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dyn10 - 2.032 DYNAMICS Problem Set No 10 Out Wednesday Due...

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2.032 DYNAMICS Fall 2004 Problem Set No. 10 Out : Wednesday, November 24, 2004 Due : Wednesday, December 1, 2004 at the beginning of class Problem 1 The system below consists of a massless hollow cylindrical tube, joined to a vertical shaft at the point O. The tube is fixed in θ -direction and θ = π/ 4. Inside the tube, moves without friction a mass m which is connected to O through a spring of stiffness k and neutral length of r 0 . Assume that the shaft is rotating with angular velocity ˙ φ about its axis. Using r , φ as generalized coordinates: (a) Reduce the problem to a one-degree-of-freedom problem for r , that has only potential active forces. (b) Find the equilibria for the reduced system and investigate the stability using Dirichlet theorem. (c) Sketch the trajectories on the ( r r ) phase plane. Select all parameters to be equal to one, including gravity g. 1 Courtesy of Prof. T. Akylas. Used with permission.
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Problem 2 (adapted from PhD Qualifying Exam 2003) Reconsider Problem 2 of PS No. 5. Using the same notation, (i) Derive the differential equation describing the motion of the bead on the ring.
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