Deck 21: Superpositions

Question 1

A platinum wire that is 1.20 m long has a radius of 0.500 mm and is fixed at both ends. In its third harmonic it vibrates at 512 Hz. The density of platinum is 21.4 × 10³ kg/m³. What is the tension in the wire?

A) 4.00 kN
B) 2.00 kN
C) 2.82 kN
D) 1.41 kN
E) 1.00 kN

Answer 1

2.82 kN

Question 2

A pipe that is 120 cm long resonates to produce sound of wavelengths 480 cm, 160 cm, and 96 cm but does not resonate at any wavelengths longer than these. This pipe is

A) open at both ends.
B) open at one end and closed at the other end.
C) closed at both ends.
D) We cannot tell because we do not know the frequency of the sound.

Answer 2

open at one end and closed at the other end.

Question 3

A wave pulse traveling to the right along a thin cord reaches a discontinuity where the rope becomes thinner and lighter. What is the orientation of the reflected and transmitted pulses?

A) Both pulses are right side up.
B) The reflected pulse returns right side up while the transmitted pulse is inverted.
C) The reflected pulse returns inverted while the transmitted pulse is right side up.
D) Both pulses are inverted.

Answer 3

Both pulses are right side up.

Question 4

A tube open at one end and closed at the other end produces sound having a fundamental frequency of 350 Hz. If you now open the closed end, the fundamental frequency becomes

A) 87.5 Hz.
B) 175 Hz.
C) 350 Hz.
D) 700 Hz.
E) 1400 Hz.

Question 5

The lowest-pitch tone to resonate in a pipe of length L that is open at both ends is 200 Hz. Which one of the following frequencies will NOT resonate in the same pipe?

A) 400 Hz
B) 600 Hz
C) 800 Hz
D) 900 Hz
E) 1000 Hz

Question 6

A standing wave is oscillating at 690 Hz on a string, as shown in the figure. What is the speed of traveling waves on this string?

A) 280 m/s
B) 410 m/s
C) 210 m/s
D) 140 m/s

Question 7

A wave pulse traveling to the right along a thin cord reaches a discontinuity where the rope becomes thicker and heavier. What is the orientation of the reflected and transmitted pulses?

A) Both pulses are right side up.
B) The reflected pulse returns right side up while the transmitted pulse is inverted.
C) The reflected pulse returns inverted while the transmitted pulse is right side up.
D) Both pulses are inverted.

Question 8

A 2.0-m string is fixed at both ends and tightened until the wave speed is 78 m/s. What is the frequency of the standing wave shown in the figure?

A) 120 Hz
B) 230 Hz
C) 350 Hz
D) 470 Hz

Question 9

The lowest-pitch tone to resonate in a pipe of length L that is closed at one end and open at the other end is 200 Hz. Which one of the following frequencies will NOT resonate in the same pipe?

A) 400 Hz
B) 600 Hz
C) 1000 Hz
D) 1400 Hz
E) 1800 Hz

Question 10

A thin 2.00-m string of mass 50.0 g is fixed at both ends and under a tension of 70.0 N. If it is set into small-amplitude oscillation, what is the frequency of the first harmonic mode?

A) 6.61 Hz
B) 13.2 Hz
C) 26.5 Hz
D) 52.9 Hz

Question 11

What characteristic of sound determines the "pitch" of a musical note?

A) amplitude
B) wavelength
C) frequency
D) phase
E) intensity

Question 12

Two pure tones are sounded together and a particular beat frequency is heard. What happens to the beat frequency if the frequency of one of the tones is increased?

A) It increases.
B) It decreases.
C) It does not change.
D) It becomes zero.
E) We cannot tell from the information given.

Question 13

A guitar string 0.650 m long has a tension of 61.0 N and a mass per unit length of 3.00 g/m.
(a) What is the speed of waves on the string when it is plucked?
(b) What is the string's fundamental frequency of vibration when plucked?

Question 14

Consider the waves on a vibrating guitar string and the sound waves the guitar produces in the surrounding air. The string waves and the sound waves must have the same

A) wavelength.
B) velocity.
C) frequency.
D) amplitude.
E) More than one of the above is true.

Question 15

In a resonating pipe that is open at one end and closed at the other end, there

A) are displacement nodes at each end.
B) are displacement antinodes at each end.
C) is a displacement node at the open end and a displacement antinode at the closed end.
D) is a displacement node at the closed end and a displacement antinode at the open end.

Question 16

A 2.00-m long piano wire with a mass per unit length of 12.0 g/m is under a tension of 8.00 kN. What is the frequency of the fundamental mode of vibration of this wire?

