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Chapter 2: Q33E (page 540)

A 75.0-cm-long wire of mass 5.625 g is tied at both ends and adjusted to a tension of 35.0 N. When it is vibrating in its second overtone, find (a) the frequency and wavelength at which it is vibrating and (b) the frequency and wavelength of the sound waves it is producing.

Short Answer

Expert verified

The frequency and wavelength of the string is 136.6Hz and 0.5m respectively and the frequency and wavelength of the string is 136.6Hz and 2.518m respectively

Step by step solution

01

STEP 1 Concept of the frequency for standing wave for the strings fixed at both ends,

Thefrequency for standing wave for the strings is given asf=nv2Lwere,f is the frequency of nth harmonic, v is the velocity of the wave,n鈫�nthharmonic (n 鈥� 1, 3, 5, ...), L is the length of the pipe.

02

Calculate the frequency and wavelength of the string

For the string

渭=mL=5.625脳10-30.75=7.5脳10-3kg/mv=F渭=35N7.5脳10-3kg/m=68.313m/s

Second overtone is third harmonic n=3

f3=nv2L=3脳65.32m/s2脳0.75m=136.6Hz位=vf=68.32136.6=0.5m

Therefore, the frequency and wavelength of the string is 136.6Hz and 0.5m respectively

03

Step 3 Calculate the frequency and wavelength of the sound

The string is the source of vibration, so, the sound has the same frequency 136.6Hz

位sound=vf=344136.6=2.518m

Therefore, the frequency and wavelength of the string is 136.6Hz and 2.518m respectively

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Most popular questions from this chapter

The portion of the string of a certain musical instrument between the bridge and upper end of the finger board (that part of the string that is free to vibrate) is \(60.0\;{\rm{cm}}\) long, and this length of the string has mass \(2.00\;{\rm{g}}\). The string sounds an \({A_4}\) note (\(440\;{\rm{Hz}}\)) when played.

(a) Where must the player put a finger (what distance x from the bridge) to play a \({D_5}\) note (\(587\;{\rm{Hz}}\))? (See Fig. E15.45.) For both the \({A_4}\) and \({D_5}\) notes, the string vibrates in its fundamental mode.

(b) Without retuning, is it possible to play a \({G_4}\) note (\(392\;{\rm{Hz}}\)) on this string? Why or why not?

(a) If two sounds differ by 5.00 dB, find the ratio of the intensity of the louder sound to that of the softer one. (b) If one sound is 100 times as intense as another, by how much do they differ in sound intensity level (in decibels)? (c) If you increase the volume of your stereo so that the intensity doubles, by how much does the sound intensity level increase?

Which of the following wave functions satisfies the wave equation, Eq. (15.12)? (a)\(y\left( {x,t} \right) = Acos\left( {kx + \omega t} \right)\) ; (b)\(y\left( {x,t} \right) = Asin\left( {kx + \omega t} \right)\) ; (c) \(y\left( {x,t} \right) = A\left( {coskx + cos\omega t} \right)\) (d) For the wave of part (b), write the equations for the transverse velocity and transverse acceleration of a particle at point x.

With what tension must a rope with length 2.50 m and mass 0.120 kg be stretched for transverse waves of frequency 40.0 Hz to have a wavelength of 0.750 m?

DATA Supernova! (a) Equation (16.30) can be written as fr=fs(1鈥�vc)12(1+vc)12

where c is the speed of light in vacuum3.0脳108m/s. Most objects move much slower than this (v/c is very small), so calculations made with Eq. (16.30) must be done carefully to avoid rounding errors. Use the binomial theorem to show that if v, Eq. (16.30) approximately reduces tofr=fs(1-vc). (b) The gas cloud known as the Crab Nebula can be seen with even a small telescope. It is the remnant of a supernova, a cataclysmic explosion of a star. (The explosion was seen on the earth on July 4,1054 C.E.) Its streamers glow with the characteristic red colour of heated hydrogen gas. In a laboratory on the earth, heated hydrogen produces red light with frequencyrole="math" localid="1668146026646" 4.568脳1014Hz; the red light received from streamers in the Crab Nebula that are pointed toward the earth has frequency4.568脳1014HzEstimate the speed with which the outer edges of the Crab Nebula are expanding. Assume that the speed of the centre of the nebula relative to the earth is negligible. (c) Assuming that the expansion speed of the Crab Nebula has been constant since the supernova that produced it, estimate the diameter of the Crab Nebula. Give your answer in meters and in light-years. (d) The angular diameter of the Crab Nebula as seen from the earth is about 5 arc-minutes(1arc鈥�min160degree)Estimate the distance (in light-years) to the Crab Nebula, and estimate the year in which the supernova actually, took place.
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