3hw3 - Physics 315 Oscillations and Waves Homework 3 Due in class on Wednesday Sept 23rd 1 Show that for a damped driven harmonic oscillator which

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Unformatted text preview: Physics 315: Oscillations and Waves Homework 3: Due in class on Wednesday, Sept. 23rd 1. Show that for a damped driven harmonic oscillator which is driven close to its resonant frequency x ( t ) ≃ X bracketleftbigg 2 ω ( ω − ω ) 4 ( ω − ω ) 2 + ν 2 bracketrightbigg cos( ω t ) + X bracketleftbigg ν ω 4 ( ω − ω ) 2 + ν 2 bracketrightbigg sin( ω t ) . Hence, demonstrate that ( x 2 ) ≃ X 2 2 bracketleftbigg ω 2 4 ( ω − ω ) 2 + ν 2 bracketrightbigg , and ( ˙ x 2 ) ≃ X 2 2 bracketleftbigg ω 4 4 ( ω − ω ) 2 + ν 2 bracketrightbigg . Finally, show that ( E ) = 1 2 k X 2 bracketleftbigg ω 2 4 ( ω − ω ) 2 + ν 2 bracketrightbigg , where E is the sum of the kinetic and potential energies of the oscillating mass. 2. Consider the circuit shown below. Suppose that V ( t ) = ˆ V cos( ω t ). Demon- strate that I ( t ) = ˆ I sin( ω t ) and I 1 ( t ) = ˆ I 1 sin( ω t ), where ˆ I = parenleftBigg ˆ V ω L parenrightBigg parenleftbigg ω 2 − ω 2 2 ω 2 −...
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This note was uploaded on 11/02/2010 for the course PHY 59372 taught by Professor Fitzpatrick during the Spring '10 term at University of Texas at Austin.

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3hw3 - Physics 315 Oscillations and Waves Homework 3 Due in class on Wednesday Sept 23rd 1 Show that for a damped driven harmonic oscillator which

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