A) 204 Hz
B) 102 Hz
C) 408 Hz
D) 510 Hz
E) 153 Hz

Question 17

Consider a pipe of length L that is open at both ends. What are the wavelengths of the three lowest-pitch tones produced by this pipe?

A) 4L, 2L, L
B) 2L, L, L/2
C) 2L, L, 2L/3
D) 4L, 4L/3, 4L/5
E) 2.0L, L, L/2

Question 18

A thin taut string is fixed at both ends and stretched along the horizontal x-axis with its left end at x = 0. It is vibrating in its third OVERTONE, and the equation for the vertical displacement of any point on the string is y(x,t) = (1.22 cm) sin[(14.4 m^-1)x] cos[(166 rad/s)t].
(a) What are the frequency and wavelength of the fundamental mode of this string?
(b) How long is the string?
(c) How fast do waves travel on this string?

Question 19

In a resonating pipe that is open at both ends, there

A) are displacement nodes at each end.
B) are displacement antinodes at each end.
C) is a displacement node at one end and a displacement antinode at the other end.
D) None of the above are possible.

Question 20

A guitar string is fixed at both ends. If you tighten it to increase its tension

A) the frequencies of its vibrational modes will increase but its wavelengths will not be affected.
B) the wavelength increases but the frequency is not affected.
C) both the frequency and the wavelength increase.

Question 21

50) A 0.25-m string, vibrating in its sixth harmonic, excites a 0.96-m pipe that is open at both ends into its second overtone resonance. The speed of sound in air is 345 m/s. The common resonant frequency of the string and the pipe is closest to

A) 540 Hz.
B) 360 Hz.
C) 450 Hz.
D) 630 Hz.
E) 700 Hz.

Question 22

An air column, open at one end and closed at the other, is being designed so that its second lowest resonant frequency is 440 Hz. What should be the length of the column if the speed of sound in air is 340 m/s?

A) 0.386 m
B) 0.772 m
C) 1.16 m
D) 0.193 m
E) 0.580 m

Question 23

The sound from a single source can reach point O by two different paths. One path is 20.0 m long and the second path is 21.0 m long. The sound destructively interferes at point O. What is the minimum frequency of the source if the speed of sound is 340 m/s?

A) 340 Hz
B) 6800 Hz
C) 520 Hz
D) 680 Hz
E) 170 Hz

Question 24

A 1.0-g string that is 0.64 m long is fixed at both ends and is under tension. This string produces a 100-Hz tone when it vibrates in the third harmonic. The speed of sound in air is 344 m/s. The tension in the string, in is closest to

A) 2.8 N.
B) 2.3 N.
C) 1.8 N.
D) 3.4 N.
E) 3.9 N.

Question 25

An organ pipe open at both ends has two successive harmonics with frequencies of 210 Hz and 240 Hz. What is the length of the pipe? The speed of sound is 344 m/s in air.

A) 5.25 m
B) 5.73 m
C) 2.76 m
D) 4.90 m
E) 3.62 m

Question 26

A string 40.0 cm long of mass 8.50 g is fixed at both ends and is under a tension of 425 N. When this string is vibrating in its third OVERTONE, you observe that it causes a nearby pipe, open at both ends, to resonate in its third HARMONIC. The speed of sound in the room is 344 m/s.
(a) How long is the pipe?
(b) What is the fundamental frequency of the pipe?

Question 27

The speed of sound in the air inside a 0.640-m long gas column is 340 m/s. What is the fundamental resonant frequency of this air column if it is
(a) open at one end and closed at the other end?
(b) open at both ends?

Question 28

A violin with string length 32 cm and string density 1.5 g/cm resonates with the first overtone from a 2.0-m long organ pipe with one end closed and the other end open. What is the tension in the string?

A) 1000 N
B) 110 N
C) 450 N
D) 4100 N
E) 56 N

Question 29

Two violinists are trying to tune their instruments in an orchestra. One is producing the desired frequency of 440.0 Hz. The other is producing a frequency of 448.4 Hz. By what percentage should the out-of-tune musician change the tension in his string to bring his instrument into tune at 440.0 Hz?

A) +1.9%
B) -1.9%
C) +3.7%
D) -3.7%
E) +8.4%

Question 30

A heavy stone of mass m is hung from the ceiling by a thin 8.25-g wire that is 65.0 cm long. When you gently pluck the upper end of the wire, a pulse travels down the wire and returns 7.84 ms later, having reflected off the lower end. The speed of sound in the room is 344 m/s, and the stone is heavy enough to prevent the lower end of the wire from moving. If the wire is vibrating in its second overtone, what is the frequency of the sound it will produce?

