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Chapter 2: Q16DQ (page 497)

If you stretch a rubber band and pluck it, you hear a (somewhat) musical tone. How does the frequency of this tone change as you stretch the rubber band further? (Try it!) Does this agree with Eq. (15.35) for a string fixed at both ends? Explain.

Short Answer

Expert verified

When the rubber band is stretched, the frequency of the tone produced will increases.

Step by step solution

01

Concept of frequency of a tone.

The length of the rubber band and the tension that is created inside of it determine the frequency of the tone. The frequency varies as a result of the rubber band being stretched.

02

Formula of frequency of the string.

If we stretch the string then its length increases and the tension in the spring also increase. The frequency of the tone depends upon both tension and length of the rubber band which can be represented by the expression given below,

f=12LF渭

Here, f is the frequency of the wave, L is the length of a rubber band, F is the tension in the rubber band. 渭 is the mass per unit length.

Therefore, when the rubber band is stretched, the frequency of the tone produced will increases.

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Most popular questions from this chapter

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.

A strong string of mass 3.00 g and length 2.20 m is tied to supports at each end and is vibrating in its fundamental mode. The maximum transverse speed of a point at the middle of the string is \(9.00\;{\rm{m/s}}\). The tension in the string is 330 N.

(a) What is the amplitude of the standing wave at its antinode?

(b) What is the magnitude of the maximum transverse acceleration of a point at the antinode?

Transverse waves on a string have wave speed 8 m/s, amplitude 0.07 m, and wavelength 0.32 m . The waves travel in the -x-direction, and at t=0 the x=0 end of the string has its maximum upward displacement. (a) Find the frequency, period, and wave number of these waves. (b) Write a wave function describing the wave. (c) Find the transverse displacement of a particle at x=36 m and time t=0.15 s. (d) How much time must elapse from the instant in part (c) until the particle at x=0.36 m next has maximum upward displacement?

Question:BIO Ultrasound and Infrasound. (a) Whale communication. Blue whales apparently communicate with each other using sound of frequency 17 Hz, which can be heard nearly 1000 km away in the ocean. What is the wavelength of such a sound in seawater, where the speed of sound is 1531 m/s? (b) Dolphin clicks. One type of sound that dolphins emit is a sharp click of wavelength 1.5 cm in the ocean. What is the frequency of such clicks? (c) Dog whistles. One brand of dog whistles claims a frequency of 25 kHz for its product. What is the wavelength of this sound? (d) Bats. While bats emit a wide variety of sounds, one type emits pulses of sound having a frequency between 39 kHz and 78 kHz. What is the range of wavelengths of this sound? (e) Sonograms. Ultrasound is used to view the interior of the body, much as x rays are utilized. For sharp imagery, the wavelength of the sound should be around one-fourth (or less) the size of the objects to be viewed. Approximately what frequency of sound is needed to produce a clear image of a tumor that is 1.0 mm across if the speed of sound in the tissue is 1550 m/s?

Speed of Propagation vs. Particle Speed. (a) Show that Eq. (15.3) may be written as

\(y\left( {x,t} \right) = Acos\left[ {\frac{{2\pi }}{\lambda }\left( {x - vt} \right)} \right]\)

(b) Use \(y\left( {x,t} \right)\) to find an expression for the transverse velocity \({v_y}\)of a particle in the string on which the wave travels. (c) Find the maximum speed of a particle of the string. Under what circumstances is this equal to the propagation speed \(v\) ? Less than\(v\)? Greater than\(v\)?
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