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Ideally the counterweight would be as heavy as possible, so the living quarters are as far away from the rotation point as possible.
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With a small living quarters, the barycenter could be within the bigger object.
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Originally Posted by Wikipedia
[Search domain en.wikipedia.org/wiki/Earth-Moon_barycenter] https://en.wikipedia.org/wiki/Earth-Moon_barycenter
This is the case for the Earth-Moon system, in which the barycenter is located on average 4,671 km (2,902 mi) from the Earth's center, well within Earth's radius of 6,378 km (3,963 mi).
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Minimum of six cables in an hyperbolic paraboloid for a rigid structure under tension. Plus a Hyperloop tube for transiting?
Good point about the precession of it's orientation. Perhaps it would actually be a thick barrel shape with open ends and the unshielded parts in the inside.