P17_051 - equality of the frequencies for the two sections...

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51. (a) The frequency of the wave is the same for both sections of the wire. The wave speed and wavelength, however, are both diFerent in diFerent sections. Suppose there are n 1 loops in the aluminum section of the wire. Then, L 1 = n 1 λ 1 / 2= n 1 v 1 / 2 f ,where λ 1 is the wavelength and v 1 is the wave speed in that section. In this consideration, we have substituted λ 1 = v 1 /f ,where f is the frequency. Thus f = n 1 v 1 / 2 L 1 . A similar expression holds for the steel section: f = n 2 v 2 / 2 L 2 . Since the frequency is the same for the two sections, n 1 v 1 /L 1 = n 2 v 2 /L 2 . Now the wave speed in the aluminum section is given by v 1 = p τ/µ 1 ,where µ 1 is the linear mass density of the aluminum wire. The mass of aluminum in the wire is given by m 1 = ρ 1 AL 1 ,where ρ 1 is the mass density (mass per unit volume) for aluminum and A is the cross-sectional area of the wire. Thus µ 1 = ρ 1 AL 1 /L 1 = ρ 1 A and v 1 = p τ/ρ 1 A . A similar expression holds for the wave speed in the steel section: v 2 = p τ/ρ 2 A . We note that the cross-sectional area and the tension are the same for the two sections. The
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Unformatted text preview: equality of the frequencies for the two sections now leads to n 1 /L 1 1 = n 2 /L 2 2 , where A has been canceled from both sides. The ratio of the integers is n 2 n 1 = L 2 2 L 1 1 = (0 . 866 m) q 7 . 80 10 3 kg / m 3 (0 . 600 m) q 2 . 60 10 3 kg / m 3 = 2 . 5 . The smallest integers that have this ratio are n 1 = 2 and n 2 = 5. The frequency is f = n 1 v 1 / 2 L 1 = ( n 1 / 2 L 1 ) p / 1 A . The tension is provided by the hanging block and is = mg , where m is the mass of the block. Thus f = n 1 2 L 1 r mg 1 A = 2 2(0 . 600 m) s (10 . 0 kg)(9 . 8 m / s 2 ) (2 . 60 10 3 kg / m 3 )(1 . 00 10 6 m 2 ) = 324 Hz . (b) The standing wave pattern has two loops in the aluminum section and ve loops in the steel section, or seven loops in all. There are eight nodes, counting the end points....
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This note was uploaded on 11/12/2011 for the course PHYS 2001 taught by Professor Sprunger during the Fall '08 term at LSU.

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