Question

Question: (I) The A string on a violin has a fundamental frequency of 440 Hz. The length of the vibrating portion is 32 cm, and it has mass 0.35 g. Under what tension must the string be placed?

Short answer

The string should have $86.7\mathrm{N}$ tension.

Step-by-step solution

Step-1:-Understanding tension on a string

In order to find the tension on a string, use the relation of tension force with frequency and mass. Also, the adjustment in the tension of a string affects the string's frequency.

Step-2:-Given data

The fundamental frequency is $f_1=440\mathrm{H}\mathrm{z}$.

The length of the string is $l=32\mathrm{c}\mathrm{m}$.

The mass of the string is $m=0.35\mathrm{g}$.

Step-3:-Calculation of tension on the A string

The relation of tension on the string is given by,

$F_{\mathrm{T}}=4l{f}_1^{2}m$

Substitute the values in the above relation.

$\begin{array}{c}{F}_{\mathrm{T}}=4\times \left(32\mathrm{c}\mathrm{m}\left(\frac{1\mathrm{m}}{100\mathrm{c}\mathrm{m}}\right)\right)\times {\left(440\mathrm{H}\mathrm{z}\right)}^2\times \left(0.35\mathrm{g}\left(\frac{1\mathrm{k}\mathrm{g}}{1000\mathrm{g}}\right)\right)\\ F_{\mathrm{T}}=86.7\mathrm{N}\end{array}$

The tension of the string is $86.7\mathrm{N}$.