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PS6 - Prof Raghuveer Parthasarathy University of Oregon...

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Prof. Raghuveer Parthasarathy University of Oregon; Fall 2007 Physics 351 – Vibrations and Waves Problem Set 6 Due date: Friday, Nov. 16, 4pm. Reading: French Chapter 5. A note: This problem set is shorter than it seems. Problems 2-6 are very similar to one another, which should cement your understanding of coupled oscillators and should also be useful in analyzing the coupled oscillator systems that you build. Problem 1 is not related to coupled oscillators and could have been assigned weeks ago. (1, 6 pts. ) Phase space. It’s often informative to consider the trajectory of a system in “phase space,” in which particular physical quantities form the coordinate axes. One common set of quantities is position and velocity. For example : let’s imagine some system in which the position is given by () / x ta t = , and the velocity is given by 2 / vt x a t == ± , where t is time and a is some constant. Therefore 2 / va x =− – i.e. vx is a quadratic function of x . At time t =0, both v and x are zero, so our parabola “starts” at the origin of our phase space plot. The plot looks like: (a, 3 pts. ) Consider an undamped simple harmonic oscillator. Prove that the phase space plot (with axes x and x ± ) is an ellipse. (Remind yourself of the equation characterizing an ellipse.) (b, 2 pts. ) Draw the phase space plot for an undamped oscillator with the initial conditions 0 (0 ) t x x = = , ) 0 t v = = . Indicate the t =0 point on the plot. (b, 1 pt. ) What would the phase space plot of a damped oscillator look like? Draw it, qualitatively – you don’t have to do any math. Indicate the “t= ” point on the plot. (2 , 12 pts ) Two coupled masses. Consider two objects ( A and B ) of equal mass m connected to each other and to rigid walls by identical springs of spring constant k (see Figure). Neglect damping, and assume all motion is in a horizontal plane, so gravity is irrelevant.
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PS6 - Prof Raghuveer Parthasarathy University of Oregon...

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