The lower A on a piano has a frequency of 27.5 Hz. If the tension in the 2.0-m-long string is 304 N and one-half wavelength occupies the string, what is the mass of the string?
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Answer:
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The mass of the string is 0.05 kg.
A wave on a string moves at a specific pace because of:
where
v is the speed of the wave,
T is the tension in the string,
μ is the string linear mass density.
The wave's frequency is determined by:
f = n × v/2L
where
n is the harmonic number,
L is the length of the string,
v is the speed of the wave.
The wavelength is twice as long as the string at the fundamental frequency (n=1), so:
λ = 2L/n = 2L
The wavelength of the harmonic with a half wavelength is four times the length of the string, so:
λ = 2L/n = 4L
The harmonic with a half-wavelength frequency is:
f = n × v/2L = v/λ = v/4L
The lower A on a piano has a frequency of 27.5 Hz, therefore
v/4L = 27.5 Hz
By solving for v:
v = 4L × 27.5 Hz = 110 Hz
By substituting into the expression for the speed of the wave:
110 Hz =
By squaring both sides:
12100 = T/μ
By substituting T=304 N:
12100 = 304 N/μ
By solving for μ:
μ = 304 N / 12100 = 0.025 kg/m
The string has a linear mass density of 0.025 kg/m.
Given that the string is 2.0 m in length, the mass of the string is:
m = μ × L
m = 0.025 kg/m × 2.0 m
m = 0.05 kg
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