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Unformatted text preview: Phys 605. Homework 4 Due 5pm, Monday, October 6, 2008 Problem 41: [10 pts.] A general surface of revolution may be described in cylindrical polar coordinates ( r , φ , z ) by the function r = r ( z ). The function r ( z ) and its derivative dr/dz ≡ r ( z ) are given. Use variational calculus to find the equation for the curve that is the shortest path between two points on this surface. (a) Show clearly that the expression for the differential path length on the surface that is the result of displacement dz and dφ is given by ( ds ) 2 = (1 + r 2 )( dz ) 2 + r 2 ( dφ ) 2 . (b) Take z as the independent variable and show that the curve, φ ( z ), that is the shortest path between two points on the surface is given by φ ( z ) = φ + k Z z z p r 2 ( z ) + 1 r ( z ) p r 2 ( z ) k 2 dz . (c) Thus, show that the shortest path between two points on the surface of a cylinder of radius a is a section of a helix....
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This note was uploaded on 02/09/2012 for the course PHYS 605 taught by Professor Staff during the Fall '09 term at Ohio University Athens.
 Fall '09
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