Q. A 5 kg ball is swung around on the end of a rope that is 1 m long. It travels a circular path with a vertical orientation. If the breaking tension of the rope is 150 N, what is the maximum speed with which the ball can travel the circle?

Well, I once gave this question as a quiz in a discussion section for a calculus based physics course (mostly engineer wannabes). The average score was 4.5 out of 10.

Let's see if we can do better.



The truth about centripetal force:
This stuff about centripetal force is a bunch of huey. When I teach the subject, I don't even bring it up. It's not necessary. The truth is, there is no such thing as the centripetal force. The term is used to describe the sum of all of the real forces which act in or out of the circle. You can still use Newton's Second Law. Add up all of the real forces and set them equal to mac . The only thing that's different is the fact that there's a special equation for circular acceleration, ac = v2/r. Other than that, there is no such thing as a centripetal force. Keep it simple. One more thing - sign convention. Always have whatever points to the center of the circle be positive, whatever points out of the circle be negative.
I gave a couple of points if the student wrote down the equation for Newton's Second Law.

At this point, most students realized that a free-body-diagram (FBD) was now called for. The funny thing is, they drew the FBD on the ball in the position that's shown in the diagram that I gave. This certainly is confusion.

But is this where the ball is most likely to be when the rope breaks? Think about what you've seen in your personal experience. When on a swing, what's the tension like when you are at the highest point? If you had a hulking body pushing you, you would have experienced a nauseating jerking sensation at the top as the tension in the chains goes to zero.

Look at the FBD's at the two extremes:
At the top of the circle

At the bottom of the circle


Did you notice how, in both situations, T has a positive sign, while mg switches sign? Isn't up always positive & down negative? Not so with circular motion. The sign convention is: pointing towards the center is positive, pointing away from the center is negative.

From the equations above, you can see that the tension in the rope is greatest at the bottom of the swing. It's at that location that we should apply the information we have. We know that the rope breaks when its tension is 150 N. What speed would cause this?


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