Question

A guitar string is 90 cm long and has a fundamental frequency of 124Hz . Where should it be pressed to produce a fundamental frequency of 186Hz ?​

Answer 1

Since the both ends of the guitar string are fixed, the fundamental frequency,

$\sf{\nu=\dfrac{v}{2L}}\quad\longrightarrow\quad(1)$

because the guitar string is similar to an open pipe.

Since the velocity of the wave is constant, (1) implies,

$\sf{\nu\propto\dfrac{1}{L}}$

Then we form an equation,

$\sf{\dfrac{\nu_1}{\nu_2}=\dfrac{L_2}{L_1}}$

[Note : Here $\sf{\nu_2}$ is the new fundamental frequency. Don't confuse it with first overtone.]

In the question,

$\sf{L_1=90\ cm}\\\\\\\sf{\nu_1=124\ Hz}\\\\\\\sf{\nu_2=186\ Hz}$

We have to find $\sf{L_2.}$ Then,

$\sf{L_2}=\dfrac{L_1\nu_1}{\nu_2}\\\\\\\sf{L_2=\dfrac{90\times124}{186}}\\\\\\\sf{\Large\underline{\underline{L_2=60\ cm}}}$

So the guitar should be pressed at 60 cm from any of its ends.