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3soln3 - Physics 315 Oscillations and Waves Homework 3...

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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 0 cos( ω t ϕ ) , (1) where x 0 = ω 2 0 X 0 [( ω 2 0 ω 2 ) 2 + ν 2 ω 2 ] 1 / 2 , (2) tan ϕ = ν ω ω 2 0 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 0 cos ϕ cos( ω t ) + x 0 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 0 ω 2 [( ω 2 0 ω 2 ) 2 + ν 2 ω 2 ] 1 / 2 , (5) sin ϕ = ν ω [( ω 2 0 ω 2 ) 2 + ν 2 ω 2 ] 1 / 2 . (6) Equations (2), (4), (5), and (6) give x ( t ) = X 0 bracketleftbigg ω 2 0 ( ω 2 0 ω 2 ) ( ω 2 0 ω 2 ) 2 + ν 2 ω 2 bracketrightbigg cos( ω t )+ X 0 bracketleftbigg ω 2 0 ν ω ( ω 2 0 ω 2 ) 2 + ν 2 ω 2 bracketrightbigg sin( ω t ) . (7) Finally, in the limit ω ω 0 , it is easily seen that ω 2 0 ω 2 2 ω 0 ( ω 0 ω ) and ν ω ν ω 0 . Hence, Eq. (7) reduces to x ( t ) = X 0 bracketleftbigg 2 ω 0 ( ω 0 ω ) 4 ( ω 0 ω ) 2 + ν 2 bracketrightbigg cos( ω t ) + X 0 bracketleftbigg ν ω 0 4 ( ω 0 ω ) 2 + ν 2 bracketrightbigg sin( ω t ) . (8)
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We can write the above expression in the form x ( t ) = A cos( ω t ) + B sin( ω t ) , (9) where A = X 0 bracketleftbigg 2 ω 0 ( ω 0 ω ) 4 ( ω 0 ω ) 2 + ν 2 bracketrightbigg , (10) B = X 0 bracketleftbigg ν ω 0 4 ( ω 0 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 0 2 bracketleftbigg ω 2 0 4 ( ω 0 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 sin( ω t ) + B ω cos( ω t )] 2 ) = A 2 ω 2 ( sin 2 ( ω t ) ) − 2 A B ω 2 ( cos( ω t ) sin( ω t ) ) + B 2 ω 2 ( cos 2 ( ω t ) ) , = ω 2 2 ( A 2 + B 2 ) . (15)
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Thus, it follows from (10), (11), and (15) that ( ˙ x 2 ) = X 2 0 2 bracketleftbigg ω 4 0 4 ( ω 0 ω ) 2 + ν 2 bracketrightbigg . (16) Here, we have made use of the fact that ω ω 0 close to the resonance.
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