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Chapter 7: Circular Motion and Rotation Test Your Understanding

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In the AP Physics I Essentials book, I'm a little bit stuck on question 5 for Chapter 7 Test Your Understanding. The problem states "A marble is rolled separately down two different inclines of the same height as shown below. Compare the speed of the marble at the bottom of incline A to the speed of the marble at the bottom of incline B. Compare the time it takes the marble to reach the bottom of incline A to the time it takes to reach the bottom of incline B." There is also a picture with a ball on top of an incline that is straight, and another picture with a ball on top of an incline that is curved.

I can explain why incline B would result in greater speed resulting in faster time due to the cycloid path/Brachistochrone Curve since the marble is aided by gravity and the beginning is steeper so more PE to KE for incline B etc. etc., but I can't explain it mathematically (w/equations and all). I tried using Work-Energy theorem, but that didn't really work...Also, I'm confused on how we connect the concepts of angular motion to this problem.

I appreciate the help!

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Conservation of energy is the key here... but much also depends on what assumptions you make (which makes this a GREAT class discussion question).  If you assume frictionless, the balls won't roll, and they'll both have the same kinetic energy at the bottom, but B will get their quicker.  If you assume there is friction, you can start go get into a great discussion about rolling with or without slipping, which one will roll vs. slide first (or both), and the problem becomes considerably more complicated but is fantastic for reviewing some of the key concepts.

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