The correct answer is obvious — to stabilize the Earth's rotation — but there are a few implementation details to work out.
Close to the plot of a recent Stephenson sci-fi novel:
I'm sure kibwen had in mind a Hollywood explosion, whereby a detonated object ceases to exist, rather than turning into many smaller objects.
just build a planetary motor.
encircle the earth with ferromagnetic coils, then fire asteroids (similarly banded) at the earth in a series of near misses the magnetic drag steals or imparts angular momentum depending on the direction of the asteroid relative to the earth.