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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( t ) , (1) where x = 2 X [( 2 2 ) 2 + 2 2 ] 1 / 2 , (2) tan = 2 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 cos( t ) + x sin sin( t ) . (4) However, cos = 1 / (1 + tan 2 ) 1 / 2 and sin = tan / (1 + tan 2 ) 1 / 2 , so Eq. (3) yields cos = 2 2 [( 2 2 ) 2 + 2 2 ] 1 / 2 , (5) sin = [( 2 2 ) 2 + 2 2 ] 1 / 2 . (6) Equations (2), (4), (5), and (6) give x ( t ) = X bracketleftbigg 2 ( 2 2 ) ( 2 2 ) 2 + 2 2 bracketrightbigg cos( t )+ X bracketleftbigg 2 ( 2 2 ) 2 + 2 2 bracketrightbigg sin( t ) . (7) Finally, in the limit , it is easily seen that 2 2 2 ( ) and . Hence, Eq. (7) reduces to x ( t ) = X bracketleftbigg 2 ( ) 4 ( ) 2 + 2 bracketrightbigg cos( t ) + X bracketleftbigg 4 ( ) 2 + 2 bracketrightbigg sin( t ) . (8) We can write the above expression in the form x ( t ) = A cos( t ) + B sin( t ) , (9) where A = X bracketleftbigg 2 ( ) 4 ( ) 2 + 2 bracketrightbigg , (10) B = X bracketleftbigg 4 ( ) 2 + 2 bracketrightbigg . (11) Hence, ( x 2 ) = ( [ A cos( t ) + B sin( t )] 2 ) = A 2 ( cos 2 ( t ) ) + 2 A B ( cos( t ) sin( t ) ) + B 2 ( sin 2 ( 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 ( t ) ) = ( sin 2 ( t ) ) = 1 / 2, as well as ( cos( t ) sin( t ) ) = 0. Thus, it follows from (10), (11), and (12) that ( x 2 ) = X 2 2 bracketleftbigg 2 4 ( ) 2 + 2 bracketrightbigg . (13) According to Eq. (9), x ( t ) = A sin( t ) + B cos( t ) . (14) Hence, ( x 2 ) = ( [ A...
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