The smaller the velocity of the object, the less centripetal force you will have to apply.
The smaller the length of rope (radius), the more centripetal force you will have to apply to the rope.
Notice that the centripetal force and the centripetal acceleration are always pointing in the same direction.
If you let go of the rope (or the rope breaks) the object will no longer be kept in that circular path and it will be free to fly off on a tangent.