No need to do the math -- the answer is the same as the question "What's the period of a low equatorial orbit?", because both are about how long it takes to circle the Earth at about ground level without experiencing any net gravity. About 90 minutes for equilibrium: any faster than that, and the orbital launch velocity at the equator goes negative.

As for what would happen, it would probably be bad even by "What If" standards. Atmosphere and oceans would both bunch up at the equator as the Earth spun up, before being flung off into a ring-like structure. But that's treating the bedrock as rigid, which it very much isn't. Eventually the continental plates would buckle and split, and molten or fractured rock would start to join the ring. That of course would reduce the gravitational pull of Earth sensu strictu, so eventually a new equilibrium would be found.

Peter