Question

EXAMPLE 14.8 Harmonics of a Stretched Wire GOAL Calculate string harmonics, relate them to sound, and combine them with tensile stress. PROBLEM (a) Find the frequencies of the fundamental, second, and third harmonics of a steel wire 1.00 m long with a mass per unit length of 2.00 x 10-kg/m and under a tension of 80.0 N. (b) Find the wavelengths of the sound waves created by the vibrating wire for all three modes. Assume the speed of sound in air is 345 m/s. (c) Suppose the wire is carbon steel with a density of 7.80 x 10 kg/m, a cross-sectional area A - 2.56 x 10"m, and an elastic limit of 2.80 x 10 Pa. Find the fundamental frequency of the wire is tightened to the elastic limit. Neglect any stretching of the wire (which would slightly reduce the mass per unit length). STRATEGY (a) It's easiest to find the speed of waves on the wire then substitute Equation 14.15 to find the first harmonic. The next two are multiples of the first, given by Equation 14.17. (b) The frequencies of the sound waves are the same as the frequencies of the vibrating wire, but the wavelengths are different. Use f , where is the speed of sound in air, to find the wavelength in air. (c) Find the force corresponding to the elastic limit and substitute it into Equation 14.16 SOLUTION (A) Find the first three harmonies at the given tension Calculate the speed of the wave on the SON " - 2.00 * 10² m/s wire: Find the wire's fundamental frequency: 1,- -2.092.00 m/s. 1.00 x 102 Hz Find the next two harmonics by 12-2,- 2.00 102H2.- 3/, - 3.00 x 10 Hz multiplication: (8) Find the wavelength of the sound waves produced

Answer 1

19 = 1.19 m = 2120 X 103 kylm F elin + Is 36212.18 3 2.2 % v=191.88 mis fundamental frequency F, = 2 2 2x1119 F, = 80162 H2 second , frequency Fe = 2F, = 2*80.62 F2 = 141.24 HZ third frequency f = 3. = 3x20.62 f = 241.86 42 1b) v sound = 344 mls * = 4.267 m suund 344 f2 - 11-24 A2 = 2.133 m "sound 344 - 241.86 73 = 10422 m © ME 7188x103 kg/m3 A = 2144 x167 ruz P= 2.60X108 pa P= 5) F-PA= 2160x102 x 2:44 x16 7 F-63.44N Fundamental frequency corresponding to the elantic linit f = 1 L J f = txing 1 2 3 4 1 2 163 = 1.6 21:19 fie 71.3498 42