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$\cdot$ Find the fundamental frequency and the frequency of thefirst three overtones of a pipe 45.0 $\mathrm{cm}$ long (a) if the pipe isopen at both ends; (b) if the pipe is closed at one end. (c) Foreach of the preceding cases, what is the number of the highest harmonic that may be heard by a person who can hear frequencies from 20 $\mathrm{Hz}$ to $20,000 \mathrm{Hz}$ ?

a) 382 $\mathrm{Hz}$, 1147 $\mathrm{Hz}$, 1529 $\mathrm{Hz}$b) 191 $\mathrm{Hz}$, 573 $\mathrm{Hz}$, 956 $\mathrm{H} 7$, 1338 $\mathrm{Hz}$

Physics 101 Mechanics

Chapter 12

Mechanical Waves and Sound

Periodic Motion

Mechanical Waves

Sound and Hearing

Cornell University

Simon Fraser University

McMaster University

Lectures

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(a) Find the length of an …

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A pipe 0.60 $\mathrm{m}$ l…

so here we know the length equals 45 centimeters, or we can say the length is equal to a 450.45 meters now for part A. They want the fundamental frequency such that the pipe is open at both ends, So this will be the velocity divided by two l hear it the velocities that's going to be the speed of sound with an air so 344 meters per second, divided by two times point for five. And this is giving us 382 hertz. Remember that this is for a pipe that is open at both ends. The the 1st 3 overtones will be the second harmonic. This will simply be two times 3 82 so 7 64 hurts. The third harmonic will be three times the fundamental frequency, or 1,146 hertz, and then the fourth harmonic or the third overtone will be equal to four times the fundamental frequency. So 1,528 hurts now for part B. They want the fundamental frequency such that it is closed at one end. So if it's closed at one end That means that the freak that the wavelength is equal to four times the length so it will be the fundamental frequency of the pipe that has closed at one end will be equal to the speed of sound, with an air divided by four times the length or the wavelength. And this will equal 344 divided by four times 40.45 And we're getting 191 hurts. Note that this is exactly half of this of 3 82 And so we can say that the frequency of the first overtone the pipe when the pipe is closed at one end and is not goingto end can only take on odd and treasures not even and odd. So the first overtone is going to be F sub three instead of F sub, too, and this is simply going to be three times the fundamental frequency. So 573 hurts, and then f sub five the second, the second overtone will be equal to five times 191. So 955 hertz and then the fourth overtone well, second through first rather than third overtone will be f sub seven and this will be seven times 191. So 1,337 hertz. So again, 1st 2nd and third overtone now for part. See, they're asking us if if someone can here a frequency of 20,000 hertz. What harmonic is that? So for the open pipe, the harmonic is going to be equal to the frequency of that harmonic divided by the fundamental frequency. So this will be 20,000 hertz divided by 382 hertz, and this is equal in 52. So the highest harmonic that can be heard is the 52nd harmonic and then here closed that one end. This means that and that can only equal f sub and divided by f f the fundamental frequency. However, here it's going to be 20,000 divided by 1 91 and this will be 104 now notice if it's closed at one end and can only take odd integers. So this is an even into jher. Therefore, on equals one less than this. So it becomes alright, Esso and equals 103. So that would be the highest that the Ah, listen, er can hear if the pipe is closed at one end. This would be the highest the listener can hear. If the pipe is open at both ends, that is the end of the solution. Thank you for watching.

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