ph409_test2ANS - Physics 409/Classical Mechanics Name Test...

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Physics 409/Classical Mechanics Test #2 30 September 2010 Name Answer all parts of each of the two questions. 50 points total. The points value of each part is indicated. Begin the answer to each question on a new sheet of paper. Clearly define all coordinates and variables. Be sure to include sufficient detail in each answer so that your logic is clear to me. (1) ( 25 pts.) Find an expression in the form of a quadrature for the minimum time t 12 needed by a particle to travel between two points on the surface of a right circular cylinder of radius a , oriented with its symmetry axis (z-axis) perpendicular to the earth’s surface. The particle is subject to uniform gravitational acceleration g acting in the z direction. Solve this problem in the following steps. Label each step clearly so I can find it easily. (a)(5 pts.) Write a general expression for the differential arc length ds using cylindrical coordinates ( ρ , φ , z ) where φ is the azimuthal angle measured in the x-y plane from the x axis. ( Credit will be given only for solutions using cylindrical coordinates. ) ANS: ds = ( d ρ 2 + ρ 2 d φ 2 + dz 2 ) ½ (b)(5 pts.) Simplify your expression for ds by applying appropriate constraints to restrict the path to the surface of the cylinder and use this result to write an expression for the differential time needed to travel an infinitesimal distance. You may assume that the particle is initially at rest at z = 0 when it is released. ANS : ρ = a , so ds = ( a 2 d φ 2 + dz 2 ) ½ , dt = ds /v, where v = (2 gz ) ½ , where the positive z axis has been chosen to point downward.(For positive z upward, v=( ! 2 gz ) ½ .) (c)(5 pts.)
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ph409_test2ANS - Physics 409/Classical Mechanics Name Test...

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