My friend asked me if a 150-200 km wide island (more precisely, a lithospheric plate of such size) that rotates insanely fast by the geological terms, something around one rotation per week, is possible. I don't think it is possible via ordinary physics though, since the energy needed for that would likely just melt the whole surroundings and vaporize the seas. But can this exist with some handwaves or artificial solutions?
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Made out of rock, check.
200km wide, check.
Rotating once per week... umm... Would you settle for rotating every 2.3 days?
https://en.wikipedia.org/wiki/Pumice_raft
https://www.thesun.co.uk/tech/9784204/worlds-biggest-pumice-raft-under-water-volcano/
All you need is some floating rocks, and a natural gyre in the ocean.
This one is only about the size of Manhattan.
If you want even more solid rock, as in an actual tectonic plate spinning around, you are out of luck.
Tectonic plates as a rule do not engage in rotating motions, their primary movement is linear, slamming into each other or pulling apart or rubbing along each other.
All three of these activities tend to be accompanied by plenty of earthquakes, very often volcanoes, and very little movement. Even the most hasty of tectonic movement, such as the mid-Atlantic rift opening up, proceeds at a rate of less than 5 cm per year.
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$\begingroup$ Pumice rafts are essentially glorified floating dust. They also disperse rapidly, in days to weeks. I’m not sure that an “ephemeral stain that you can’t even stand on” satisfy OPs definitively of “island”. $\endgroup$– GimelistCommented Dec 28, 2020 at 21:06
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$\begingroup$ If rock isn't a requirement, I remember in my younger days sitting off a beautiful lake in Finland and watching the wind casually push a half-mile-square island across it. All the OP needs are trees with strong, inter-tangling roots and the build up of enough biomass to make the island a couple of meters thick (and buoyant). Suspension-of-disbelief would allow it to survive a hurricane, but maybe not a tsunami. But, I'm not sure the pumice would survive a tsunami. That might actually be a limiting factor to the believablity of the idea. $\endgroup$– JBHCommented Dec 29, 2020 at 0:33