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The fundamental frequency of a sonometer wire carrying a block of mass ' $M$ ' and density ' $\rho$ ' is ' $n$ ' Hz. When the block is completely immersed in a liquid of density ' $\sigma$ ' then the new frequency will be
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Verified Answer
The correct answer is:
$\mathrm{n}\left[\frac{\rho-\sigma}{\rho}\right]^{\frac{1}{2}}$
$$
\begin{aligned}
& \mathrm{n} \propto \sqrt{\mathrm{T}} \\
& \mathrm{T}=\mathrm{mg}=\rho \mathrm{Vg} \\
\therefore \quad \mathrm{T} & \propto \sqrt{\rho \mathrm{Vg}}
\end{aligned}
$$
After immersion in the liquid,
$$
\begin{array}{ll}
\therefore & \frac{\mathrm{n}_2}{\mathrm{n}} \propto \frac{\sqrt{\mathrm{V}(\rho-\sigma) \mathrm{g}}}{\sqrt{\mathrm{V} \rho \mathrm{g}}} \\
\therefore & \mathrm{n}_2=\mathrm{n}\left[\frac{\rho-\sigma}{\rho}\right]^{\frac{1}{2}}
\end{array}
$$
\begin{aligned}
& \mathrm{n} \propto \sqrt{\mathrm{T}} \\
& \mathrm{T}=\mathrm{mg}=\rho \mathrm{Vg} \\
\therefore \quad \mathrm{T} & \propto \sqrt{\rho \mathrm{Vg}}
\end{aligned}
$$
After immersion in the liquid,
$$
\begin{array}{ll}
\therefore & \frac{\mathrm{n}_2}{\mathrm{n}} \propto \frac{\sqrt{\mathrm{V}(\rho-\sigma) \mathrm{g}}}{\sqrt{\mathrm{V} \rho \mathrm{g}}} \\
\therefore & \mathrm{n}_2=\mathrm{n}\left[\frac{\rho-\sigma}{\rho}\right]^{\frac{1}{2}}
\end{array}
$$
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