P214soln2

P214soln2 - Prepared by Girma 1 Physics 214 Spring 2006...

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Unformatted text preview: Prepared by Girma 1 Physics 214 Spring 2006 Solution to Homework ] 2 Problem 1. ( a ). First we compute the real part and simplify x ( t ). Recall that a general complex number a + ib , where a and b are real, can be written as a + ib = √ a 2 + b 2 e i tan- 1 ( b/a ) . Thus x ( t ) = < [(5 + 7 i ) e iwt ] mm = < [ p 5 2 + 7 2 e i tan- 1 (7 / 5) e iwt ] mm = p 5 2 + 7 2 cos( wt + tan- 1 (7 / 5)) mm (1) v ( t ) = dx ( t ) dt =- p 5 2 + 7 2 w sin( wt + tan- 1 (7 / 5)) mm (2) x (10- 4 s ) = p 5 2 + 7 2 cos(10 4 rad/s 10- 4 s + tan- 1 (7 / 5)) mm =- 3 . 2 mm (3) v (10- 4 s ) =- p 5 2 + 7 2 10 4 s- 1 sin(10 4 rad/s 10- 4 s + tan- 1 (7 / 5)) mm =- 8 . × 10 4 mm/s (4) ( b ). We read off from Equation (1) that A = √ 5 2 + 7 2 mm = 8 . 6 mm ( c ). See Figure (1).-4-2 2 4 6 Re LBracket1 x Smile RBracket1 LParen1 mm RParen1 7.25 7.5 7.75 8 8.25 8.5 8.75 9 Im LBracket1 x Smile RBracket1 LParen1 mm RParen1 t Equal 5 Plus 7i t Equal 10 Minus 4 s Minus 3.2 Plus 8.0 Imagi Figure 1: This plot shows the complex position ˘ x ( t ) = (5+7 i ) e iwt for ω = 10 4 rad/s and ≤ t ≤ 10- 4 s . The arrow on the right points to the position at t = 0 and the arrow on the left points to the position at t = 10- 4 s . Problem 2. Using the expression for the complex amplitude which was derived in the lecture and recalling again that a complex number a + ib can be written as a + ib = √ a 2 + b 2 e i tan- 1 ( b/a ) , where now a = f m ( ω 2 o- ω 2 ) / (( ω 2 o- ω 2 ) 2 + γω ) and b =- f m γω/ (( ω 2 o-...
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P214soln2 - Prepared by Girma 1 Physics 214 Spring 2006...

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