Hmm… I dropped a (2*pi)[sup]2[/sup] in my estimate of the temperature change, now that I think on it, which is a factor of 40. That’s not so good, and instead of 1/2 a degree, I’m getting 20 degrees instead. That’s a lot.
I still have no idea what the specific heat of the Earth is, though…
You have it backward, the moon is tidally locked to the earth which is why the moon always has the same side facing us.
Okay, fine, so I’m obsessed. Rock seems to in general have a specific heat about 5 times smaller than that of water, so my answer bumps up ANOTHER factor of 5, to something in the 100 K range. Which is still not enough to melt the planet but is plenty to make things more than uncomfortable for us measly humans.
Oops, that was just a typo. I actually knew that the moon is tidally locked to the earth but not vice versa, and in the course of rearranging my sentence, I managed to add an extra “not” to it.
Which is not to say that I’m wrong in saying that the moon would be affected by the earth not spinning. If I have any credibility left in this thread, which I rather doubt at this point. :smack:
g8rguy, I have a physics textbook that gives the specific heat of marble as 860 and iron as 448 J/kg*K. Also, I found references on the web to the specific heat of granite being about 803. Those might give better estimates of what the specific heat of the earth should be – certainly much less than water. So there’s another factor of five or ten to increase the temperature. Now you’re talking 100 to 200 degrees.
On preview, I see that g8rguy has already addressed this, but I spent enough time looking this up while I should’ve been working that I’m going to post this anyway. 
PS- If it’s any consolation g8rguy, Carl Sagan calculated that the temperature would increase about 240 K at sea level (see Appendix 2 of his book “Broca’s Brain,” 1976, ISBN 0-345-33689-5). I’d say your calculation is in pretty good company.
This is not conservation of momentum. This is conservation of energy.
The kinetic energy of the earth, due to the earth spinning, is equal to (1/2) * I * [symbol]w[/symbol][sup]2[/sup]
where:
I = moment of inertia of the earth (with respect to its axis of rotation)
[symbol]w[/symbol] = angular velocity of the earth.
Quick calculation:
Moment of inertia of earth:
(1) Assume Earth to be a sphere or radius 6370 km
Moment of inertia of a sphere: (2/5) * m * r[sup]2[/sup]
m = mass of earth = 5.98 * 10^24 kg
So,
I = (2/5) m r[sup]2[/sup]
I = (2/5) (5.98 * 10^24 kg) (6.37 * 10^6 m)[sup]2[/sup]
I = 9.71 * 10^37 kgm[sup]2[/sup]
And, Rotational Kinetic energy of Earth:
angular velocity of earth = 2[symbol]p[/symbol] radians per day * 1 day per 86400 seconds
angular velocity of earth = 2[symbol]p[/symbol]/86400 rad/s = 7.27 * 10^-5 rad/s
So,
E = (1/2) I ([symbol]w[/symbol])[sup]2[/sup]
E = (1/2) (9.71 * 10^37 kgm[sup]2[/sup])kg [(7.27 * 10^-5)rad/s][sup]2[/sup]
E = 2.56 * 10^29 kg m[sup]2[/sup] / s[sup]2[/sup]
E = 2.56 * 10^29 Joules
The kinetic energy (rotational) of the earth is 2.56 * 10^29 Joules. Or, 2.56 * 10^20 GigaJoules. That’s a lot. The biggest hydroelectic dam I can think of offhand makes about 2 Gigajoules per second (GigaWatts, for the keen). At that rate, the LG-2 (Robert-Bourassa) Dam would need more than 10^20 years (or, about a hundred billion billion years) to generate as much energy as the Earth has locked up in its rotation.
As Sam stone suggested, in stopping the Earth, you transform this energy into something else. What, I don’t know. Under ordinary circumstances, Sam’s suggestion holds; it’s turned into heat. Think of a brake disk on a car. Its’s rotationg really fast, you apply friction, and it makes itself and the stuff around it (brake pads and air) hot.
Maybe nuclear fusion occurs, with all the energy suddenly available. Beats me… it really depends on how you decide to go about stopping the rotation. If you had a giant pair of brake pads, and applied them on opposite sides at the equator, you could make enough heat to do just about anything you want.
As for conservation of momentum: A different way of looking at things, that has next to nothing to do with heat. To stop the earth, you would need to apply a force to it that acts opposite to its direction of rotation. This force, applied for a certain time, is called an ‘impulse’. To calculate the angular momentum of the Earth requires some linear algebra, or some calculus, or some easy pre-fab formulas that I don’t have with me right now. Suffice it to say, it’s a lot.
You can’t really use momentum equations to discuss changes in energy states. Heat being a creature not of momentum, but of energy.
References ( :: Digs around through old textbooks :: )
Rotational Kinetic Energy Equation, Moment of Inertia formula for a solid sphere: Vector mechanics for engineers, Third SI metric edition (F. P. Beer and E. R. Johnston)
Radius and mass of the earth: Physics for Scientists and Engineers, Fourth Edition (P. A. Tipler)
Eh, the angular momentum of the earth is dead easy, assuming one isn’t as algebraically inept as I seem to be (it’s nice to be within a factor of 2 or so of Carl Sagan, but it’s pathetic that I missed a factor of 40 the first time around!). It’s just L = Iw. That is, with our assumptions, L = 2/5 MR[sup]2[/sup]w. Actually, Goldstein quotes a value for the moment of inertia of the earth as ~1/3 MR[sup]2[/sup], the 1/3 being more or less empirical and arising due to the greater density of the core.
