Answer :
The frequency of the violin string when played with a finger positioned one-fourth of the string length from the neck end is 860 Hz.
When a violinist puts a finger down on a string, they effectively shorten the vibrating length of the string, causing it to produce a higher pitch. In this case, the finger is placed one-fourth of the string length from the neck end.
To find the new frequency, we can use the formula:
[tex]\[ f' = \frac{f}{n} \][/tex]
Where:
- [tex]\( f' \)[/tex] = new frequency
- [tex]\( f \)[/tex] = original frequency
- [tex]\( n \)[/tex] = fraction of the string length that the finger is placed from the end
Given that the finger is placed one-fourth of the string length from the neck end ([tex]\( n = \frac{1}{4} \)[/tex]), and the original frequency [tex]\( f = 430 \)[/tex] Hz, we can calculate the new frequency as follows:
[tex]\[ f' = \frac{430}{\frac{1}{4}} \][/tex]
[tex]\[ f' = 430 \times 4 \][/tex]
[tex]\[ f' = 1720 \][/tex]
However, this calculation gives us the frequency of the first harmonic, which is not the fundamental frequency we are seeking. The fundamental frequency is the lowest resonant frequency of the string.
Since we know that the finger placement creates a new node, dividing by 2 will give us the fundamental frequency.
[tex]\[ f_{fundamental} = \frac{1720}{2} \][/tex]
[tex]\[ f_{fundamental} = 860 \, \text{Hz} \][/tex]
Therefore, the frequency of the string when played with a finger positioned one-fourth of the string length from the neck end is 860 Hz.
