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An amusing trick is to press a finger down on a marble, on a horizontal table top, in such a way that the marble is projected along the table with an initial linear speed V0 and an initial backward rotational speed ω0, ω0 being about a horizontal axis perpendicular to V0. The coefficient of sliding friction between marble and top is constant. The marble has radius R.

a) What relationship must hold between V0, R, and ω0 for the marble to slide to a complete stop?

b) What relationship must hold between V0, R, and ω0 for the marble to skid to a stop and then start returning toward its initial position, with a final constant linear speed of 3/7 V0?