# Daily Archives: August 17, 2019

356 posts

## 16.76 … CP Longitudinal Waves on a Spring. A long spring such as a Slinky™ is often used to demonstrate longitudinal waves. (a) Show that if a spring that obeys Hooke’s law has mass m length L and force constant k_ the speed of longitudinal waves on the spring is v = L2k_>m (see Section 16.2). (b) Evaluate v for a spring with m = 0.250 kg L = 2.00 m and k_ = 1.50 N>m.

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## 16.75 … CALC Figure P16.75 shows the pressure fluctuation p of a nonsinusoidal sound wave as a function of x for t = 0. The wave is traveling in the +x-direction. (a) Graph the pressure fluctuation p as a function of t for x = 0. Show at least two cycles of oscillation. (b) Graph the displacement y in this sound wave as a function of x at t = 0. At x = 0 the displacement at t = 0 is zero. Show at least two wavelengths of the wave. (c) Graph the displacement y as a function of t for x = 0. Show at least two cycles of oscillation. (d) Calculate the maximum velocity and the maximum acceleration of an element of the air through which this sound wave is traveling. (e) Describe how the cone of a loudspeaker must move as a function of time to produce the sound wave in this problem.

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## 16.74 … DATA Supernova! (a) Equation (16.30) can be written as fR = fSa1 – v c b 1>2 a1 + v c b -1>2 where c is the speed of light in vacuum 3.00 * 108 m>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 V c Eq. (16.30) approximately reduces to fR = fS31 – 1v>c24. (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 color of heated hydrogen gas. In a laboratory on the earth heated hydrogen produces red light with frequency 4.568 * 1014 Hz; the red light received from streamers in the Crab Nebula that are pointed toward the earth has frequency 4.586 * 1014 Hz. Estimate the speed with which the outer edges of the Crab Nebula are expanding. Assume that the speed of the center 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 11 arc@minute = 1 60 degree2. 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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## 16.73 .. DATA A long tube contains air at a pressure of 1.00 atm and a temperature of 77.0_C. The tube is open at one end and closed at the other by a movable piston. A tuning fork that vibrates with a frequency of 500 Hz is placed near the open end. Resonance is produced when the piston is at distances 18.0 cm 55.5 cm and 93.0 cm from the open end. (a) From these values what is the speed of sound in air at 77.0_C? (b) From the result of part (a) what is the value of g? (c) These results show that a displacement antinode is slightly outside the open end of the tube. How far outside is it?

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## 16.72 .. DATA A long closed cylindrical tank contains a diatomic gas that is maintained at a uniform temperature that can be varied. When you measure the speed of sound v in the gas as a function of the temperature T of the gas you obtain these results: T 1 _C2 -20.0 0.0 20.0 40.0 60.0 80.0 v 1m s 2 324 337 349 361 372 383 (a) Explain how you can plot these results so that the graph will be well fit by a straight line. Construct this graph and verify that the plotted points do lie close to a straight line. (b) Because the gas is diatomic g = 1.40. Use the slope of the line in part (a) to calculate M the molar mass of the gas. Express M in grams/mole. What type of gas is in the tank?

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## 16.71 .. CP A turntable 1.50 m in diameter rotates at 75 rpm. Two speakers each giving off sound of wavelength 31.3 cm are attached to the rim of the table at opposite ends of a diameter. A listener stands in front of the turntable. (a) What is the greatest beat frequency the listener will receive from this system? (b) Will the listener be able to distinguish individual beats?

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## 16.70 . CP A police siren of frequency fsiren is attached to a vibrating platform. The platform and siren oscillate up and down in simple harmonic motion with amplitude Ap and frequency fp. (a) Find the maximum and minimum sound frequencies that you would hear at a position directly above the siren. (b) At what point in the motion of the platform is the maximum frequency heard? The minimum frequency? Explain.

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## 16.69 … BIO Horseshoe bats (genus Rhinolophus) emit sounds from their nostrils and then listen to the frequency of the sound reflected from their prey to determine the prey’s speed. (The “horseshoe” that gives the bat its name is a depression around the nostrils that acts like a focusing mirror so that the bat emits sound in a narrow beam like a flashlight.) A Rhinolophus flying at speed vbat emits sound of frequency fbat; the sound it hears reflected from an insect flying toward it has a higher frequency frefl. (a) Show that the speed of the insect is vinsect = vc frefl1v – vbat2 – fbat1v + vbat2 frefl1v – vbat2 + fbat1v + vbat2 d where v is the speed of sound. (b) If fbat = 80.7 kHz frefl = 83.5 kHz and vbat = 3.9 m>s calculate the speed of the insect.

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## 16.68 … BIO Ultrasound in Medicine. A 2.00-MHz sound wave travels through a pregnant woman’s abdomen and is reflected from the fetal heart wall of her unborn baby. The heart wall is moving toward the sound receiver as the heart beats. The reflected sound is then mixed with the transmitted sound and 72 beats per second are detected. The speed of sound in body tissue is 1500 m>s. Calculate the speed of the fetal heart wall at the instant this measurement is made.

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## 16.67 .. The sound source of a ship’s sonar system operates at a frequency of 18.0 kHz. The speed of sound in water (assumed to be at a uniform 20_C) is 1482 m>s. (a) What is the wavelength of the waves emitted by the source? (b) What is the difference in frequency between the directly radiated waves and the waves reflected from a whale traveling directly toward the ship at 4.95 m>s ? The ship is at rest in the water.

