Answer :
Final answer:
The ratio of the mass of the A string to that of the E string on a violin is approximately 0.682.
Explanation:
The ratio of the mass of the A string to that of the E string can be determined using the formula for the speed of a wave on a string. The speed of a wave on a string is given by the equation v = √(F/μ), where v is the speed of the wave, F is the tension in the string, and μ is the linear mass density of the string. Since the frequency of a wave is directly proportional to the speed of the wave, we can write the equation f = (1/2L)√(F/μ), where f is the frequency, L is the length of the string, F is the tension in the string, and μ is the linear mass density of the string.
Let's assume that the lengths of the A and E strings are the same, so we can cancel out the term (1/2L). We can then write the equation as f₁/μ₁ = f₂/μ₂, where f₁ and f₂ are the frequencies of the A and E strings respectively, and μ₁ and μ₂ are the linear mass densities of the A and E strings respectively.
Plugging in the given frequencies, f₁ = 450 Hz and f₂ = 659 Hz, we can solve for the ratio of the linear mass densities: μ₁/μ₂ = f₁/f₂ = 450/659 ≈ 0.682.
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