12.91. Model: Assume that the marble does not slip as it rolls down the track and around a loop-the-loop. The mechanical energy of the marble is conserved. Visualize:Solve:The ball’s center of mass moves in a circle of radius .Rr−The free-body diagram on the marble at its highest position shows that Newton’s second law for the marble is 21mvmgnRr+=−The minimum height (h) that the track must have for the marble to make it around the loop-the-loop occurs when the normal force of the track on the marble tends to zero. Then the weight will provide the centripetal acceleration needed for the circular motion. For
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