely define a signal s(t) unless we have a firm boundary condition, such as s(0) = 0, To take a simple example the signal function (sin qt + sin 2qt) has precisely the same frequency spectrum as (sin qt + cos 2qt)  equal spikes at f = q and f = 2q  but they are quite different functions.

To take a simple example the signal function (sin qt + sin 2qt) has precisely the same frequency spectrum as (sin qt + cos 2qt)  equal spikes at f = q and f = 2q  but they are quite different functions.
The Fourier spectra of these are actually quite different. For both signals you get four spikes: at +/q and +/2q. In the first case, they're weighted by i/2, i/2, i/2, i/2, respectively. In the second case, they're weighted by i/2, i/2, 1/2, 1/2, respectively. You get spikes at the same frequencies, but the weights are different, so it's a change in Fourier spectrum.
At some point, it comes down to the definition. I would say the frequency in the above case has changed, since I'm using the Fourier definition of frequency. The fact that you get (different) complex weights allows you to invert the transform to get the signals back.

The FT is an integral (of sorts).
If I integrate something then differentiate it again I lose information because I don't know the "constant of integration".
I think that zero frequency information is lost in the same way.

Zero frequency information shouldn't be lost via the FT. It's what we call a "DC term" in frequency analysis. If I have a signal s(t)+Constant, then the FT of that is equal to the Fourier transform of s(t) plus a zerofrequency component of amplitude Constant.
If you invert the Fourier transform, you get exactly the original signal back.
This is because the FT is a definite integral (with infinite limits), and the inverse FT is also a definite integral. There is no derivative being taken.
The FT and it's inverse do run into some issues with some signals. I know that figuring out the class of functions it fails on is very difficult, but what was taught to me in physics and optics was that for almost every physical signal, the FT works. (I do think it has problems representing an instantaneous phase change in a sine wave, for example, but this isn't physical.)

I do think it has problems representing an instantaneous phase change in a sine wave, for example, but this isn't physical.
I would guess that's because there is "no time" involved, in which case it's no longer a continuous function. It's really a different wave, or a wave that is in two places at the same time, so everything goes haywire.
Presumably that does not happen as soon as you introduce any sort of slope into the function.

That's not exactly true. You can Fourier transform back and forth from a step function or a "top hat" without losing information, even though they're discontinuous. I could have just done the computation wrong for the sin/cos, which is possible, or there might be something special about it.
By the way, I'm talking about doing the calculation analytically here. Of course, trying to do a discrete FT of a step is going to have some issues, since you can't sample the step with perfect resolution.

since you can't sample the step with perfect resolution.
Ah, right! Doesn't that boil down to giving it a certain amount of slope that doesn't really exist, in which case a step function would really be a ramp?

since you can't sample the step with perfect resolution.
Ah, right! Doesn't that boil down to giving it a certain amount of slope that doesn't really exist, in which case a step function would really be a ramp?
Something like that, I think. When you actually sample the function, you have to do a discrete Fourier transform, which isn't quite the same as the continuous integral. If the function has sharp features that you can't resolve perfectly with your sampling, it throws off the result a little bit.

Something like that, I think. When you actually sample the function, you have to do a discrete Fourier transform, which isn't quite the same as the continuous integral. If the function has sharp features that you can't resolve perfectly with your sampling, it throws off the result a little bit.
Right, but what about a "vertical" section of the signal? If it really is vertical, there is no phase angle between the start and end of the vertical section.

Something like that, I think. When you actually sample the function, you have to do a discrete Fourier transform, which isn't quite the same as the continuous integral. If the function has sharp features that you can't resolve perfectly with your sampling, it throws off the result a little bit.
Right, but what about a "vertical" section of the signal? If it really is vertical, there is no phase angle between the start and end of the vertical section.
The way I'm picturing this, the signal is sin(t) to the left of the discontinuity and sin(t+phi) to the right. The phase of the sine shifts by phi at the discontinuity.

The way I'm picturing this, the signal is sin(t) to the left of the discontinuity and sin(t+phi) to the right. The phase of the sine shifts by phi at the discontinuity.
Ewe! You mean a DC section. I'm not sure that constitutes a phase shift exactly. I think you have to instantaneously "jump" from one sine wave to another sine wave of the same frequency but different phase, and that results in an "infinite" slope rather than no slope.

Maybe we're at odds with each other on terminology. Usually when I see DC, I think of a constant added to the signal. I would tell you that sin(t)+C has a DC component of C.
When I think of phase of a sine function, I think of whatever's in the parentheses. For sin(t), the phase is t, which increases linearly and continuously with time. If it jumps from sin(t) to sin(t+phi) suddenly, that's a discontinuous change in phase, which I would probably call a phase jump or phase discontinuity, although I assumed (perhaps wrongly) that's what we meant by the term phase shift.
At any rate, I think I get your point. Your sampling density can only put a lower limit on the slope. If you know it was sin(t) on one side and sin(t+phi) on the other, you don't know if it took the entire interval to climb or if it jumped in a fraction of the interval. Your discrete Fourier transform will look identical whether it did either of those, you you've lost information. (I believe the usual discrete Fourier transform assumes linear changes in phase across intervals.)

I'm pretty sure an instantaneous phase change constitutes a vertical "jump". If the signal had a horizontal section and you analysed only that section, there would be no frequency components at all, whereas the vertical jump tends towards infinite frequencies.
EDIT: Of course, if you do the "jump" at precisely the right time, the signal would just have a sharp bend in it.

I'm pretty sure an instantaneous phase change constitutes a vertical "jump". If the signal had a horizontal section and you analysed only that section, there would be no frequency components at all, whereas the vertical jump tends towards infinite frequencies.
EDIT: Of course, if you do the "jump" at precisely the right time, the signal would just have a sharp bend in it.
Yep. I agree. The DC term I was talking about is a flat, horizontal component of the signal, or a flat, horizontal component added to a changing signal, which just shifts it's amplitude globally by a certain amount. A discontinuous jump in the signal would have infinite slope.
But if you sampled the signal, you'd only be able to say that it jumped between samples by a certain amount. You couldn't tell if that was an instantaneous jump with infinite slope, or a steep but linear rise. For that reason, sampling and doing a discrete version of the Fourier transform can lose information if you aren't careful about sampling finely enough. (And you can never sample vertical segments densely enough.)
By the way, one of the basic rules of determining sampling density is to use Nyquist sampling:
http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem ,
which essentially says that you have to sample your signal more finely if you want to accurately recover high frequency components.

Great! Well, I think this cow has been more than adequately flogged.

Great! Well, I think this cow has been more than adequately flogged.
We went from cow farts to Fourier transforms in just 6 pages of posts!