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374bhw13 - HW#13 Phys374Spring 2007 Due before class...

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HW#13 —Phys374—Spring 2007 Prof. Ted Jacobson Due before class, Wednesday, May 9, 2007 Room 4115, (301)405-6020 www.physics.umd.edu/grt/taj/374b/ [email protected] 1. Consider a mass m suspended by a spring with spring constant k from another mass m which is suspended from a fixed support by another spring with spring constant k . Let y 1 and y 2 denote the displacements from their equilibrium positions of the top and bottom masses respectively, with the downward direction taken as positive. (a) Write out Newton’s second law governing the displacement of each of the two masses from their equilibrium positions. (b) Combine the two Newton’s law equations into a single 2 × 2 matrix equation ¨ y = A y , and specify the components of the matrix A in units with k = m = 1. (c) Determine the normal mode amplitudes and frequencies by finding the eigen- vectors and eigenvalues of A by hand . ( Hint : To check yourself, the squared frequencies are (3 ± 5) / 2.) (d) Describe or indicate with arrows the nature of the two normal mode motions, showing both direction and relative amplitude of the motion of each mass. Indi- cate which normal mode has the higher frequency. Using the concept of effective
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