set5 - 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 Fifth Problem Set for Physics 664 (Theoretical Mechanics) Spring quarter 2004 Important dates: May 11 9:30am-10:18am midterm exam, May 31 no class, Jun 9 9:30am-11:18am final exam Due date: Thursday, May 6 12. Shortest path on a volcano 12 points You are hiking on a volcano given by the equation z = 1- q x 2 + y 2 . You are at point (- 1 , , 0) and want to get to the other side, namely to the point (1 , , 0), along the shortest possible path. x y z a) Express the infinitesimal line element d s = d x 2 + d y 2 + d z 2 in cylindrical coordinates ( r, , z ) defined by x = r cos and y = r sin . b) Write down the length of an arbitrary path (specified by r ( ), why?) on the volcano as an integral over . c) Find a differential equation for the path r ( ) of minimal length. d) Solve the differential equation for your hiking problem. [Hint: in order to solve the differential equation, it is helpful to reformulate it in terms of the variable...
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set5 - Department of Physics Prof. R. Bundschuh The Ohio...

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