Final answer:
Centripetal force is crucial for maintaining a circular path and is directed towards the axis of rotation as per Newton's second law. Gravity often provides this force in astronomical contexts, as with planets orbiting the sun.
Step-by-step explanation:
To maintain a circular path, the minimum speed of a ball at position i is determined by centripetal force. This is because an object in uniform circular motion undergoes constant acceleration by changing direction. According to Newton's second law of motion, there must be a constant net external force acting on the object, which in this case is the centripetal force. This force is always directed towards the axis of rotation. When considering a circular orbit, such as a small mass m orbiting a large mass M, gravity can supply the necessary centripetal force to keep mass m in orbit.
Additionally, Newton's law of universal gravitation helps explain the paths of celestial objects, like a planet moving around the sun, which is also an application of centripetal force. The speed and radius of an object's circular path are at play when we consider the centripetal force required to maintain that path. If the radius increases while keeping the force constant, the velocity of the particle must also increase to maintain the same centripetal force.