Question

A piano wire $0.5m$ long and mass $5gm$ is stretched by a tension of $400N$. The number of highest overtone that can be heard by a person is

• A. $160$
• B. $99$
• C. $140$
• D. $120$

Answer 1

The correct option is B $99$

Given, $l=0.5m$

$m=5g=5\times {10}^{-3}kg,T=400N$ .

Fundamental frequency of the wire is given by ,

${\nu }_1=\frac{1}{2l}\sqrt{Tl/m}$ ,

${\nu }_1=\frac{1}{2\times 0.5}\sqrt{\frac{400\times 0.5}{5\times {10}^{-3}}}$

or ${\nu }_1=200Hz$

Now , let $nth$ overtone (highest) can be heard by a person , as the highest frequency that can be heard by human ear is $20,000Hz$ therefore frequency of $nth$ overtone should be equal to $20,000Hz$ ,

Hence , frequency of nth overtone$=20,000$ ,

or $(n+1){\nu }_1=20000$ ,

or $(n+1)200=20000$ ,

or $n+1=100$ ,

or $n=100-1=99$