Q. An object of density $2000 , k-gm^{-3}$ is hung from a thin light wire. The fundamental frequency of the transverse waves in the wire is $200, Hz$. If the object is immersed in water such that half of its volume is submerged, then the fundamental frequency of the transverse waves in the wire is

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An object of density $2000 , k-gm^{-3}$ is hung from a thin light wire. The fundamental frequency of the transverse waves in the wire is $200, Hz$. If the object is immersed in water such that half of its volume is submerged, then the fundamental frequency of the transverse waves in the wire is

Solution:

Fundamental frequency, initially

$n_{1}=frac{1}{2 l} sqrt{frac{T}{mu}}=frac{1}{2 l} sqrt{frac{V g(2000)}{mu}}$

Fundamental frequency, finally

$n_{2}=frac{1}{2 l} sqrt{frac{operatorname{Vg}left(2000-frac{1000}{2}right)}{mu}}$

(here we applied loss of weight due to upthrust)

$=frac{1}{2 l} sqrt{frac{V g times 1500}{mu}}$

$Rightarrow frac{n_{1}}{n_{2}}=sqrt{frac{2000}{1500}}=sqrt{frac{4}{3}}$

$Rightarrow n_{2}=frac{n_{1} times sqrt{3}}{2}=100 sqrt{3}, Hz $

$=173.2, Hz$