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3hw3 - Physics 315 Oscillations and Waves Homework 3 Due in...

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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 0 bracketleftbigg 2 ω 0 ( ω 0 ω ) 4 ( ω 0 ω ) 2 + ν 2 bracketrightbigg cos( ω t ) + X 0 bracketleftbigg ν ω 0 4 ( ω 0 ω ) 2 + ν 2 bracketrightbigg sin( ω t ) . Hence, demonstrate that ( x 2 ) ≃ X 2 0 2 bracketleftbigg ω 2 0 4 ( ω 0 ω ) 2 + ν 2 bracketrightbigg , and ( ˙ x 2 ) ≃ X 2 0 2 bracketleftbigg ω 4 0 4 ( ω 0 ω ) 2 + ν 2 bracketrightbigg . Finally, show that ( E ) = 1 2 k X 2 0 bracketleftbigg ω 2 0 4 ( ω 0 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 0 ( t ) = ˆ I 0 sin( ω t ) and I 1 ( t ) = ˆ I 1 sin( ω t ), where ˆ I 0 = parenleftBigg ˆ V ω L parenrightBigg parenleftbigg ω 2 0 ω 2 2 ω 2 0 ω 2 parenrightbigg , ˆ I 1 = parenleftBigg ˆ V ω L parenrightBigg parenleftbigg ω 2 0 2 ω 2 0 ω 2
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