Given the earth’s current rotational speed and conserving angular momentum, how fast would the earth be spinning if it it’s diameter were shrunk down to the size of a bowling ball (while, of course, maintaining it’s current mass)? Or is there a physical limitation to such a thing, like surpassing C?
Angular momentum equals the product of angular velcity and moment of inertia. The moment of inertia of a sphere is proportional to r[sup]3[/sup]. Without going through the arithmetic I’m pretty sure it would fly apart long before it got down to the size of a bowling ball.
It’s cold out today so I’m not golfing. That leaves me with some time so I did the arithmetic.
The way I figure it if the earth were the size of a bowling ball it would be spinning at about 10[sup]19[/sup] rpm. Thats 10,000,000,000,000,000,000.
I believe I was right when I said it would fly apart.
I knew it would be fast, but holy crap. Although, I think I have a disc drive that gets close to that.
Now, I can’t help but wonder how small the Earth would have to get before it DID fly apart… and how fast it’d be spinning.
well, internet sites give max rpm for 10" grinding wheels as about 2500 rpm. I suppose that means the speed for flying apart about is around 10000 rpm.
Unless I screwed up the arithmetic if earth has the same strength as a grinding wheel it would fly apart when the radius reached 840 miles and the rotation speed just over 27 rpm.
Well if you shrunk it past the size of a bowling ball down to about 9mm you’d solve the problem of it flying apart – by turning it into a black hole!
I’m not going to sit down and work out the numbers, but the ultimate limit would be when the rotation rate at the equator equals escape velocity (assuming the tensile forces holding it together are negligable). Of course, it’s going to oblate and flatten out before that, and that’ll change the rotation rate, blah blah blah. Basically, it’ll come apart like a cheap gold watch long before it ever gets to that size.
There’s an additional complication, though. Compressing the Earth to a sphere the size of a bowling ball would make it denser than neutronium. You’d end up with a black hole, albeit one rotating so fast that it’ll have an enormously distorted ergosphere. (You still wouldn’t be able to use it for time travel, as the tidal forces would rip atoms to component particles, but at least in theory there would be spacelike paths through time–both forward and back–around it.)
All in all, I think it’s a bad plan. Maybe you could just do the seeds-growing-under-an-ultraviolet-lamp for your science fair project?
Stranger
sigh But, that’s what I did last year.
Except that you couldn’t do that, without shedding a lot of angular momentum. The Earth’s dimensionless angular momentum parameter ( Lc/(Gm[sup]2[/sup] ) is 890, (approximating the Earth as uniform density) while for a black hole, it can’t be any higher than 1. And if you’re shedding angular momentum anyway, there are much easier ways to prevent the Earth from flying apart.
Back to the flying-apart, I get that, for a given mass and angular momentum, the ratio of centrifugal force to gravity for a gravitationally-bound sphere is inversely proportional to radius. This ratio is currently equal to 0.00343956008, so the Earth would fly apart once you shrunk it down to about 1/300th of its current radius.
Can we assume that a planet-Earth-turned-bowling-ball would pack a tremendous amount of “English”? (And I thought I had an unpredictable left hook before!)
Wait a sec, black holes are collapsed stars. Stars spin. Why don’t they go flying apart before they get small enough to be black holes?
Yeh, I was kinda thinking the same thing. What are we missing?
Note the “unless you shed a lot of angular momentum” disclaimer. In a supernova, a star does shed a lot of angular momentum, in the debris that’s blasted out by the explosion. Which incidentally guarantees, for most stars, that there will be an explosion (i.e., not everything ends up down the hole), since there has to be some matter to carry away the angular momentum, and almost all stars spin fast enough to need to shed at least some. Even at that, most black holes formed through astrophysical processes like supernovae are expected to be spinning at very close to the maximum possible rate, upwards of .95 or even .99.
Gravity
I’m not sure that any of our ideas of when the earth would fly apart are all that great. The earth is a liquid ball with a BB in the center and surrounded by a relatively thin shell. And the shell is cracked and broken up into a bunch of pieces that slide around on the surface of the liquid. I think that it would fly apart long before it got down to my number, or any of the others, or even very much flattening.
Afraid of a little naked singularity, eh? You’re such a prude! Causality isn’t all it’s cracked up to be, you know.
There you go, David, with your “facts,” bringing a discordant note of unwelcome reality into this discussion. Didn’t you get the memo? We’re an empire now, and when we act, we create our own reality.
Stranger
There’s always this…
Well, you know the English were all over the place even before the Great Momentum Transfer. Now they’re heading out into the galaxy along with sundry bits of tectonic debris…
Just trying to do my insignificant bit.
I think the collapse into a black hole is very rapid. Maybe the star does lose a lot of material but has enough left to continue the collapse because it occurs in such a short time.