Question

A $30\text{cm}$ violin string under a tension of $25\text{N}$ has a mass per unit length of $2.5\text{gm/cm}$. What is the highest resonance frequency of the string that can be heard by a person able to hear frequencies upto $20000\text{Hz}$?

• A. $20000\text{Hz}$
• B. $1999\text{Hz}$
• C. $1997\text{Hz}$
• D. $19973\text{Hz}$

Answer 1

The correct option is A $20000\text{Hz}$

Given that
$l=30\text{cm}=0.3\text{m}$
$T=25\text{N}$
$\mu =2.5\text{g/cm}=0.25\text{kg/m}$

So, fundamental frequency $\left(f\right)$,

$f=\frac{v}{2l}=\frac{1}{2l}\sqrt{\frac{T}{\mu }}=\frac{\sqrt{\frac{25}{0.25}}}{2\times 0.3}$

$\Rightarrow f=\frac{\sqrt{100}}{2\times 0.3}=\frac{50}{3}\text{Hz}$

Let $n^{th}$ harmonic be the highest frequency heard by a person, then

$\frac{50}{3}\times n=20000\Rightarrow n=1200$

Hence, ${1200}^{th}$ harmonic is the highest frequency heard by the person.

$\therefore f_{max}=1200\times \frac{50}{3}=20000\text{Hz}$

Hence, $\left(A\right)$ is the correct answer.