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Dear Frank, I wasn’t the person who first responded to your question, but I can answer this part for you. Ben was solving for the motion of the yoyo near the bottom where it turns around. Measuring the height from the lowest point with “z”, he determined that at the time t = -Pi/(2w) the height was z = +r , at the time t = 0 , the height was z = 0 , and at the time t = +Pi/ (2w) the height was again z = +r . The question now is to determine what "z" is as a general function of "t". Since z=0 when t=0, and it is symmetric for positive and negative t, you can assume it has a simple quadratic form, that is, z(t) = a t^2 . All you need now is the value for the constant “a”. Well, when t = Pi/(2w) we have z = a [Pi/(2w)]^2 = a Pi^2 / 4w^2 , but we also know that at this time z = r , so we have aPi^2 / 4w^2 = r , or a = 4w^2 r / Pi^2 , this leaves z(t) = (4w^2 r / Pi^2) t^2 or z(t) = 4w^2 r t^2 / Pi^2 ,which was the result Ben got. I hope that helps, Frank, and good luck with your project! Thank you for your interest. Sincerely, Jim Guinn Georgia Perimeter College

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