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The frequency of a sonometer wire is $100 Hz$. When the weights producing the tensions are completely immersed in water, the frequency becomes $80 Hz$ and on immersing the weights in a certain liquid, the frequency becomes $60 Hz$. The specific gravity of the liquid is
$1.42$
$1.77$
$1.82$
$1.21$
Solution
(b)
As we know, frequency
$f \propto \sqrt{m g}$ or $f \propto \sqrt{g}$
In water, $f _{ w }=0.8 f _{ air }$
$\frac{ g ^{\prime}}{ g }(0.8)^2=0.64$
$\Rightarrow 1-\frac{\rho_{ w }}{\rho_{ m }}=0.64$
$\Rightarrow \frac{\rho_{ W }}{\rho_{ m }}=0.36$
In liquid, $\frac{ g ^{\prime}}{ g }=(0.6)^2=0.36$
$1-\frac{\rho_1}{\rho_{ m }}=0.36 \frac{\rho_1}{\rho_{ m }}=0.64$
From eq. $(1)$ and $(2)$
$\frac{\rho_l}{\rho_n}=\frac{0.64}{0.36} \quad \therefore \quad \rho_l=1.77$
