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Assignment_1_09_revised

# Assignment_1_09_revised - TA's Name Section Your Name...

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TA's Name:____________________ Section: ____ Your Name: _________________________________ Physics 214 Assignment 1 Concepts: complex numbers force law for SHM complex exponentials damped oscillations oscillations driven oscillations and resonance restoring forces decay time and resonance width simple harmonic motion oscillator equations Reading: AG Lecture Notes on Oscillations (from the website); Y&F, Vol. 1, Chapter 13 Assignment: Due in lecture on Tuesday, January 27. Please turn in this sheet stapled to the top of your work. A. Math Warm-Up Problems 1. (a) What are the real and imaginary parts of the function z(t) = r e i( ω t + φ) ? (b) In (a) you assumed that ω is real. But suppose that it has an imaginary component as well, i.e., ω = ω 0 + i ω 1 . What is the real part of z(t) in this case? (c) From your answer to (a), we can represent a real sinusoidal oscillation using the real part of a complex exponential, x(t) = Re [z]. Evaluate (i) dz/dt; (iii) dx/dt = d/dt { Re[z] }; and (ii) Re {dz/dt}. (d) Consider a second order differential equation of the form a d 2 x/dt 2 + b dx/dt + cx = 0 where a, b and c are constants. Show that x(t) = Re { [z(t)] }, where z(t) is as in (a), is a solution to this equation for a particular value of ω . Determine that ω . What are its real and imaginary parts? Hint: From your answer to part (c), we see that the operations of taking the time derivative and taking the real part of a complex exponential function of time commute.

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Assignment_1_09_revised - TA's Name Section Your Name...

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