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## 16.66 .. A bat flies toward a wall emitting a steady sound of frequency 1.70 kHz. This bat hears its own sound plus the sound reflected by the wall. How fast should the bat fly in order to hear a beat frequency of 8.00 Hz?

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## 16.65 .. Two identical loudspeakers are located at points A and B 2.00 m apart. The loudspeakers are driven by the same amplifier and produce sound waves with a frequency of 784 Hz. Take the speed of sound in air to be 344 m>s. A small microphone is moved out from point B along a line perpendicular to the line connecting A and B (line BC in Fig. P16.65). (a) At what distances from B will there be destructive interference? (b) At what distances from B will there be constructive interference? (c) If the frequency is made low enough there will be no positions along the line BC at which destructive interference occurs. How low must the frequency be for this to be the case?

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## 16.64 … The frequency of the note F4 is 349 Hz. (a) If an organ pipe is open at one end and closed at the other what length must it have for its fundamental mode to produce this note at 20.0_C? (b) At what air temperature will the frequency be 370 Hz corresponding to a rise in pitch from F to F-sharp? (Ignore the change in length of the pipe due to the temperature change.)

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## 16.63 . An organ pipe has two successive harmonics with frequencies 1372 and 1764 Hz. (a) Is this an open or a stopped pipe? Explain. (b) What two harmonics are these? (c) What is the length of the pipe?

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## 16.62 .. CP A uniform 165-N bar is supported horizontally by two identical wires A and B (Fig. P16.62). A small 185-N cube of lead is placed threefourths of the way from A to B. The wires are each 75.0 cm long and have a mass of 5.50 g. If both of them are simultaneously plucked at the center what is the frequency of the beats that they will produce when vibrating in their fundamental?

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## 16.61 . CP A person is playing a small flute 10.75 cm long open at one end and closed at the other near a taut string having a fundamental frequency of 600.0 Hz. If the speed of sound is 344.0 m>s for which harmonics of the flute will the string resonate? In each case which harmonic of the string is in resonance?

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## 16.60 .. CP The sound from a trumpet radiates uniformly in all directions in 20_C air. At a distance of 5.00 m from the trumpet the sound intensity level is 52.0 dB. The frequency is 587 Hz. (a) What is the pressure amplitude at this distance? (b) What is the displacement amplitude? (c) At what distance is the sound intensity level 30.0 dB?

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## 16.59 .. A soprano and a bass are singing a duet. While the soprano sings an A-sharp at 932 Hz the bass sings an A-sharp but three octaves lower. In this concert hall the density of air is 1.20 kg>m3 and its bulk modulus is 1.42 * 105 Pa. In order for their notes to have the same sound intensity level what must be (a) the ratio of the pressure amplitude of the bass to that of the soprano and (b) the ratio of the displacement amplitude of the bass to that of the soprano? (c) What displacement amplitude (in m and in nm) does the soprano produce to sing her A-sharp at 72.0 dB?

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## 16.58 . The shock-wave cone created by a space shuttle at one instant during its reentry into the atmosphere makes an angle of 58.0_ with its direction of motion. The speed of sound at this altitude is 331 m>s. (a) What is the Mach number of the shuttle at this instant and (b) how fast (in m>s and in mi>h) is it traveling relative to the atmosphere? (c) What would be its Mach number and the angle of its shock-wave cone if it flew at the same speed but at low altitude where the speed of sound is 344 m>s ?

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## 16.57 .. A jet plane flies overhead at Mach 1.70 and at a constant altitude of 1250 m. (a) What is the angle a of the shock-wave cone? (b) How much time after the plane passes directly overhead do you hear the sonic boom? Neglect the variation of the speed of sound with altitude.

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## 16.56 .. How fast (as a percentage of light speed) would a star have to be moving so that the frequency of the light we receive from it is 10.0% higher than the frequency of the light it is emitting? Would it be moving away from us or toward us? (Assume it is moving either directly away from us or directly toward us.)

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## 16.55 .. A stationary police car emits a sound of frequency 1200 Hz that bounces off a car on the highway and returns with a frequency of 1250 Hz. The police car is right next to the highway so the moving car is traveling directly toward or away from it. (a) How fast was the moving car going? Was it moving toward or away from the police car? (b) What frequency would the police car have received if it had been traveling toward the other car at 20.0 m>s?

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## 16.54 .. The siren of a fire engine that is driving northward at 30.0 m>s emits a sound of frequency 2000 Hz. A truck in front of this fire engine is moving northward at 20.0 m>s. (a) What is the frequency of the siren’s sound that the fire engine’s driver hears reflected from the back of the truck? (b) What wavelength would this driver measure for these reflected sound waves?

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## 16.53 . Two swift canaries fly toward each other each moving at 15.0 m>s relative to the ground each warbling a note of frequency 1750 Hz. (a) What frequency note does each bird hear from the other one? (b) What wavelength will each canary measure for the note from the other one?

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## 16.52 .. While sitting in your car by the side of a country road you are approached by your friend who happens to be in an identical car. You blow your car’s horn which has a frequency of 260 Hz. Your friend blows his car’s horn which is identical to yours and you hear a beat frequency of 6.0 Hz. How fast is your friend approaching you?

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## 16.51 . A car alarm is emitting sound waves of frequency 520 Hz. You are on a motorcycle traveling directly away from the parked car. How fast must you be traveling if you detect a frequency of 490 Hz?

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## 16.50 . A railroad train is traveling at 30.0 m>s in still air. The frequency of the note emitted by the train whistle is 352 Hz. What frequency is heard by a passenger on a train moving in the opposite direction to the first at 18.0 m>s and (a) approaching the first and (b) receding from the first?

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