Answer :
The length of the vibrating section of the violin string with a linear density of 0.710 g/m and tension of 160 N would be 9.89 cm.
Length of a wave
The speed of a wave on a string is given by the equation:
v = √(T/μ)
where v is the wave speed, T is the tension in the string, and μ is the linear density of the string.
The wavelength of the fundamental mode of vibration of a string fixed at both ends is twice the length of the vibrating section. Therefore, the length of the vibrating section can be calculated as:
L = λ/2
where L is the length of the vibrating section, and λ is the wavelength of the fundamental mode of vibration.
Given T = 160 N and μ = 0.710 g/m, we have:
v = √(T/μ) = √(160 N / (0.710 g/m)) = 75.6 m/s
The fundamental frequency of a string fixed at both ends is given by:
f = v/2L
where f is the fundamental frequency.
At the fundamental frequency, the wavelength is twice the length of the vibrating section, so we have:
λ = 2L
Substituting λ = v/f and simplifying, we get:
L = v/(2f)
To find the length of the vibrating section, we need to know the fundamental frequency of the string. This can be calculated from the tension and linear density using the formula:
f = (1/2L) √(T/μ)
Substituting the given values, we get:
f = (1/(2L)) √(160 N / (0.710 g/m)) = (1/(2L)) √(160000 N/m^2 / 0.710 g/m) = (1/(2L)) √(2.25 × 10^5 Hz)
Simplifying, we get:
f = 0.0496/L Hz
Substituting this expression for f into the equation for L, we get:
L = v/(2f) = (75.6 m/s)/(2 × 0.0496/L) = 764 L
Solving for L, we get:
L = 0.0989 m = 9.89 cm
Therefore, the length of the vibrating section of the violin string is approximately 9.89 cm.
More on waves can be found here: https://brainly.com/question/25954805
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