On the earth, at the equator, how fast would the world have to be spinning in order for a 100# weight to float? (gravity vs. circumferential force). Would speed needed depend on weight, shape, or any other factor? Are there examples in the universe of things spinning fast enough that the matter in them is pulsing in and out?
Regardless of the weight, the earth would have to be rotating fast enough that a day would be about 84 minutes to neutralize gravity. Of course, the Earth can’t spin that fast and remain a solid body.
So even if you got an alien super-technology to do the spinning up, you’d run into the problem that physics wouldn’t consider your 100# weight any different from any of the loose bits of Earth surface, and barely different from what we consider solid.
ETA: ninja’d by naita
Following AndyL: Which, not coincidentally, is about how long a low Earth orbit takes.
Something that has zero effective weight while sitting on the surface of a rapidly rotating planet is in effect orbiting the planet. Which can’t happen in real life because the material of the outer layers of the planet itself would also be in orbit.
so, things weigh more or less at the poles?
Things weigh slightly more at the poles than they do at the equator, for two related reasons. First, there’s the direct effect of the rotation, which lightens things at the equator; second the earth’s rotation distorts the shape of the earth, which means that objects at the poles are closer to the center of the Earth than objects at the equator are.
Is the mass of a planet derived from noting its speed of rotation around the sun and its distance from the sun? Any slower, it would fall into the sun, any faster, it would fly off into a more distant orbit?
If by “derived” you mean “can be figured out by using” then yes.
If by “derived” you mean “is caused by” then no.
The planet masses whatever it masses. If it settles into a stable orbit the size of the orbit is caused by the mass and velocity. Not the other way around.
Interesting … what we’re talking about is orbital speed … and Wikipedia actually calculated this out as 17,672 mph … and a period of 1h 24m 18s …
Which is probably where poster #2 got 84 minutes. 
And unless the planet has a mass comparable to the Sun, the velocity of the planet is essentially unaffected by the mass of the planet. For example, for the Earth, the orbital velocity is 29.78911 km/s. For a 1 kg object at the same distance from the sun as the Earth has, orbital velocity would be 29.78906 km/s.
In other words, in reality, the earth will flatten out and become more oblate as it spins faster. (The earth is already a slightly oblate spheroid due to its rotation.) But as it flattens out, the equatorial radius increases, so it would take less rotation speed to achieve “weightlessness” at the equator. I don’t know off hand what speed this would happen, but it would make a good question for a freshman astronomy class.
No, that has nothing to do with the planet’s mass. Its orbital speed as it goes around the Sun only depends on the Sun’s mass and how far the planet is from the Sun.
Think about it - if an object’s orbit depended on the object’s own mass, then when the object splits into two, they go into different orbits than the original object. Which doesn’t make sense. When an astronaut steps out from the Space Station and lets go, he/she doesn’t fall into a different orbit. He/she would continue to float right next to it.
And this also gives the answer to another question that is often asked.
Say that you dug a tunnel of the right curvature from New York to London, evacuated the air, and dropped a ball into it? How long would it take to arrive?
42 minutes, or half an orbit. This is true for any such tunnel of any length.
Most large bodies in the universe have no tensile strength to speak of. Hence, if you spun them up so that centripetal acceleration matched gravity, they would just fall apart (as said by other posters).
However, a body like a nickel-iron asteroid has at least a little tensile strength, and not much gravity. I’m sure there are examples of these spinning at roughly the rate needed to counteract gravity at the equator (though, since the equator isn’t going to be a nice circle, there’s only going to be certain altitudes where this works).
If you go small enough, an object is almost always going to have this property. A bowling ball in orbit has a tiny but non-zero amount of gravity; even a tiny amount of spin will exceed that acceleration.
Note that any body which is (roughly) spherical due to gravity is, almost by definition, not going to have this property. Past a certain size, there’s no material which is strong enough to resist the forces of gravity–in astrophysics, this is called being in hydrostatic equilibrium (hydro- because the material is acting as a fluid). Centrifugal forces which are on par with gravity will also exceed the material strength and cause the body to fall apart.
The OP said “float.” We float megaton ships all day, every day. We need to know the density and medium of the 100# weight. Is it in a vaccuum? Air? Water?
Do some reading about millisecond pulsars.
From Wikipedia:
"Pulsar PSR J1748-2446ad, discovered in 2005, is, as of 2012, the fastest-spinning pulsar currently known, spinning 716 times a second.
Current theories of neutron star structure and evolution predict that pulsars would break apart if they spun at a rate of c. 1500 rotations per second or more, …
…
However, in early 2007 data from the Rossi X-ray Timing Explorer and INTEGRAL spacecraft discovered a neutron star XTE J1739-285 rotating at 1122 Hz [RPS]. …"
At these rates, they are losing energy thru gravity radiation. Which slows down the rotation.
You’re not going to find anything natural that’s faster, I think.
There’s arguably a similar limit for black holes: Black holes can rotate, but there’s a maximum amount of rotation (compared to its mass) an object can have and still be a black hole. Fortunately, it appears to be impossible to bring a black hole above this limit, since that would result in a naked singularity, but it’s expected that most black holes in the Universe (at least, the ones of stellar size or larger) are probably very close to the limit, at around 97% or so.
[emphasis added]
Actually, this is true for a straight path. The path with the shortest time would be curved and, for points that are not antipodes, would take less than 42 minutes.
Barlennan would know the answer.