Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique.

| 56 | Concept Questions Both drawings show the same square, at one corner of which an observer O is stationed. Two loudspeakers are located at corners of the square, either as in drawing 1 or as in drawing 2. The speakers produce the same single-frequency tone in either drawing and are in phase. Constructive interference occurs in drawing 1, but destructive interference occurs in drawing 2. (a) Will only certain frequencies lead to the constructive interference in drawing 1, or will it occur for any frequency at all? (b) Will only certain frequencies lead to the destructive interference in drawing 2, or will it occur for any frequency at all? Justify your answers.
Problem One side of the square has a length of L = 0.75 m. The speed of sound is 343 m/s. Find the single smallest frequency that will produce both constructive interference in drawing 1 and destructive interference in drawing 2.

| 57 | Concept Questions(a) When sound emerges from a loudspeaker, is the diffraction angle determined by the wavelength, the diameter of the speaker, or a combination of these two factors? (b) How is the wavelength of a sound related to its frequency? Explain your answers.
Problem The following two lists give diameters and sound frequencies for three loudspeakers. Pair each diameter with a frequency, so that the diffraction angle is the same for each of the speakers. The speed of sound is 343 m/s. Find the common diffraction angle.

Diameter, D | Frequency, f
0.050 m | 6.0 Khz
0.10 m | 4.0 kHz
0.15 m | 12.0 kHz

| 58 | Concept Questions Two cars have identical horns, each emitting a frequency fs. One of the cars is moving toward a bystander waiting at a corner, and the other is parked. The two horns sound simultaneously. (a) From the moving horn, does the bystander hear a frequency that is greater than, less than, or equal to fs? (b) From the stationary horn, does the bystander hear a frequency that is greater than, less than, or equal to fs? (c) Does the bystander hear a beat frequency from the combined sound of the two horns? Account for your answers.
Problem The frequency that the horns emit is fs = 395 Hz. The speed of the moving car is 12.0 m/s, and the speed of sound is 343 m/s. What is the beat frequency heard by the bystander?

| 59 | Concept Questions Two wires are stretched between two fixed supports and have the same length. On wire A there is a second-harmonic standing wave whose frequency is 660 Hz. However, the same frequency of 660 Hz is the third harmonic on wire B. (a) Is the fundamental frequency of wire A greater than, less than, or equal to the fundamental frequency of wire B? Explain. (b) How is the fundamental frequency related to the length L of the wire and the speed v at which individual waves travel back and forth on the wire? (c) Do the individual waves travel on wire A with a greater, smaller, or the same speed as on wire B? Give your reasoning.
Problem The common length of the wires is 1.2 m. Find the speed at which individual waves travel on each wire. Verify that your answer is consistent with your answers to the Concept Questions.

| 60 | Concept Questions A copper block is suspended in air from a wire in Part 1 of the drawing. A container of mercury is then raised up around the block as in Part 2. (a) The fundamental frequency of the wire is given by Equation 17.3 with n = 1: f1 = v/(2L). How is the speed v at which individual waves travel on the wire related to the tension in the wire? (b) Is the tension in the wire in Part 2 less than, greater than, or equal to the tension in Part 1 ? (c) Is the fundamental frequency of the wire in Part 2 less than, greater than, or equal to the fundamental frequency in Part 1? Justify each of your answers.

Problem In Part 2 of the drawing some (50.0 %) of the block’s volume is submerged in the mercury. The density of copper is 8890 kg/m3, and the density of mercury is 13 600 kg/m3. Find the ratio of the fundamental frequency of the wire in Part 2 to the fundamental frequency in Part 1. Check to see that your answer is consistent with your answers to the Concept Questions.

| 61 | Concept Questions One method for measuring the speed of sound uses standing waves. A cylindrical tube is open at both ends, and one end admits sound from a tuning fork. A movable plunger is inserted into the other end at a distance L from the end of the tube where the tuning fork is. For a fixed frequency, the plunger is moved until the smallest value of L is measured that allows a standing wave to be formed. (a) When a standing wave is formed in the tube, is there a displacement node or antinode at the end of the tube where the tuning fork is, and is there a displacement node or antinode at the plunger? (b) How is the smallest value of L related to the wavelength of the sound? Explain your answers.
Problem The tuning fork produces a 485-Hz tone, and the smallest value observed for L is 0.264 m. What is the speed of the sound in the gas in the tube?

| 62 | Concept Questions A listener is standing in front of two speakers that are producing sound of the same frequency and amplitude, except that they are vibrating out of phase. Initially, the distance between the listener and each speaker is the same (see the drawing). (a) What is meant by the phrase “the speakers are vibrating out of phase?” (b) When the sound waves reach the listener, would they exhibit constructive or destructive interference? Provide a reason for your answer. (c) When the listener begins to move sideways (to the right in the drawing), would the sound intensity increase or decrease? Why?
Problem As the listener moves sideways, the sound intensity gradually changes. When the distance x in the drawing is 0.92 m, the change reaches the maximum amount (either loud to soft, or soft to loud). Using the data shown in the drawing and 343 m/s for the speed of sound, determine the frequency of the sound from the speakers.

| 63 | Concept Questions Standing waves are set up on two strings fixed at each end, as shown in the drawing. The tensions in the strings are the same, and each string has the same mass per unit length. However, one string is longer than the other, (a) Do the waves on the longer string have a larger speed, a smaller speed, or the same speed as those on the shorter string? Justify your answer. (b) Will the longer string vibrate at a higher frequency, a lower frequency, or the same frequency as the shorter string? Provide a reason for your answer. (c) Will the beat frequency produced by the two standing waves increase or decrease if the longer string is increased in length? Why?
Problem The two strings in the drawing have the same tension and mass per unit length, but they differ in length by 0.57 cm. The waves on the shorter string propagate with a speed of 41.8 m/s, and the fundamental frequency of the shorter string is 225 Hz. Determine the beat frequency produced by the two standing waves.

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