p605mx_f07

p605mx_f07 - Phys 605. Midterm Exam October 15, 2007...

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Unformatted text preview: Phys 605. Midterm Exam October 15, 2007 Problem 1: [45 pts.]A cylinder of radius R , mass M 1 , and moment of inertia (about its central axis) I = 1 2 M 1 R 2 is rolling without slipping on an incline with angle with respect to the horizontal. The incline itself has mass M 2 and is free to slide without friction on a horizontal surface. Use s and x 2 as generalized coordinates as shown in the figure below. 2 x 2 s n o frictio n R M M 1 (a) [10 pts] Clearly show that the Lagrangian is given by: L = 1 2 ( M 1 + M 2 ) x 2 2 + 3 4 M 1 s 2 + M 1 (cos ) s x 2- M 1 g (sin ) s (b) [8 pts] Find any constants of motion and indicate the physical quantity each constant represents. (c) [8 pts] Find the differential equations of motion for this system. (You do not need to solve these equations!) (d) [7 pts] Now suppose a new constraint is added that forces x 2 to vary with time as: x 2 ( t ) = A sin( t ), where A and are given constants. Find the new Lagrangian....
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