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If we treat the Earth as an isolated system then both its linear and angular momenta will remain constant. To answer your question you need only consider the angular momentum. The angular momentum is given by: $$ L = I\omega $$ Since $L$ is a constant, if the moment of inertia changes from $I_1$ to $I_2$ then we have: $$ I_1\omega_1 = I_2\omega_2 $$ and ...

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