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Suppose engineers built a rotating space station similar to Space Station V from the film 2001: A Space Odyssey (circa 1968), but with a large sphere, instead of a ring? Could this be rotated or pushed through space in such a way as to give it mostly consistent gravity on all parts of the surface within the sphere?

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Spinning station: Centrifugal force points away from the axis of rotation and increases with distance. So "gravity" is the greatest at the equator and decreases to zero at the poles. Also, unless you are exactly on the equator, gravity doesn't point straight down relative to the floor. Walking from the equator to the pole feels like walking uphill. Still, this is probably your best bet.

Uniformly accelerating station: A rocket or something produces thrust in a constant direction (say from the south to north pole). The "gravity" points in the same direction and has the same magnitude everywhere. I mean the same direction in an absolute sense, not relative to the floor. So someone at the south pole feels like they're at the bottom of a bowl and someone at the north pole feels like they are hanging from the top of a dome. Not particularly pleasant but you could probably come up with some exciting sports.

Other: You could try a mixture of applied thrust and spinning but this is very likely to end up with an uncontrolled tumbling motion that is no good for anybody. And even if you did manage to stabilise it it would not produce a uniform (as in radially outward) gravity.

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