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A spinning superconducting sphere generates a magnetic field (the London field) while excluding external magnetic fields from its interior (the Meissner effect), so that a torque on the sphere results if there are magnetic or superconducting materials nearby. We calculate the torque on such a sphere inside a concentric, infinitely long superconducting cylinder. We also calculate the torque on such a sphere inside a concentric, spherical superconducting shell in which there is a single circular hole.