A large yo-yo stands on a table. A rope wrapped around the yo-yo's axle is pulled horizontally to the right, with the rope coming off the yo-yo above the axle. In which direction does the yo-yo move?
To the right.
The situation is repeated but with the rope coming off the yo-yo below the axle. If the rope is pulled to the right, which way will the yo-yo move now?
To the right.
If the yo-yo's axle is half the radius of the yo-yo, and the yo-yo moves a distance L to the right when the rope is pulled from above the axle, how far does the end of the rope move?
One way to answer this is to determine the translational and rotational components separately, and then combine the results.
If the wheel was translating only, the rope would not wind or un-wind and moving the yo-yo a certain distance would require the end of the rope to move the same distance.
If the wheel was rotating only, the distance moved by the end of the rope equals the distance moved by a point on the outside of the axle.
For rolling without slipping, the rotational speed of a point on the outer edge of the yo-yo equals vt, the translational speed of the center of the yo-yo. If the axle is half the yo-yo's radius, a point on the outer edge of the axle has a rotational speed equal to half the yo-yo's translational speed. Above the axle, where the rope is unwinding, the net velocity is 1.5 vt, because the rotational and translational velocities have the same direction.
If the yo-yo moves a distance L, the end of the rope moves a distance 1.5 L.