Answer :
Final answer:
To find the tension in the violin string resonating in the fundamental mode, we need to calculate the fundamental frequency first using the given length of the string and the speed of sound in air. Then, we can use the relationship between the third overtone and the fundamental frequency to find the tension in the string using the formula T = (4 * L^2 * μ * f^2).
Explanation:
To solve this problem, we need to use the relationship between the fundamental and the third overtone of a resonating string. In this case, the violin string is resonating in the fundamental mode, so we can use the equation f3 = 3*f1.
We are given the length of the violin string (32.0 cm) and the speed of sound in air (343 m/s), so we can calculate the fundamental frequency (f1). Then, using the relationship between f3 and f1, we can find the tension in the string.
First, we need to calculate the fundamental frequency (f1) using the formula v = f1 * λ, where v is the speed of sound (343 m/s) and λ is the wavelength of the fundamental mode (2 * L, where L is the length of the string). Plugging in the values, we get f1 = v / (2 * L). Substituting the values, we get f1 = 343 / (2 * 0.32) = 535.9375 Hz.
Next, we can calculate the third overtone frequency (f3) using the equation f3 = 3 * f1. Substituting the value of f1, we get f3 = 3 * 535.9375 = 1607.8125 Hz.
Now, to find the tension in the string, we can use the formula f = 1 / (2 * L * √(T / μ)) where f is the frequency, L is the length of the string, T is the tension, and μ is the linear mass density (string density). Rearranging the formula to solve for T, we get T = (4 * L^2 * μ * f^2).
Plugging in the values, we get T = (4 * 0.32^2 * 1.5 * 10^-3 * 1607.8125^2) = 27.58 N.
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