Question

Suppose that the strings on a violin are stretched with the same tension and each has the same length between its two fixed ends. The musical notes and corresponding fundamental frequencies of two of these strings are G (196.0 Hz) and E (659.3 Hz). The linear density of the E string is 3.47 10-4 kg/m. What is the linear density of the G string?

Answer 1

Answer:

$3.93\times10^-^3kg.m^-^1$

Explanation:

The length,$L$ of the string , fundamental frequency $(f)$ and the tension $(F)$ on the string are related as:

$L=\frac{1}{2f_1}\sqrt{\frac{F}{(m/L)}}\\\\\frac{2L}{\sqrt F}=\frac{1}{f_1\sqrt{(m/L)}}$

#Since both E and G have the same length and tension on them:

$\frac{1}{f_1_,_G\sqrt{(m/L)_G}}=\frac{1}{f_1_,_E\sqrt{(m/L)_E}}$

Where $(m/L)_i$ are the linear densities, $f_1_,_i$ the fundamental frequencies.

#taking square and inverse on both sides, we have:

$f^2_1_,_G(m/L)_G=f^2_1_,_E(m/L)_E\\\\(m/L)_G=\frac{f^2_1_,_E}{f^2_1_,_G}(m/L)_E\\\\(m/L)_G=\frac{(659.3^2)}{(196^2)}\times 3.47\times 10^-^4=3.93\times10^-^3kg.m^-^1$

Hence, the linear density of the G string is 0.00393kg/m