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Use the worked example above to help you solve this problem. Figure a shows a simple apparatus for demonstrating resonance in a tube. A long tube open at both ends is partially submerged in a beaker of water, and a vibrating tuning fork of unknown frequency is placed near the top of the tube. The length of the air column, L, is adjusted by moving the tube vertically. The sound waves generated by the fork are reinforced when the length of the air column corresponds to one of the resonant frequencies of the tube. Suppose the smallest value of L for which a peak occurs in the sound intensity is 8.83 cm. Take the speed of sound to be 341 m/s. (a) With this measurement, determine the frequency of the tuning fork. Hz

Use the worked example above to help you solve this problem. Figure a shows a simple apparatus for demonstrating resonance in a tube. A long tube open at both ends is partially submerged in a beaker of water, and a vibrating tuning fork of unknown frequency is placed near the top of the tube. The length of the air column, L, is adjusted by moving the tube vertically. The sound waves generated by the fork are reinforced when the length of the air column corresponds to one of the resonant frequencies of the tube. Suppose the smallest value of L for which a peak occurs in the sound intensity is 8.83 cm. Take the speed of sound to be 341 m/s. (a) With this measurement, determine the frequency of the tuning fork. Hz

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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PROBLEM Figure a shows a simple apparatus for demonstrating resonance in a tube. A long tube open at both ends is partially submerged in a beaker of water, and a vibrating tuning fork of unknown frequency is placed near the top of the tube. The length of the air column, L, is adjusted by moving the tube vertically. The sound waves generated by the fork are reinforced when the length of the air column corresponds to one of the resonant frequencies of the tube. Suppose the smallest value of L for which a peak occurs in the sound intensity is 9.00 cm. (a) With this measurement, determine the frequency of the tuning fork. (b) Find the wavelength and the next two air-column lengths giving resonance. Take the speed of sound to be 345 m/s. SOLUTION (A) Find the frequency of the tuning fork. = 1, v = 345 m/s, and Substitute n = L₁ = 9.00 x 10-² m. STRATEGY Once the tube is in the water, the setup is the same as a pipe closed at one end. For part (a), substitute values for v and L into the one ended harmonics equation with n= 1 and find the frequency of the tuning fork. (b) The next resonance maximum occurs when the water level is low enough to allow a second node, which is another half-wavelength in distance. The third resonance occurs when the third node is reached, requiring yet another half-wavelength of distance. The frequency in each case is the same because it's generated by the tuning fork. Add a half-wavelength of distance to L₁ to get the next resonance position. f₁ Add another half-wavelength to L₂ to obtain the third resonance position. = (B) Find the wavelength and the next two water levels giving resonance. Calculate the wavelength, using the fact that, for a tube open at one end, = 4L for the fundamental. λ = Water = 4L₁ V 345 m/s 4L₁ 4(9.00 x 10-² cm) = First resonance (a) Apparatus for demonstrating the resonance of sound waves in a tube closed at one end. The length L of the air column is varied by moving the tube vertically while it is partially submerged in water. (b) The first three resonances of the system. 4(9.00 x 10-2 m): Second resonance (third harmonic) 958 Hz = 0.360 m Third resonance (fifth harmonic) L₂ = L₁ + 1/2 = 0.090 m + 0.180 m = 0.270 m L3 = L₂ + 1/2 = 0.270 m + 0.180 m = 0.450 m
Transcribed Image Text:PROBLEM Figure a shows a simple apparatus for demonstrating resonance in a tube. A long tube open at both ends is partially submerged in a beaker of water, and a vibrating tuning fork of unknown frequency is placed near the top of the tube. The length of the air column, L, is adjusted by moving the tube vertically. The sound waves generated by the fork are reinforced when the length of the air column corresponds to one of the resonant frequencies of the tube. Suppose the smallest value of L for which a peak occurs in the sound intensity is 9.00 cm. (a) With this measurement, determine the frequency of the tuning fork. (b) Find the wavelength and the next two air-column lengths giving resonance. Take the speed of sound to be 345 m/s. SOLUTION (A) Find the frequency of the tuning fork. = 1, v = 345 m/s, and Substitute n = L₁ = 9.00 x 10-² m. STRATEGY Once the tube is in the water, the setup is the same as a pipe closed at one end. For part (a), substitute values for v and L into the one ended harmonics equation with n= 1 and find the frequency of the tuning fork. (b) The next resonance maximum occurs when the water level is low enough to allow a second node, which is another half-wavelength in distance. The third resonance occurs when the third node is reached, requiring yet another half-wavelength of distance. The frequency in each case is the same because it's generated by the tuning fork. Add a half-wavelength of distance to L₁ to get the next resonance position. f₁ Add another half-wavelength to L₂ to obtain the third resonance position. = (B) Find the wavelength and the next two water levels giving resonance. Calculate the wavelength, using the fact that, for a tube open at one end, = 4L for the fundamental. λ = Water = 4L₁ V 345 m/s 4L₁ 4(9.00 x 10-² cm) = First resonance (a) Apparatus for demonstrating the resonance of sound waves in a tube closed at one end. The length L of the air column is varied by moving the tube vertically while it is partially submerged in water. (b) The first three resonances of the system. 4(9.00 x 10-2 m): Second resonance (third harmonic) 958 Hz = 0.360 m Third resonance (fifth harmonic) L₂ = L₁ + 1/2 = 0.090 m + 0.180 m = 0.270 m L3 = L₂ + 1/2 = 0.270 m + 0.180 m = 0.450 m
PRACTICE IT Use the worked example above to help you solve this problem. Figure a shows a simple apparatus for demonstrating resonance in a tube. A long tube open at both ends is partially submerged in a beaker of water, and a vibrating tuning fork of unknown frequency is placed near the top of the tube. The length of the air column, L, is adjusted by moving the tube vertically. The sound waves generated by the fork are reinforced when the length of the air column corresponds to one of the resonant frequencies of the tube. Suppose the smallest value of L for which a peak occurs in the sound intensity is 8.83 cm. Take the speed of sound to be 341 m/s. (a) With this measurement, determine the frequency of the tuning fork. = Hz (b) Find the wavelength and the next two air-column lengths giving resonance. λ = L₂ = L3= ε ε ε EXERCISE An unknown gas is introduced into the aforementioned apparatus using the same tuning fork, and the first resonance occurs when the air column is 5.93 cm long. Find the speed of sound in the gas. m/s sound HINTS: GETTING STARTED I I'M STUCK!
Transcribed Image Text:PRACTICE IT Use the worked example above to help you solve this problem. Figure a shows a simple apparatus for demonstrating resonance in a tube. A long tube open at both ends is partially submerged in a beaker of water, and a vibrating tuning fork of unknown frequency is placed near the top of the tube. The length of the air column, L, is adjusted by moving the tube vertically. The sound waves generated by the fork are reinforced when the length of the air column corresponds to one of the resonant frequencies of the tube. Suppose the smallest value of L for which a peak occurs in the sound intensity is 8.83 cm. Take the speed of sound to be 341 m/s. (a) With this measurement, determine the frequency of the tuning fork. = Hz (b) Find the wavelength and the next two air-column lengths giving resonance. λ = L₂ = L3= ε ε ε EXERCISE An unknown gas is introduced into the aforementioned apparatus using the same tuning fork, and the first resonance occurs when the air column is 5.93 cm long. Find the speed of sound in the gas. m/s sound HINTS: GETTING STARTED I I'M STUCK!
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