Using these, I get L ~ 610[sup]33[/sup] Js. I’m not sure how informative this number is, but it’s a number.
oops. I missed gr8guy’s first post up there where he mentions kinetic energy, and most of the posts that followed because I was busy typing.
Let’s see, here…
MegaDave has some good points too.
Now, like gr8guy was saying, if you made all the earth’s roational kinetic energy into heat, you could heat up something (say the earth) to the tune of
m * S(C dT)
Where (because I haven’t got the symbol font all worked out yet)
m is the mass of whatever you’re heating, and
S(C dT) is the integral of (C dT) on the Temperature range in question.
C being the specific heat (or “heat capacity”) in Joules per kilogram-Kelvin
T being Temperature in Kelvins.
Now, can we assume that the Earth has a constant specific heat? Umm… that depends on the temperature range… and on the material… The atmosphere sure doesn’t, over a range of more than a few hundred degrees… and the water sure doesn’t, especially once it boils… and, for all the gas turbines I’ve analysed, I’ve never done a rock turbine… so, I dunno about the rest of the earth… but if we assume it does, then the calculations get much easier…
So, yes, it does 
Umm… what’s that specific heat? Hard to say. If the earth was all water, it would be about 4178 (off the top of my head) J/kgK, like gr8guy said. Of course, boiiling the whole ocean would take a lot of energy. A LOT of energy… so we need to round up our estimate… and rocks… anybody know the specific heat of silicon, oxygen, iron, and nickel?
Hmm. Well, like gr8guy said, E = mC (T2 - T1) But as soon as you start getting changes of state (oceans boiling, rocks melting) you lose lots of energy just to make them change state. And don’t foget, a lot of the earth is already molten rock, which has a very different specific heat than if it were solid.
Another factor to consider: a lot of this heat might just radiate off into space, rather than penetrate down to the mantle and heat it up.
But, if we assume that none of it radiates away, and all of the Rotational Kinetic energy is turned into heat used to warm up the earth…
We should really figure out the mass of the ocean (or the volume and density), and the mass of the ice caps, if we’re really nitpicky, and do this in five steps:
First, heat up all the rocks, and all the air, all the watrer, and all the ice, to zero Celsius.
Second, apply the latent heat of fusion to find out how much energy it takes to melt all the ice.
Third, heat the rocks, air, water, and water that used to be ice up to 100 Celsius.
Fourth, apply the latent heat of vapourisation to the water and water that used to be ice to find out how much energy it takes to boil all the water.
Fifth, heat everything until you run out of energy.
(This approach neglects other materials that have a state-change on the temperature range ordianrily experienced at the earth’s surfface… there aren’t many, and the amonuts in which they appear are megligible, I’d guess)
A final problem: Not all of the earth is at the same temperature right now… there are places where it’s -50 Celsius, and places where it’s almost +50 Celsius. And, just a few miles down into the crust, it’s very warm indeed. I’m not sure how to take that into account.
On preview, I see that gr8guy has caculated the angular momentum of the Earth. Yeah, that’s the prefab formula I wanted. I wasn’t completely sure there was no exponent on I or Omega. Boy, I must be getting old… my memory’s starting to go. 
To boil down what wolfstu said: The rotation of the Earth holds both energy and angular momentum, and both must be conserved. It’s easy to conserve energy; you just dump any extra left over into heat. Angular momentum is a bit trickier. If you stopped the Earth, you would have to end up with either something else rotating that wasn’t before, or something that was rotating in the opposite direction stopping. Depending on how we stop the planet, this may or may not be a problem: Presumably, something from outside the planet is interacting with it to cause this stoppage, and we might be able to hand off all our angular momentum to that thing.
Chronos, of course, is right on.
(I couldn’t think of a good example to demonstrate conservation of momentum in the case of stopping to earth… and then I got distracted with the math)
Thanks for the much easier to read summary.
On the other hand,
If the world should stop revolving, spinning slowly down to die,
I’d spend the end with you,
And when the world was through,
Then one by one, the stars would all go out
And you and I would simply fly away.
David Gates wouldn’t lie to you, would he?
And why would God suspend almost every law of physics in order to allow some minor general win a war? What evidence is there that God has ever played around with the laws of physics. There are many manifestations of physics that have astonished man, but no proof that God goes around playing God
This question comes up a lot when discussing Velikovsky, because one of his theories was that the biblical day with no night was caused by Mars careening past the Earth and stopping its rotation for a day. How it starts up again, I do not remember.
I knew a professor of physics who was totally convinced that Velikovsky was right, and lent me a manuscript of a book he was writing on the subject. It sucked, really badly. I was astonished that a professor of physics would buy into it.
I don’t recall ever seeing the book published.
I’d stop the world and melt with you!