Can cow farts make the Earth rotate faster?

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Offline Geezer

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Can cow farts make the Earth rotate faster?
« Reply #100 on: 29/07/2011 21:07:47 »
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.
« Last Edit: 29/07/2011 21:24:07 by Geezer »
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Offline Bored chemist

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Can cow farts make the Earth rotate faster?
« Reply #101 on: 29/07/2011 21:33:59 »
"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.
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Offline Geezer

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« Reply #102 on: 29/07/2011 21:59:48 »
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 wake-up time determines the spectrum of frequencies produced during the phase shift.

For example, if you adjusted the wake-up 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.   

« Last Edit: 30/07/2011 02:17:08 by Geezer »
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Offline damocles

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Can cow farts make the Earth rotate faster?
« Reply #103 on: 30/07/2011 01:39:13 »
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.
1 4 6 4 1
4 4 9 4 4     
a perfect perfect square square
6 9 6 9 6
4 4 9 4 4
1 4 6 4 1

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Offline Geezer

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« Reply #104 on: 30/07/2011 02:27:36 »

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 "no-no", but I didn't think anyone else was actually awake [;D]
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Offline Bored chemist

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« Reply #105 on: 31/07/2011 11:26:47 »
" but unless we pick an infinite number of cycles"
At last! You have got it! A single one-off 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.

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Offline Geezer

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« Reply #106 on: 01/08/2011 16:59:08 »
A single one-off event in the whole of time doesn't produce a frequency shift.

It's not a one-off event. As you pointed out yourself, it's a phase shift, and we can't change phase without changing frequency.

It might be a one-off 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.
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Offline JP

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Can cow farts make the Earth rotate faster?
« Reply #107 on: 01/08/2011 17:27:27 »
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.

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Offline Geezer

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« Reply #108 on: 01/08/2011 17:52:49 »
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.
« Last Edit: 01/08/2011 17:54:56 by Geezer »
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Offline JP

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« Reply #109 on: 01/08/2011 18:03:11 »
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.

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Offline Bored chemist

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« Reply #110 on: 01/08/2011 19:48:05 »
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.
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Offline Geezer

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« Reply #111 on: 01/08/2011 20:44:03 »
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 non-sinusoidal "frequency". You can change its phase, but not without changing its "frequency".

c) A non-sinusoidal function composed of more than one sinusoidal "frequency". You can change the phase of the function by altering the amplitudes of the sinusoidal frequencies.

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Offline JP

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« Reply #112 on: 01/08/2011 21:48:04 »
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).

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Offline Geezer

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« Reply #113 on: 01/08/2011 22:04:36 »
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.
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Offline JP

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« Reply #114 on: 01/08/2011 22:59:57 »
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.

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Offline Geezer

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« Reply #115 on: 02/08/2011 02:56:15 »
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)   
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Offline JP

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« Reply #116 on: 02/08/2011 13:52:08 »
"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?
« Last Edit: 02/08/2011 14:19:49 by JP »

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Offline Geezer

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« Reply #117 on: 02/08/2011 17:26:07 »
For your case, I believe it was 2*pi radians and you started right when the cows farted?

That would work.
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Offline JP

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« Reply #118 on: 02/08/2011 18:43:54 »
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. 

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Offline Bored chemist

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« Reply #119 on: 02/08/2011 20:04:37 »
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 super-hurricanes 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.
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Offline Geezer

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« Reply #120 on: 02/08/2011 21:09:52 »
Oh, I see where you might be going wrong BC.

Quote
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 non-sine 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.
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Offline JP

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« Reply #121 on: 02/08/2011 21:13:26 »
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 time-frequency analysis, so I can assure you I know precisely what I'm talking about on this one. 

