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Q17.5-50PE

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College Physics (Urone)
Found in: Page 595
College Physics (Urone)

College Physics (Urone)

Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000

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Short Answer

The ear canal resonates like a tube closed at one end. (See Figure \(17.39\).) If ear canals range in length from \(1.80\)to\(2.60\;{\rm{cm}}\)in an average population, what is the range of fundamental resonant frequencies? Take air temperature to be\(37.0{\rm{^\circ C}}\), which is the same as body temperature. How does this result correlate with the intensity versus frequency graph (Figure \(17.37\) of the human ear?

The range of the fundamental frequencies is\({\rm{3}}{\rm{.39}}\;{\rm{kHz}}\)to\({\rm{4}}{\rm{.9}}\;{\rm{kHz}}\)

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Given Data

The temperature is\({{\rm{t}}_{\rm{1}}}{\rm{ = 37}}{\rm{.0^\circ C = }}\left( {{\rm{37 + 273}}} \right)\;{\rm{K = 310}}\;{\rm{K}}\)

The length of the canal ranges from \({{\rm{l}}_{\rm{1}}}{\rm{ = 1}}{\rm{.80}}\;{\rm{cm = 0}}{\rm{.018}}\;{\rm{m}}\) to \({{\rm{l}}_{\rm{2}}}{\rm{ = 2}}{\rm{.60}}\;{\rm{cm = 0}}{\rm{.026}}\;{\rm{m}}\)

The diagram of Human Ear is shown below:

The human ear

The diagram of intensity Vs frequency is shown below:

The intensity-frequency graph

Calculation of the ratio of the frequencies  

For a closed tube, the resonance frequencies are,

\({{\rm{f}}_{\rm{n}}}{\rm{ = n}}\frac{{\rm{v}}}{{{\rm{4l}}}}{\rm{,}}\;{\rm{n = 1,3,}}\;{\rm{5,}}\;....\)

For an open tube, the resonance frequencies are,

\({{\rm{f}}_{\rm{n}}}{\rm{ = n}}\frac{{\rm{v}}}{{{\rm{2l}}}}{\rm{,}}\;{\rm{n = 1,2,}}\;{\rm{3,}}\;....\)

The speed of sound at\({t_1}\) temperature is

\(\begin{array}{c}{\rm{v = 331}}\sqrt {\frac{{{{\rm{t}}_{\rm{1}}}}}{{{\rm{273}}}}} \\{\rm{ = 331}}\sqrt {\frac{{{\rm{310}}}}{{{\rm{273}}}}} \\{\rm{ = 352}}{\rm{.7}}\;{\rm{m/s}}\end{array}\)

The frequencies are,

\(\begin{array}{c}{{\rm{f}}_{\rm{1}}}{\rm{ = }}\frac{{{\rm{352}}{\rm{.72}}}}{{{\rm{4 \times 0}}{\rm{.018}}}}\\{\rm{ = 4898}}{\rm{.89}}\;{\rm{Hz}}\\{\rm{ = 4}}{\rm{.9}}\;{\rm{kHz}}\end{array}\)

\(\begin{array}{c}{{\rm{f}}_{\rm{2}}}{\rm{ = }}\frac{{{\rm{352}}{\rm{.72}}}}{{{\rm{4 \times 0}}{\rm{.026}}}}\\{\rm{ = 3391}}{\rm{.54}}\;{\rm{Hz}}\\{\rm{ = 3}}{\rm{.39}}\;{\rm{kHz}}\end{array}\)

Our ear can hear\({\rm{ - 20}}\;{\rm{dB}}\) to\({\rm{10}}\;{\rm{dB}}\) with\({\rm{0}}\;{\rm{phon}}\), \({\rm{10}}\;{\rm{dB}}\)to\({\rm{40}}\;{\rm{dB}}\)with maximum of\({\rm{40}}\;{\rm{phon}}\) and\({\rm{40}}\;{\rm{dB}}\)to\({\rm{60}}\;{\rm{dB}}\)with maximum of\({\rm{60}}\;{\rm{phon}}\).

Most popular questions for Physics Textbooks

a) What is the speed of sound in a medium where a 100 kHz frequency produces a 5.96 cm wavelength? (b) Which substance in Table 17.1 is this likely to be?

(a) Students in a physics lab are asked to find the length of an air column in a tube closed at one end that has a fundamental frequency of 256 Hz. They hold the tube vertically and fill it with water to the top, then lower the water while a 256 Hz tuning fork is rung and listen for the first resonance. What is the air temperature if the resonance occurs for a length of 0.336 m?

(b) At what length will they observe the second resonance (first overtone)?

(a) Find the intensity in watts per meter squared of a \({\rm{60}}{\rm{.0 Hz}}\) sound having a loudness of \({\bf{60}}{\rm{ }}{\bf{phons}}\). (b) Find the intensity in watts per meter squared of a \({\rm{10,000 Hz}}\) sound having a loudness of \({\bf{60}}{\rm{ }}{\bf{phons}}\).

Two eagles fly directly toward one another, the first at \({\rm{15}}{\rm{.0}}\;{\rm{m/s}}\) and the second at \({\rm{20}}{\rm{.0}}\;{\rm{m/s}}\). Both screech, the first one emitting a frequency of \({\rm{3200}}\;{\rm{Hz}}\) and the second one emitting a frequency of \({\rm{3800}}\;{\rm{Hz}}\). What frequencies do they receive if the speed of sound is \({\rm{330}}\;{\rm{m/s}}\)?

The frequencies to which the ear responds vary by a factor of \[{\rm{1}}{{\rm{0}}^{\rm{3}}}\]. Suppose the speedometer on your car measured speeds differing by the same factor of \[{\rm{1}}{{\rm{0}}^{\rm{3}}}\], and the greatest speed it reads is \[{\rm{90}}{\rm{.0}}\;{\rm{mi/h}}\]. What would be the slowest nonzero speed it could read?
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