set4 - Department of Physics Prof. R. Bundschuh The Ohio...

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Unformatted text preview: Department of Physics Prof. R. Bundschuh The Ohio State University Fourth Problem Set for Physics 664 (Theoretical Mechanics) Spring quarter 2004 Important dates: Apr 27 9:30am-10:18am & May 11 9:30am-10:18am midterm exams, May 31 no class, Jun 9 9:30am-11:18am final exam Due date: Thursday, Apr 29 10. Motion on the cycloid 14 points In class we found that the fastest motion from a point ( x 1 , y 1 ) to a point ( x 2 , y 2 ) in a gravi- tational potential is along the cycloid parameterized by x = x 1 + a (1- cos ) and y = y 1 + a ( - sin ) where the x-axis points downward and the parameter a has to be adjusted such that the curve goes through the point ( x 2 , y 2 ). This result was derived by minimizing the functional T [ y ( x )] x 2 Z x 1 1 + y 2 2 g ( x- x 1 ) ! 1 2 d x which measures the total time along the curve y ( x ) from x 1 to x 2 . In this problem we want to calculate how long it takes to get from any intermediate point ( x 3 , y 3 ) on the cycloid to the minimum of the cycloid. For simplicity we choose our coordinates such thatthe minimum of the cycloid....
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This note was uploaded on 07/17/2008 for the course PHYS 664 taught by Professor Bundschuh during the Fall '04 term at Ohio State.

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set4 - Department of Physics Prof. R. Bundschuh The Ohio...

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