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Why would'nt this work?
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What about the mass of that stick? Inertial doesn't care for your little silly games.
Neither do the two gravity wells the stick spans. And the earth and moon are moving relative to each other, someone would probably get their head knocked off by that stick. Before it eventually falls to the earth with quite a bit of force because earth's gravity well will win. Then it'll eventually settle into a giant teeter totter, assuming it is rigid enough to survive the impact.