The ice at the North Pole floats. The ice displaces as much water, by weight, as the ice weighs. So the weight distribution change would be zero. No change in the Earth’s rotation.
The answer in the South Pole is not quite as simple, because the ice is partly on top of rock. (For the part that is not on top of rock, the answer is the same as for the North Pole: No change.)
But the rock complicates things. You might think that if the South Pole melts, all the weight of that water would shift to other parts of the world. But that doesn’t take in account that the mountains themselves are floating in a sea of Earth’s molten core. That is, in a sense the South Pole is very much like the North Pole: The weight is in an equilibrium, a sort of “sea level” for the Earth’s crust. If melted ice moved away, the melted rock underneath would rise up a similar amount. But that wouldn’t happen instantaneously, for awhile the Earth would be unusually heavy at the equator. That would slow the rotation of the Earth. But eventually equilibrium would be established, the South Pole would rise back, and the Earth’s rotation — which had been only slightly changed — would go back to what it was.
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