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SilasBarta comments on Open Thread June 2010, Part 3 - Less Wrong
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Right: you know the laws of physics. You don't know your mass though, and you don't know any object that has a known mass. I posit this because, in the history of science, they made certain measurements that aren't possible in a two-planet universe, and to assume you can calibrate to those measurements would assume away the problem.
But still, in the rotating scenario the attractive force wouldn't be perpendicular to the planet's surface, and this can be established without knowing the gravitational constant. If the planet is spherical and you already know what is perpendicular, of course.