476_PartUniversity Physics Solution

476_PartUniversity Physics Solution - Mechanical Waves...

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Mechanical Waves 15-13 15.41. IDENTIFY: Compare ( , ) yx t given in the problem to Eq.(15.28). From the frequency and wavelength for the third harmonic find these values for the eighth harmonic. (a) SET UP: The third harmonic standing wave pattern is sketched in Figure 15.41. Figure 15.41 EXECUTE: (b) Eq. (15.28) gives the general equation for a standing wave on a string: SW (, ) ( s i n ) s i n A k x t ω = SW 2, AA = so SW /2 (5.60 cm)/2 2.80 cm == = (c) The sketch in part (a) shows that 3( /2). L λ = 2/, k πλ = 2/ k π = Comparison of ( , ) given in the problem to Eq. (15.28) gives 0.0340 rad/cm. k = So, 2 /(0.0340 rad/cm) 184.8 cm 3( / 2) 277 cm L (d) 185 cm, = from part (c) 50.0 rad/s = so 7.96 Hz f period 1/ 0.126 s Tf 1470 cm/s vf (e) SW /s i n c o s y vd y d t A k x t , max SW (50.0 rad/s)(5.60 cm) 280 cm/s y vA = (f) 31 7.96 Hz 3 , f f so 1 2.65 Hz f = is the fundamental 81 82 1 . 2 H z ; ff 88 2 133 rad/s f ωπ / (1470 cm/s)/(21.2 Hz) 69.3 cm = and 2 / 0.0906 rad/cm k = = ( , ) (5.60 cm)sin([0.0906 rad/cm] )sin([133 rad/s] ) x t = EVALUATE: The wavelength and frequency of the standing wave equals the wavelength and frequency of the two traveling waves that combine to form the standing wave. In the 8th harmonic the frequency and wave number are larger than in the 3rd harmonic. 15.42. IDENTIFY: Compare the ( , ) yxt specified in the problem to the general form of Eq.(15.28). SET UP: The comparison gives SW 4.44 mm A = , 32.5 rad/m k = and 754 rad/s = . EXECUTE: (a) 11 SW 22 (4.44 mm) 2.22 mm = . (b) 0.193 m. 32.5 rad m k = (c) 754 rad s 120 Hz f ππ = . (d) 754 rad s 23.2 m s. 32.5 rad m v k = (e) If the wave traveling in the x + direction is written as 1 (,) c o s ( ) , yx t A k =− then the wave traveling in the -direction x is 2 c o s ( ) t A k + , where 2.22 mm A = from part (a), 32.5 rad m k = and 754 rad s = . The harmonic cannot be determined because the length of the string is not specified. EVALUATE: The two traveling waves that produce the standing wave are identical except for their direction of propagation. 15.43. (a) IDENTIFY and SET UP: Use the angular frequency and wave number for the traveling waves in Eq.(15.28) for the standing wave. EXECUTE: The traveling wave is ( , ) (2.30 mm)cos([6.98 rad/m] ) [742 rad/s] ) x t = + 2.30 mm A = so SW 4.60 mm; A = 6.98 rad/m k = and 742 rad/s = The general equation for a standing wave is SW ( , ) ( sin )sin , A k x t = so ( , ) (4.60 mm)sin([6.98 rad/m] )sin([742 rad/s] ) x t = (b) IDENTIFY and SET UP: Compare the wavelength to the length of the rope in order to identify the harmonic. EXECUTE: 1.35 m L = (from Exercise 15.24) 2/ 0 . 9 0 0 m k 3( / 2), L = so this is the 3rd harmonic
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15-14 Chapter 15 (c) For this 3rd harmonic, /2 118 Hz f ω π == 31 3 f f = so 1 (118 Hz)/3 39.3 Hz f EVALUATE: The wavelength and frequency of the standing wave equals the wavelength and frequency of the two traveling waves that combine to form the standing wave. The n th harmonic has n node-to-node segments and the node-to-node distance is /2, λ so the relation between L and for the n th harmonic is ( Ln =/ 2 ).
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This note was uploaded on 07/16/2011 for the course PHY 2053 taught by Professor Buchler during the Spring '06 term at University of Florida.

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476_PartUniversity Physics Solution - Mechanical Waves...

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