Question

The two highest-pitch strings on a violin are tuned to 450 Hz (the A string) and 659 Hz (the E string). What is the ratio of the mass of the A string to that of the E string? Violin strings are all the same length and under essentially the same tension.

Answer 1

Answer: The ratio is 2.14

Explanation:

Please see the attachments below

Answer 2

The ratio of the mass of the A string to that of the E string is 2.14.

The given parameters;

• frequency of string A, f₁ = 450 Hz
• frequency of string E, f₂ = 659 Hz

The frequency of wave on the string at a given mass per unit length is calculated as follows;

$F = \frac{1}{\lambda} \sqrt{\frac{T}{\mu} } \\\\F_1\sqrt{\mu_1} = F_2\sqrt{\mu _2} \\\\F_A\sqrt{\mu_A} = F_E\sqrt{\mu _E}\\\\\sqrt{\frac{\mu_A}{\mu_ E} } = \frac{F_E}{F_A} \\\\\frac{\mu_A}{\mu_ E} = \frac{F_E^2}{F_A^2} \\\\\frac{M_A/L}{M_E/L} = (\frac{F_E}{F_A} )^2\\\\\frac{M_A}{M_E} = (\frac{F_E}{F_A} )^2\\\\\frac{M_A}{M_E} = (\frac{659}{450} )^2\\\\\frac{M_A}{M_E} = 2.14$

Thus, the ratio of the mass of the A string to that of the E string is 2.14.

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