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Unformatted text preview: Physics 315: Oscillations and Waves Homework 2: Solutions 1. The solution to the damped harmonic oscillator equation can be written x ( t ) = x e − ν t/ 2 cos( ω 1 t ) + parenleftbigg v + ν x / 2 ω 1 parenrightbigg e − ν t/ 2 sin( ω 1 t ) , where ω 1 = radicalbig ω 2 − ν 2 / 4. Here, it is assumed that ν ≤ 2 ω . Furthermore, x = x (0) and v = ˙ x (0). Taking the limit ν → 2 ω , which is equivalent to taking the limit ω 1 → 0, we find that x ( t ) → x e − ν t/ 2 + parenleftbigg v + ν x / 2 ω 1 parenrightbigg e − ν t/ 2 ω 1 t, since cos θ ≃ 1 and sin θ ≃ θ when  θ  ≪ 1. Hence, x ( t ) → ( x + [ v + ( ν/ 2) x ] t ) e − ν t/ 2 . 2. Let I 1 ( t ), I 2 ( t ), and I 3 ( t ) be the currents flowing in the left, middle, and right legs of the circuit, respectively. If I ( t ) = I cos( ω t ) is the current fed into the circuit then conservation of current requires that I ( t ) = I 1 ( t ) + I 2 ( t ) + I 3 ( t ) . Since the three legs of the circuit are connected in parallel, the potential drops across them are the same. The potential drop across the left leg is L dI 1 dt , whereas the potential drop across the middle leg is R I 2 , and the potential drop across the right leg is integraldisplay t I 3 ( t ′ ) dt ′ slashbigg C. Hence, the common potential drop across all three legs is V ( t ) = L dI 1 dt = R I 2 = integraldisplay t I 3 ( t ′ ) dt ′ slashbigg C. It follows that I 2 = L R dI 1 dt , and L C d 2 I 1 dt 2 = I 3 = I − I 1 − I 2 = I − I 1 − L R dI 1 dt . Thus, d 2 I 1 dt 2 + ν dI 1 dt + ω 2 I 1 = ω 2 I cos( ω t ) , (1) where ω = 1 / √ L C and ν = 1 /R C . This is the driven damped harmonic oscillator equation. The resonant frequency is ω = 1 √ L C , whereas the quality factor takes the form Q f = ω ν = R radicalbig L/C ....
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This note was uploaded on 10/04/2011 for the course PHY 315 taught by Professor Staff during the Spring '08 term at University of Texas.
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