Question

Question: Each string on a violin is tuned to a frequency $\mathbf{1}\frac{\mathbf{1}}{\mathbf{2}}$ times that of its neighbor. The four equal-length strings are to be placed under the same tension; what must be the mass per unit length of each string relative to that if the lowest string?

Short answer

The masses of string relative to string A are

${\mu }_B=0.44{\mu }_A$, ${\mu }_C=0.20{\mu }_A$, ${\mu }_D=0.088{\mu }_A$.

Step-by-step solution

Determination of relative mass

The string's frequency is proportional to the square root of the ratio of tensile force and mass of the string. Using this relation, the relative mass of the remaining string can be determined with respect to the first string.

Step 2:Given information 

The string of a violin is tuned to a frequency $1\frac{1}{2}$ times

Find the relative mass of each string with respect to the lowest string

Let the frequencies of four strings in wire are $f_A$, $f_B$, $f_C$ and $f_D$and among them $f_A$ is the lowest. The length of all the strings is same and have the same tension.

For the string $B$,

$\begin{array}{c}{f}_B=1.5f_A\\ \frac{1}{2l}\sqrt{\frac{F_T}{{\mu }_B}}=\left(1.5\right)\frac{1}{2l}\sqrt{\frac{F_T}{{\mu }_A}}\\ {\mu }_B=\frac{{\mu }_A}{{\left(1.5\right)}^2}\\ =0.44{\mu }_A\end{array}$

For the string C,

$\begin{array}{c}{f}_C=1.5f_B\\ f_C={\left(1.5\right)}^2{f}_B\\ \frac{1}{2l}\sqrt{\frac{F_T}{{\mu }_C}}={\left(1.5\right)}^2\frac{1}{2l}\sqrt{\frac{F_T}{{\mu }_C}}\\ {\mu }_C=\frac{{\mu }_A}{{\left(1.5\right)}^4}\\ {\mu }_C=0.20{\mu }_A\end{array}$

For the string D,

$\begin{array}{c}{f}_D=1.5f_C\\ f_D={\left(1.5\right)}^3{F}_A\\ \frac{1}{2l}\sqrt{\frac{F_T}{{\mu }_D}}={\left(1.5\right)}^3\frac{1}{2l}\sqrt{\frac{F_T}{{\mu }_A}}\\ {\mu }_D=\frac{{\mu }_A}{{\left(1.5\right)}^6}\\ {\mu }_D=0.088{\mu }_A\end{array}$

Thus, the masses of the string relative to string A are ${\mu }_B=0.44{\mu }_A$, ${\mu }_C=0.20{\mu }_A$, and ${\mu }_D=0.088{\mu }_A$.