Given,
The length of the string, L₁=25.4 cm
The fundamental frequency, f₁=440 Hz
The new frequency, f₂=523.3 Hz
The frequency of a standing wave is related to the length of the string as,
[tex]f=\frac{v}{2L}[/tex]On rearranging the above equation,
[tex]\begin{gathered} fL=\frac{v}{2} \\ \Rightarrow fL=\text{ constant} \\ \Rightarrow f_1L_1=f_2L_2 \end{gathered}[/tex]Where L₂ is the shortened length of the string.
On substituting the known values,
[tex]\begin{gathered} 440\times25.4=523.3\times L_2 \\ L_2=\frac{440\times25.4}{523.3} \\ =21.36\text{ cm} \end{gathered}[/tex]Therefore the shortened length of the string in order to produce the required frequency is 21.36 cm.
