Question

A particular guitar wire is $30.0cm$ long and vibrates at a frequency of $196Hz$ when no finger is placed on it. The next higher notes on the scale are $220Hz,247Hz,262Hz\&294Hz$. How far from the end of the string must the finger be placed to play these notes?

Answer 1

Step$1$. Given Data:

$$\begin{gathered} \Rightarrow l_1=30cm \\ or{l}_1=0.3m \\ \Rightarrow f_1=196Hz \\ \Rightarrow f_2=220Hz \\ \Rightarrow f_3=247Hz \\ \Rightarrow f_4=262Hz \end{gathered}$$

Step$2$. We know that:

$$\begin{gathered} f\propto \frac{1}{l} \\ orf=\frac{v}{l} \end{gathered}$$ (where, $v$= constant for a medium)

Step$3$. Now:

$$\begin{gathered} \Rightarrow \frac{f_1}{f_2}=\frac{l_2}{l_1} \\ \Rightarrow l_2=26.7cm \end{gathered}$$

Again,

$f_3=247Hz$

Step 4. Further:

$$\begin{gathered} \Rightarrow \frac{f_3}{f_1}=\frac{l_1}{l_3} \\ \Rightarrow \frac{f_3}{f_1}=\frac{0.3}{l_3} \\ \Rightarrow l_3=0.224m \\ or{l}_3=22.4cm \end{gathered}$$

In the same way, we have

$\Rightarrow l_4=20cm$

Hence, the finger should be placed at $26.7cm,22.4cm$ & $20cm$ to play these notes respectively.