Physics 325 Homework 6 - m = 10kg, k = 250 N/m, and c = 60...

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Physics 325, Fall 2010 Homework Assignment #6 Due Thursday 7 October at 11:15 am Note that this homework is the last you will be assigned before the Frst Midterm Exam. I wish to post the solutions in time for them to be of use to you in your review. The solutions will be posted in the afternoon on ±riday 8 October and, consequently, no late homework solutions will be accepted after noon on that ±riday 1) (10 points) For an underdamped harmonic oscillator with β ω , show that the driving frequency for which the steady-state amplitude is one-half of the steady-state amplitude at the resonance frequency is given by ω f ω ± β 3. What is the phase shift δ between the driving force and the resulting oscillator’s position? 2) (15 points) Solve the di±erential equation of motion of the damped harmonic oscillator driven by an exponentially decreasing harmonic force F ext ( t ) = F 0 e - αt cos ω f t Hint: e - αt cos ω f t = Re( e - αt + f t ) = Re( e γt ), where γ = α + f . Assume a solution of the form Ae γt - . 3) (15 points) A damped harmonic oscillator with
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Unformatted text preview: m = 10kg, k = 250 N/m, and c = 60 kg/s is subject to a driving force given by F cosωt , where F = 48 N. a) What value of ω results in steady-state oscillations with maximum amplitude? For the conditions determined in part a), b) What is the maximum amplitude? c) What is the phase shift? 4) (15 points) Taylor 5.44 Another interpretation of the Q of a resonance comes from the following: Consider the motion of a driven damped oscillator after any transients have died out, and suppose that it is being driven close to resonance, so you can set ω f = ω , a) Show that the oscillator’s total energy (kinetic plus potential) is E = 1 2 mω 2 f A 2 . b) Show that the energy Δ E dis dissipated during one cycle by the damping force F dmp is 2 πmβω f A 2 . (Remember that the rate at which a force does work is Fν .) c) Hence show that Q is 2 π times the ratio E/ Δ E dis . 55 Points Total...
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This note was uploaded on 10/06/2011 for the course PHYS 325 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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