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Offline Geezer

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« Reply #122 on: 03/08/2011 01:18:24 »
It's really just another example of FMF (Fart Modulated Frequency).
« Last Edit: 03/08/2011 01:20:44 by Geezer »
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Offline Bored chemist

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« Reply #123 on: 03/08/2011 21:13:06 »
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.
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Offline Airthumbs

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« Reply #124 on: 03/08/2011 23:10:41 »
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]
Any intelligent fool can make things bigger and more complex... It takes a touch of genius - and a lot of courage to move in the opposite direction. (Einstein)

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Offline Geezer

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« Reply #125 on: 03/08/2011 23:20:47 »
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? 
There ain'ta no sanity clause, and there ain'ta no centrifugal force Šther.

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Offline Geezer

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« Reply #126 on: 03/08/2011 23:32:21 »
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.
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Offline Bored chemist

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« Reply #127 on: 04/08/2011 06:59:54 »
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).
« Last Edit: 04/08/2011 07:04:41 by Bored chemist »
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Offline Geezer

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Can cow farts make the Earth rotate faster?
« Reply #128 on: 04/08/2011 08:43:31 »
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?
« Last Edit: 04/08/2011 08:56:29 by Geezer »
There ain'ta no sanity clause, and there ain'ta no centrifugal force Šther.

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Offline Bored chemist

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Can cow farts make the Earth rotate faster?
« Reply #129 on: 04/08/2011 19:56:44 »
If it did it briefly enough I might.
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Offline JP

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Can cow farts make the Earth rotate faster?
« Reply #130 on: 04/08/2011 20:36:21 »
If it only happens once, you can chalk it up to observational error and ignore it, right?  :)

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Offline Geezer

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« Reply #131 on: 04/08/2011 21:23:42 »
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.
« Last Edit: 04/08/2011 21:55:38 by Geezer »
There ain'ta no sanity clause, and there ain'ta no centrifugal force Šther.

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Can cow farts make the Earth rotate faster?
« Reply #132 on: 04/08/2011 22:36:02 »
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."

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Offline JP

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Can cow farts make the Earth rotate faster?
« Reply #133 on: 04/08/2011 22:55:39 »
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?
« Last Edit: 04/08/2011 23:12:19 by JP »

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Offline damocles

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Can cow farts make the Earth rotate faster?
« Reply #134 on: 05/08/2011 02:28:55 »
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.
1 4 6 4 1
4 4 9 4 4     
a perfect perfect square square
6 9 6 9 6
4 4 9 4 4
1 4 6 4 1

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Can cow farts make the Earth rotate faster?
« Reply #135 on: 05/08/2011 12:55:03 »
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. 

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« Reply #136 on: 05/08/2011 18:06:35 »
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.
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« Reply #137 on: 05/08/2011 19:18:46 »
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 zero-frequency 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.)

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Offline Geezer

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Can cow farts make the Earth rotate faster?
« Reply #138 on: 05/08/2011 19:58:01 »
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.
There ain'ta no sanity clause, and there ain'ta no centrifugal force Šther.

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« Reply #139 on: 05/08/2011 21:35:34 »
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.

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« Reply #140 on: 06/08/2011 01:54:17 »
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?
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« Reply #141 on: 06/08/2011 12:43:59 »
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.

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« Reply #142 on: 06/08/2011 17:54:50 »
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.
There ain'ta no sanity clause, and there ain'ta no centrifugal force Šther.

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« Reply #143 on: 08/08/2011 14:40:16 »
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.

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Offline Geezer

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« Reply #144 on: 08/08/2011 18:06:15 »

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.
There ain'ta no sanity clause, and there ain'ta no centrifugal force Šther.

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Offline JP

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« Reply #145 on: 08/08/2011 18:52:28 »
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.)

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Offline Geezer

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« Reply #146 on: 08/08/2011 21:46:17 »
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.
« Last Edit: 08/08/2011 22:15:52 by Geezer »
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« Reply #147 on: 08/08/2011 22:51:18 »
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.

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Offline Geezer

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« Reply #148 on: 09/08/2011 03:45:06 »
Great! Well, I think this cow has been more than adequately flogged.
There ain'ta no sanity clause, and there ain'ta no centrifugal force Šther.

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« Reply #149 on: 09/08/2011 16:44:39 »
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!