A) 128 Hz
B) 191 Hz
C) 255 Hz
D) 383 Hz
E) 765 Hz

Question 31

A pipe is 0.90 m long and is open at one end but closed at the other end. If it resonates with a tone whose wavelength is 0.72 m, what is the wavelength of the next higher overtone in this pipe?

A) 0.36 m
B) 0.40 m
C) 0.45 m
D) 0.51 m
E) 0.58 m

Question 32

Two stereo speakers mounted 4.52 m apart on a wall emit identical in-phase sound waves. You are standing at the opposite wall of the room at a point directly between the two speakers. You walk 2.11 m parallel to the wall, to a location where you first notice that the sound intensity drops to zero. If the wall along which you are walking is 10.7 m from the wall with the speakers, what is the wavelength of the sound waves? CAREFUL! The distance to the wall is NOT much greater than the distance between the speakers.

A) 1.7 m
B) 2.1 m
C) 2.6 m
D) 2.9 m

Question 33

A string, 0.28 m long and vibrating in its third harmonic, excites an open pipe that is 0.82 m long into its second overtone resonance. The speed of sound in air is 345 m/s. The speed of transverse waves on the string is closest to

A) 120 m/s.
B) 110 m/s.
C) 100 m/s.
D) 98 m/s.
E) 91 m/s.

Question 34

Standing waves of frequency 57 Hz are produced on a string that has mass per unit length 0.0160 kg/m. With what tension must the string be stretched between two supports if adjacent nodes in the standing wave are to be 0.71 meters apart?

Question 35

A heavy stone of mass m is hung from the ceiling by a thin 8.25-g wire that is 65.0 cm long. When you gently pluck the upper end of the wire, a pulse travels down the wire and returns 7.84 ms later, having reflected off the lower end. The speed of sound in the room is 344 m/s, and the stone is heavy enough to prevent the lower end of the wire from moving. If the wire is vibrating in its second overtone, what is the wavelength of the sound it will produce?

A) 0.217 m
B) 0.433 m
C) 0.650 m
D) 0.899 m
E) 1.35 m

Question 36

A 1.30-m long gas column that is open at one end and closed at the other end has a fundamental resonant frequency 80.0 Hz. What is the speed of sound in this gas?

A) 104 m/s
B) 61.5 m/s
C) 26.0 m/s
D) 246 m/s
E) 416 m/s

Question 37

One of the harmonics of a column of air open at one end and closed at the other has a frequency of 448 Hz and the next higher harmonic has a frequency of 576 Hz. What is the fundamental frequency of the air column?

A) 32 Hz
B) 64 Hz
C) 88 Hz
D) 128 Hz
E) 256 Hz

Question 38

A pipe that is 0.46 m long and open at both ends vibrates in the second overtone with a frequency of 1150 Hz. In this situation, the distance from the center of the pipe to the nearest antinode is closest to

A) 7.7 cm.
B) 3.8 cm.
C) 12 cm.
D) 15 cm.
E) zero.

Question 39

Two in-phase loudspeakers that emit sound with the same frequency are placed along a wall and are separated by a distance of 5.00 m. A person is standing 12.0 m away from the wall, equidistant from the loudspeakers. When the person moves 1.00 m parallel to the wall, she experiences destructive interference for the first time. What is the frequency of the sound? The speed of sound in air is 343 m/s.

A) 211 Hz
B) 256 Hz
C) 422 Hz
D) 512 Hz
E) 674 Hz

Question 40

Two in-phase loudspeakers that emit sound with the same frequency are placed along a wall and are separated by a distance of 8.00 m. A person is standing 12.0 m away from the wall, equidistant from the loudspeakers. When the person moves 3.00 m parallel to the wall, she experiences destructive interference for the second time. What is the frequency of the sound? The speed of sound in the room is 343 m/s. CAREFUL! The distance to the wall is NOT much greater than the distance between the speakers.

A) 278 Hz
B) 422 Hz
C) 452 Hz
D) 562 Hz
E) 694 Hz

Question 41

Two strings of identical material and radius are stretched with the same tension with their ends fixed, but one string is 8.0 mm longer than the other. Waves on these strings propagate at 420 m/s. The fundamental frequency of the longer string is 528 Hz. What is the beat frequency when each string is vibrating at its fundamental frequency?

A) 22 Hz
B) 11 Hz
C) 16 Hz
D) 5.5 Hz
E) 27 Hz

Question 42

Two identical loudspeakers that are 5.00 m apart and face toward each other are driven in phase by the same oscillator at a frequency of 875 Hz. The speed of sound in the room is 344 m/s. If you start out standing midway between the speakers, find the shortest distance you can walk toward either speaker in order to hear a minimum of sound.

