Answer :
For the violin string with a mass of 0.45 g and a length of 32 cm, the wavelength of the first harmonic is approximately 72.73 cm, the wavelength of the third harmonic is approximately 24.24 cm, the velocity of the wave in the string is approximately 140.63 m/s, and the tension in the string should be approximately 31.81 N.
To determine the wavelength of the first harmonic (fundamental mode) of the violin string, we can use the formula λ = 2L, where λ represents the wavelength and L is the length of the string. Substituting the given values (L = 32 cm), we find that the wavelength of the first harmonic is approximately 72.73 cm.
For the wavelength of the third harmonic, we can use the formula λ = 2L/n, where n represents the harmonic number. For the third harmonic, n = 3. Substituting the given length (L = 32 cm), we find that the wavelength of the third harmonic is approximately 24.24 cm.
The velocity of the wave in the string can be calculated using the formula v = fλ, where v represents the velocity, f is the frequency, and λ is the wavelength. Given that the frequency (f) is 44 Hz and we have already determined the wavelength of the first harmonic (λ = 72.73 cm), we can find that the velocity of the wave in the string is approximately 140.63 m/s.
To find the tension in the string, we can use the formula T = μv^2, where T represents the tension, μ is the linear mass density (mass per unit length), and v is the velocity of the wave. The linear mass density (μ) is calculated by dividing the mass of the string (0.45 g) by its length (32 cm). Substituting the given values, we find that the tension in the string should be approximately 31.81 N.
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