A string is stretched horizontally, and one end is connected to the tip of a mechanical oscillator. The other end extends over a pulley and supports a total hanging mass of 300 grams. The string is L = 1.74 meters long, from the point of attachment to the oscillator to the point of contact with the pulley. The patterns shown in the figure (NOT ACTUAL SIZE) are observed at the frequencies indicated. a. Make a (large) graph of wavelength vs. period for the first 3 patterns. Include units on your axes. b. Using your graph, determine the wave speed on this stretched string? Include units. c. Estimate the mass per unit length of the string, and please include units! d. What is the frequency f4 of the pattern shown?

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Chapter16: Waves
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Problem 16.7CYU: Check Your Understanding The equations for the wavelengths and the frequencies of the modes of a...
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Standing Waves
Given the total string length L, pictures of a few standing wave patterns, and the frequencies
at which they are observed, deduce some quantities, given sufficient information. You should
be able to graph wavelength vs. period data to find quantities such as the mass per unit
length of the string, the tension in the string, or the mass hanging from the end. All graphs
that are used to determine a slope must be sufficiently large (at least 1/2 page).
EXAMPLE
A string is stretched horizontally, and one end is connected to the tip of a mechanical
oscillator. The other end extends over a pulley and supports a total hanging mass of 300
grams. The string is L = 1.74 meters long, from the point of attachment to the oscillator to the
point of contact with the pulley. The patterns shown in the figure (NOT ACTUAL SIZE) are
observed at the frequencies indicated.
a. Make a (large) graph of wavelength vs. period for the first 3 patterns. Include units on your
axes.
b. Using your graph, determine the wave speed on this stretched string? Include units.
c. Estimate the mass per unit length of the string, and please include units!
d. What is the frequency f4 of the pattern shown?
23.6 Hz
59.0 Hz
11.8 Hz
f4 = ??? Hz
L
Rough solution: from the graph, we find v = 41 m/s.
From the fact that the string supports a hanging mass of 300 grams, we know the string
tension is (0.300kg) (9.8 m/s²)=2.94 newtons.
We know how to relate wave speed, tension, and mass per unit length, so we solve for
1.75e-3 kg/m.
Looking at the pattern, the wavelength is 1.16 meters, v is about 41 m/s, solve for
frequency of about 35 Hz. (You could also be asked for the corresponding period T.)
EXAMPLE 2
Suppose you knew the mass per unit length of the string, 1.75e-3 kg/m, and its total length,
1.74 m, but didn't know the hanging mass. Using the same diagrams as above, you should
know how to find the tension, and the amount of hanging mass (0.300kg).
EXAMPLE 3
A string is under a tension of 120.0 N. A 1.600 m length of the string has a mass of 5.700
grams.
a. What is the speed of a transverse wave of wavelength 0.6000 m in this string? (183.5 m/s)
b. The wavelength is 0.60 m. What is the frequency of the wave? (305.9 Hz)
Transcribed Image Text:Standing Waves Given the total string length L, pictures of a few standing wave patterns, and the frequencies at which they are observed, deduce some quantities, given sufficient information. You should be able to graph wavelength vs. period data to find quantities such as the mass per unit length of the string, the tension in the string, or the mass hanging from the end. All graphs that are used to determine a slope must be sufficiently large (at least 1/2 page). EXAMPLE A string is stretched horizontally, and one end is connected to the tip of a mechanical oscillator. The other end extends over a pulley and supports a total hanging mass of 300 grams. The string is L = 1.74 meters long, from the point of attachment to the oscillator to the point of contact with the pulley. The patterns shown in the figure (NOT ACTUAL SIZE) are observed at the frequencies indicated. a. Make a (large) graph of wavelength vs. period for the first 3 patterns. Include units on your axes. b. Using your graph, determine the wave speed on this stretched string? Include units. c. Estimate the mass per unit length of the string, and please include units! d. What is the frequency f4 of the pattern shown? 23.6 Hz 59.0 Hz 11.8 Hz f4 = ??? Hz L Rough solution: from the graph, we find v = 41 m/s. From the fact that the string supports a hanging mass of 300 grams, we know the string tension is (0.300kg) (9.8 m/s²)=2.94 newtons. We know how to relate wave speed, tension, and mass per unit length, so we solve for 1.75e-3 kg/m. Looking at the pattern, the wavelength is 1.16 meters, v is about 41 m/s, solve for frequency of about 35 Hz. (You could also be asked for the corresponding period T.) EXAMPLE 2 Suppose you knew the mass per unit length of the string, 1.75e-3 kg/m, and its total length, 1.74 m, but didn't know the hanging mass. Using the same diagrams as above, you should know how to find the tension, and the amount of hanging mass (0.300kg). EXAMPLE 3 A string is under a tension of 120.0 N. A 1.600 m length of the string has a mass of 5.700 grams. a. What is the speed of a transverse wave of wavelength 0.6000 m in this string? (183.5 m/s) b. The wavelength is 0.60 m. What is the frequency of the wave? (305.9 Hz)
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