A) 0.0983 m
B) 0.197 m
C) 0.295 m
D) 0.393 m
E) 0.590 m

Question 43

Two in-phase loudspeakers are some distance apart. They emit sound with a frequency of 1536 Hz. You move between the speakers, along the line joining them, at a constant speed of 2.8 m/s. What beat frequency do you observe? The speed of sound in the room is 330 m/s.

A) 13 Hz
B) 431 Hz
C) 26 Hz
D) 4.7 Hz
E) 118 Hz

Question 44

A puddle of water has a thin film of gasoline floating on it. A beam of light is shining perpendicular on the film. If the wavelength of light incident on the film is 560 nm and the indices of refraction of gasoline and water are 1.40 and 1.33, respectively, what is the minimum thickness of the film to see a bright reflection?

A) 100 nm
B) 200 nm
C) 300 nm
D) 400 nm
E) 500 nm

Question 45

A 360-nm thick oil film floats on the surface of the water. The indices of refraction of the oil and the water are 1.5 and 1.33, respectively. The surface of the oil is illuminated from above at normal incidence with white light. What TWO wavelengths of light in the 400-nm to 800-nm wavelength band are most strongly reflected?

A) 410 nm and 700 nm
B) 430 nm and 720 nm
C) 450 nm and 740 nm
D) 470 nm and 760 nm
E) 490 nm and 780 nm

Question 46

Radio station KBOB broadcasts at a frequency of 85.7 MHz on your dial using radio waves that travel at 3.00 × 10⁸ m/s. Since most of the station's audience is due south of the transmitter, the managers of KBOB don't want to waste any energy broadcasting to the east and west. They decide to build two towers, transmitting in phase at exactly the same frequency, aligned on an east-west axis. For engineering reasons, the two towers must be AT LEAST 10.0 m apart. What is the shortest distance between the towers that will eliminate all broadcast power to the east and west?

Question 47

A person is hearing two sound waves simultaneously. One has a period of 1.50 ms and the other one a period of 1.54 ms. What is the period of the beat due to these two waves?

A) 0.040 ms
B) 1.5 ms
C) 3.0 ms
D) 58 ms
E) 330 ms

Question 48

Two extremely flat glass plates, 10.0 cm wide, touch at one end but are separated by a thin wire at the other end, forming a wedge. Light with a wavelength of 450 nm shines almost perpendicularly on the glass and forms fringes which are 1.80 mm apart. What is the diameter of the wire?

A) 25.0 µm
B) 17.5 µm
C) 20.0 µm
D) 12.5 µm
E) 10.0 µm

Question 49

A coating is being applied to reduce the reflectivity of a pane of glass to light with a frequency of 5.75 × 10¹⁴ Hz that is incident normally on the pane. If the material has an index of refraction of 1.375 and the glass has an index of refraction of 1.537, what is the minimum thickness the coating should have? (c = 3.00 × 10⁸ m/s)

A) 60.0 nm
B) 94.9 nm
C) 145 nm
D) 65.2 nm
E) 80.1 nm

Question 50

Light of frequency 6.00 × 10¹⁴ Hz illuminates a soap film (n = 1.33) having air on both sides of it. When viewing the film by reflected light, what is the minimum thickness of the film that will give an interference maximum when the light is incident normally on it? (c = 3.00 × 10⁸ m/s)

A) 24.0 nm
B) 94.0 nm
C) 188 nm
D) 279 nm
E) 376 nm

Question 51

A thin layer of oil (n = 1.25) is on top of a puddle of water (n = 1.33). If normally incident 500-nm light is strongly reflected, what is the minimum nonzero thickness of the oil layer?

A) 200 nm
B) 250 nm
C) 100 nm
D) 400 nm
E) 150 nm

Question 52

Two loudspeakers placed 6.0 m apart are driven in phase by an audio oscillator whose frequency range is 2193 Hz to 2967 Hz. A point P is located 4.4 m from one loudspeaker and 3.6 m from the other. The speed of sound is 344 m/s. The frequency produced by the oscillator, for which constructive interference of sound occurs at point P, is closest to

A) 2580 Hz.
B) 2903 Hz.
C) 2473 Hz.
D) 2795 Hz.
E) 2688 Hz.

Question 53

Two motors in a factory are running at slightly different rates. One runs at 825 rpm and the other at 786 rpm. You hear the sound intensity increase and then decrease periodically due to wave interference. How much time elapses between successive maxima of the sound intensity?

