An object of density $2000 \mathrm{~kg}-\mathrm{m}^{-3}$ is hung from a thin light wire. The fundamental…
An object of density $2000 \mathrm{~kg}-\mathrm{m}^{-3}$ is hung from a thin light wire. The fundamental frequency of the transverse waves in the wire is $200 \mathrm{~Hz}$. 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
200 Hz
173.2 Hz
100 Hz
141.4 Hz
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{V g\left(2000-\frac{1000}{2}\right)}{\mu}}
$
(here we applied loss of weight due to upthrust)
$
\begin{aligned}
& =\frac{1}{2 l} \sqrt{\frac{V g \times 1500}{\mu}} \\
\Rightarrow \quad \frac{n_1}{n_2} & =\sqrt{\frac{2000}{1500}}=\sqrt{\frac{4}{3}} \\
\Rightarrow \quad n_2 & =\frac{n_1 \times \sqrt{3}}{2}=100 \sqrt{3} \mathrm{~Hz}=173.2 \mathrm{~Hz}
\end{aligned}
$