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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 t/ 2 cos( 1 t ) + parenleftbigg v + x / 2 1 parenrightbigg e t/ 2 sin( 1 t ) , where 1 = radicalbig 2 2 / 4. Here, it is assumed that 2 . Furthermore, x = x (0) and v = x (0). Taking the limit 2 , which is equivalent to taking the limit 1 0, we find that x ( t ) x e t/ 2 + parenleftbigg v + x / 2 1 parenrightbigg e t/ 2 1 t, since cos 1 and sin when | | 1. Hence, x ( t ) ( x + [ v + ( / 2) x ] t ) e 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( 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 + dI 1 dt + 2 I 1 = 2 I cos( t ) , (1) where = 1 / L C and = 1 /R C . This is the driven damped harmonic oscillator equation. The resonant frequency is = 1 L C , whereas the quality factor takes the form Q f = = R radicalbig L/C ....
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- Spring '08