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2soln2 - Physics 315 Oscillations and Waves Homework 2...

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Physics 315: Oscillations and Waves Homework 2: Solutions 1. The solution to the damped harmonic oscillator equation can be written x ( t ) = x 0 e ν t/ 2 cos( ω 1 t ) + parenleftbigg v 0 + ν x 0 / 2 ω 1 parenrightbigg e ν t/ 2 sin( ω 1 t ) , where ω 1 = radicalbig ω 2 0 ν 2 / 4. Here, it is assumed that ν 2 ω 0 . Furthermore, x 0 = x (0) and v 0 = ˙ x (0). Taking the limit ν 2 ω 0 , which is equivalent to taking the limit ω 1 0, we find that x ( t ) x 0 e ν t/ 2 + parenleftbigg v 0 + ν x 0 / 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 0 + [ v 0 + ( ν/ 2) x 0 ] 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 0 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 0 I 3 ( t ) dt slashbigg C.
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Hence, the common potential drop across all three legs is V ( t ) = L dI 1 dt = R I 2 = integraldisplay t 0 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 0 I 1 = ω 2 0 I 0 cos( ω t ) , (1) where ω 0 = 1 / L C and ν = 1 /R C . This is the driven damped harmonic oscillator equation. The resonant frequency is ω 0 = 1 L C , whereas the quality factor takes the form Q f = ω 0 ν = R radicalbig L/C . At the resonant frequency ( ω = ω 0 ), the first and third terms on the left- hand side of Eq. (1) cancel (since d 2 /dt 2 ≡ − ω 2 ), and we are left with ν dI 1 dt = ω 2 0 I 0 cos( ω 0 t ) .
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