In the scenario you described, where a ball is tied to a rope and spun in a constant circular motion, the ball experiences a centripetal force directed toward the center of the circular path. This force is provided by the tension in the rope, which constantly pulls the ball inward, allowing it to maintain its circular motion.
When the ball is spinning fast enough, the tension in the rope can provide the necessary centripetal force to counteract the force of gravity, and the ball remains in the circular path without falling. In this case, the gravitational force and the tension in the rope are balanced.
However, if the ball's velocity magnitude falls below a certain point, the tension in the rope may no longer be sufficient to counteract the force of gravity. When this happens, the ball cannot maintain its circular path, and it will start to fall downward due to the unbalanced force of gravity.
The minimum velocity required for the ball to maintain the circular motion is determined by the balance between the centripetal force provided by the tension in the rope and the force of gravity pulling the ball downward. If the tension decreases or the velocity decreases below a certain threshold, the gravitational force will become dominant, and the ball will no longer remain in the circular path.
This behavior is similar to the concept of "banking" in curved roads or tracks. The banking angle of the curve is designed such that the gravitational force and the normal force between the vehicle and the road provide the necessary centripetal force for the vehicle to safely negotiate the curve without skidding or sliding off the track.
So, in summary, when the ball's velocity magnitude is below a certain point, the gravitational force becomes dominant because the tension in the rope is not sufficient to counteract it, and the ball will start to fall.