Homework7 - choice oF ω In doing this problem I would...

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Physics 321 – Spring 2006 Homework #7, Due at beginning of class Wednesday Mar 15. 1. [8 pts] A “triangle wave” can be defned by F ( t ) = 1 - 2 ω | t | For - π/ω < t < + π/ω , with F ( t ) defned at all other values oF the time t by the property oF having period 2 π/ω . (a) ±ind the ±ourier series representation oF the F ( t ). Express your answer BOTH in exponential Form and in the Form oF sines and/or cosines. But do the exponential Form frst (as usual, it’s easier), and then get the sines+cosines Form From that. (b) Solve the driven damped oscillator equation ¨ x + 2 β ˙ x + x = F ( t ) in the Form oF an infnite series. (Note that For convenience, units oF time have been chosen such that the natural Frequency oF the oscillator is equal to 1.) You will probably fnd the exponential Form oF your answer to part (a) more convenient. (c) Plot the solution x ( t ) over a time interval oF two periods: 0 < t < 4 π/ω For the case β = 0 . 1, with ω = 1 / 3, 1, 2. (Make three separate plots—one For each
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Unformatted text preview: choice oF ω .) In doing this problem, I would preFer that you keep the entire infnite series in your answers. But it would be acceptable to keep only the Frequencies up through 5 ω . You can neglect the solutions to the homogeneous equation, which contain two arbitrary constants, because those “transient” e²ects go away like e-βt iF you wait long enough. 2. [4 pts] Marion & Thornton, problem 4-3 (same in 4th edition). Draw the phase space diagram ( ˙ x vs. x ) right below the potential energy plot, as is done in ±igure 4-5 in the book, so it is easy to see the correspondence between the two diagrams. 3. [4 pts] Marion & Thornton, problem 4-6 (same in 4th edition). Use conservation oF energy to calculate ˙ θ as a Function oF θ . 4. [4 pts] Marion & Thornton, problem 4-8 (same in 4th edition). (Last updated 3/02/2006.)...
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This note was uploaded on 07/25/2008 for the course PHY 321 taught by Professor B.pope during the Spring '08 term at Michigan State University.

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