A) 1.5 s
B) 1.4 s
C) 1.7 s
D) 1.8 s

Question 54

A carousel that is 5.00 m in radius has a pair of 600-Hz sirens mounted on posts at opposite ends of a diameter. The carousel rotates with an angular velocity of 0.800 rad/s. A stationary listener is located at a distance from the carousel. The speed of sound is 350 m/s. The maximum beat frequency of the sirens at the position of the listener is closest to

A) 6 Hz.
B) 8 Hz.
C) 10 Hz.
D) 12 Hz.
E) 14 Hz.

Question 55

Two harmonic sound waves reach an observer simultaneously. The observer hears the sound intensity rise and fall with a time of 0.200 s between the maximum intensity and the succeeding minimum intensity. What is the difference in frequency of the two sound waves?

A) 10.0 Hz
B) 0.200 Hz
C) 5.00 Hz
D) 2.50 Hz
E) 1.25 Hz

Question 56

The tension in each of two strings is adjusted so that both vibrate at exactly 666 Hz. The tension in one of the strings is then increased slightly. As a result, six beats per second are heard when both strings vibrate. What is the new frequency of the string that was tightened?

A) 672 Hz
B) 660 Hz
C) 669 Hz
D) 663 Hz

Question 57

A policeman in a stationary car measures the speed of approaching cars by means of an ultrasonic device that emits a sound with a frequency of 41.2 kHz. A car is approaching him at a speed of 33.0 m/s. The wave is reflected by the car and interferes with the emitted sound producing beats. What is the frequency of the beats? The speed of sound in air is 330 m/s.

A) 9.2 kHz
B) 4.1 kHz
C) 4.6 kHz
D) 1.2 kHz
E) 8.2 kHz

Question 58

White light is incident normally on a thin soap film having an index of refraction of 1.34. It reflects with an interference maximum at 684 nm and an interference minimum at 570 nm with no minima between those two values. The film has air on both sides of it. What is the thickness of the soap film?

A) 766 nm
B) 627 nm
C) 638 nm
D) 894 nm
E) 510 nm

Question 59

A bat emits a sound at a frequency of 30.0 kHz as it approaches a wall. The bat detects beats such that the frequency of the echo is 900 Hz higher than the frequency the bat is emitting. The speed of sound in air is 340 m/s. The speed of the bat is closest to

A) 20.0 m/s.
B) 530 m/s.
C) 10.0 m/s.
D) 30.0 m/s.
E) 5.02 m/s.

Question 60

A piece of glass has a thin film of gasoline floating on it. A beam of light is shining perpendicular on the film. If the wavelength of light incident on the film is 560 nm and the indices of refraction of gasoline and glass are 1.40 and 1.50, respectively, what is the minimum nonzero thickness of the film to see a bright reflection?

A) 500 nm
B) 400 nm
C) 300 nm
D) 200 nm
E) 100 nm

Question 61

Two optically flat glass plates, 16.0 cm long, are in contact at one end and separated by 0.0200 mm at the other end. The space between the plates is occupied by oil with index of refraction 1.45. The index of refraction of the glass plates is 1.55. The plates are illuminated at normal incidence with monochromatic light, and fringes are observed. If the dark fringes are spaced 2.00 mm apart, what is the wavelength of the monochromatic light?

A) 425 nm
B) 475 nm
C) 525 nm
D) 675 nm
E) 725 nm

Question 62

Light of wavelength 425.0 nm in air falls at normal incidence on an oil film that is 850.0 nm thick. The oil is floating on a water layer 15 cm thick. The refractive index of water is 1.33, and that of the oil is 1.40. You want to add oil so that light reflected off of the top of the oil film will be canceled. What is the minimum distance that you should INCREASE the oil film?

A) 60.7 nm
B) 75.9 nm
C) 106 nm
D) 121 nm
E) 152 nm

Question 63

A soap bubble, when illuminated with light of frequency 5.11 × 10¹⁴ Hz, appears to be especially reflective. If it is surrounded by air and if its index of refraction is 1.35, what is the thinnest thickness the soap film can be? (c = 3.00 × 10⁸ m/s)

Question 64

Light is incident normally from air onto a liquid film that is on a glass plate. The liquid film is 164 nm thick, and the liquid has index of refraction 1.60. The glass has index of refraction n = 1.50. Calculate the longest visible wavelength (as measured in air) of the light for which there will be totally destructive interference between the rays reflected from the top and bottom surfaces of the film. (Assume that the visible spectrum lies between 400 and 700 nm.)