soln3(1) - Physics 315: Oscillations and Waves Homework 3:...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 7

soln3(1) - Physics 315: Oscillations and Waves Homework 3:...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online