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Answer :
The length of the vibrating section of the violin string is approximately 3.17 meters, given a linear density of 0.660 g/m and a tension of 140 N. This is calculated using the formula for the fundamental frequency of a vibrating string and rearranging it to solve for length.
Given:
Linear density (μ) = 0.660 g/m
Tension (T) = 140 N
To find the length of the vibrating section (L), we can use the formula for the fundamental frequency of a vibrating string:
[tex]f = (1/2L) * √(T/μ)[/tex]
Given that the fundamental frequency (f) for a violin string is typically 440 Hz, we can rearrange the formula to solve for L:
[tex]L = (1/2f) * √(T/μ)[/tex]
Substituting the given values:
[tex]L = (1/2 * 440) * √(140 / 0.660)[/tex]
[tex]L = (0.5 * 440) * √(212.121)[/tex]
[tex]L ≈ (220) * 14.567 ≈ 3207.5 ≈ 3.17 m[/tex]
Therefore, the length of the vibrating section of the violin string is approximately 3.17 meters.
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