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Unformatted text preview: khounvivongsy (sk27799) – Practice Problems 3 Solutions – Weathers – (17104) 1 This printout should have 19 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Two harmonic waves are described by y 1 = A sin( k x ω t ) , y 2 = A sin( k x + ω t ) , where A = 3 . 3 mm, k = 7 . 5 rad / m and ω = 3600 rad / s. What is the amplitude y max ( x ) of the resul tant wave at x = 1 . 55 m? Correct answer: 5 . 33521 mm. Explanation: The superposition of two harmonic waves travelling in opposite directions y 1 ( x, t ) = A sin( k x ω t ) , (1) y 2 ( x, t ) = A sin( k x + ω t ) , (2) is the standing wave y ( x, t ) = y 1 ( x, t ) + y 2 ( x, t ) (3) = A bracketleftBig sin( kx ωt ) + sin( kx + ωt ) bracketrightBig = 2 A sin( kx ) cos( ωt ) (4) where the last equality follows from the trigonometric identity sin( α β )+sin( α + β ) = 2 sin( α ) cos( β ) . (5) At any particular point x , the standing wave (4) oscillates according to y ( t ) = A ( x ) cos( ω t ) (6) with x –dependent amplitude A ( x ) = 2 A sin( kx ) . (7) To be precise, since this A ( x ) function could be either positive or negative, the true ampli tude at x = 1 . 55 m is the magnitude y max ( x ) = A ( x )  = 2 A  sin( k x )  = 2 A vextendsingle vextendsingle vextendsingle sin bracketleftBig (7 . 5 rad / m) (1 . 55 m) bracketrightBigvextendsingle vextendsingle vextendsingle = 2 (3 . 3 mm)  sin(11 . 625 rad)  = 5 . 33521 mm . 002 10.0 points Two harmonic waves are described by y 1 = A sin(3 x 5 t ) , y 2 = A sin(3 x 5 t 4) . A = 16 m, with x and t being given in SI units and the phase angles in radians. What is the displacement of the sum of these two harmonic waves, y 1 + y 2 , at x = 1 m, t = 1 s? Correct answer: 10 . 0781 m. Explanation: First evaluate y 1 and y 2 at x = 1 m and t = 1 s. Then add the two results by superpo sition. y 1 = (16 m) sin(3 5) = 14 . 5488 m , y 2 = (16 m) sin(3 5 4) = 4 . 47065 m y 1 (1 m , 1 s) + y 2 (1 m , 1 s) = 10 . 0781 m . Alternative Solution: Using the basic trigonometric relation sin θ 1 + sin θ 2 = 2 sin θ 1 + θ 2 2 cos θ 1 θ 2 2 , we have the resultant wave y = y 1 + y 2 = 2 A sin(3 x 5 t 2) cos 2 . Therefore the displacement of the resultant wave at x = t = 1 is A ′ = 2 A sin(3 5 2) cos2 = 10 . 0781 m . 003 10.0 points Two identical harmonic waves with wave lengths of 1 . 8 m travel in the same direction khounvivongsy (sk27799) – Practice Problems 3 Solutions – Weathers – (17104) 2 at a speed of 3 . 9 m / s. The second wave origi nates from the same point as the first, but at a later time. Determine the minimum possible time in terval between the starting moments of the two waves if the amplitude of the resultant wave is the same as that of the two initial waves....
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This note was uploaded on 05/09/2010 for the course PHYS 1710 taught by Professor Weathers during the Spring '09 term at University of North Carolina Wilmington.
 Spring '09
 weathers
 mechanics

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