Question:

When will the string become slack? When the gravitational force is …

Answer:

Suppose the ball arrives at the top of the circular motion with speed v.

If mv²/r is larger than mg, it means that gravity alone is not strong enough to force the ball into circular motion at radius r and speed v. As the ball tries to travel above the circular path, it stretches the string. The resulting tension in the string is such mg+T=mv²/r. This keeps the ball in circular motion.

If mv²/r is smaller than mg, it means that gravity by itself will already cause the ball to fall so fast that it travels below the circular path. There is nothing to stretch the string. The string becomes slack.

The condition for the string to be just slack (or just taut) is thus

mg=mv²/r

This happens when the ball is arriving at the top position with speed v such that mg provides the exact amount of centripetal force to keep the ball in circular motion at radius r and speed v. Anything faster and the string will be stretched. Anything slower and the ball will go into a parabolic fall.

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