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Unformatted text preview: Physics 315: Oscillations and Waves Homework 3: Solutions 1. According to Eqs. (3.41), (3.48), and (3.49) in the lecture notes, a damped driven harmonic oscillator varies as x ( t ) = x cos( ω t − ϕ ) , (1) where x = ω 2 X [( ω 2 − ω 2 ) 2 + ν 2 ω 2 ] 1 / 2 , (2) tan ϕ = ν ω ω 2 − ω 2 . (3) Using a standard trigonometric identity, cos( A − B ) = cos A cos B +sin A sin B , Eq. (1) can be rewritten x ( t ) = x cos ϕ cos( ω t ) + x sin ϕ sin( ω t ) . (4) However, cos ϕ = 1 / (1 + tan 2 ϕ ) 1 / 2 and sin ϕ = tan ϕ/ (1 + tan 2 ϕ ) 1 / 2 , so Eq. (3) yields cos ϕ = ω 2 − ω 2 [( ω 2 − ω 2 ) 2 + ν 2 ω 2 ] 1 / 2 , (5) sin ϕ = ν ω [( ω 2 − ω 2 ) 2 + ν 2 ω 2 ] 1 / 2 . (6) Equations (2), (4), (5), and (6) give x ( t ) = X bracketleftbigg ω 2 ( ω 2 − ω 2 ) ( ω 2 − ω 2 ) 2 + ν 2 ω 2 bracketrightbigg cos( ω t )+ X bracketleftbigg ω 2 ν ω ( ω 2 − ω 2 ) 2 + ν 2 ω 2 bracketrightbigg sin( ω t ) . (7) Finally, in the limit ω → ω , it is easily seen that ω 2 − ω 2 → 2 ω ( ω − ω ) and ν ω → ν ω . Hence, Eq. (7) reduces to x ( t ) = X bracketleftbigg 2 ω ( ω − ω ) 4 ( ω − ω ) 2 + ν 2 bracketrightbigg cos( ω t ) + X bracketleftbigg ν ω 4 ( ω − ω ) 2 + ν 2 bracketrightbigg sin( ω t ) . (8) We can write the above expression in the form x ( t ) = A cos( ω t ) + B sin( ω t ) , (9) where A = X bracketleftbigg 2 ω ( ω − ω ) 4 ( ω − ω ) 2 + ν 2 bracketrightbigg , (10) B = X bracketleftbigg ν ω 4 ( ω − ω ) 2 + ν 2 bracketrightbigg . (11) Hence, ( x 2 ) = ( [ A cos( ω t ) + B sin( ω t )] 2 ) = A 2 ( cos 2 ( ω t ) ) + 2 A B ( cos( ω t ) sin( ω t ) ) + B 2 ( sin 2 ( ω t ) ) , = 1 2 ( A 2 + B 2 ) . (12) Here, (···) denotes an average over an oscillation period, and we have made use of the standard result ( cos 2 ( ω t ) ) = ( sin 2 ( ω t ) ) = 1 / 2, as well as ( cos( ω t ) sin( ω t ) ) = 0. Thus, it follows from (10), (11), and (12) that ( x 2 ) = X 2 2 bracketleftbigg ω 2 4 ( ω − ω ) 2 + ν 2 bracketrightbigg . (13) According to Eq. (9), ˙ x ( t ) = − A ω sin( ω t ) + B ω cos( ω t ) . (14) Hence, ( ˙ x 2 ) = ( [ − ω A...
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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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