A guitar string of length $90 \mathrm{cm}$ vibrates with a fundamental frequency of $120 \mathrm{Hz}$. The length of the string producing a fundamental of $180 \mathrm{Hz}$ will be _____ $\mathrm{cm}$

The formula to calculate the fundamental frequency produced in a vibrating string is given by

$f_0=\frac{v}{2L} ...\left(1\right)$

For the first case, it can be written that

$120=\frac{v}{2\left(90\right)} ...\left(2\right)$

Similarly, for the second case, it can be written that

$180=\frac{v}{2\left(L\right)} ...\left(3\right)$

Divide equation (3) by equation (2) and solve to obtain the required length of the string.

$\begin{array}{rcl}\frac{180}{120}& =& \frac{{\displaystyle \frac{v}{2\left(L\right)}}}{{\displaystyle \frac{v}{2\left(90\right)}}}\\ & =& \frac{90}{L}\\ & \Rightarrow & L= 90\times \frac{120}{180}\\ & =& 60\end{array}$