The fundamental frequency of a sonometer wire carrying a block of mass ' $M$ ' and density ' $\rho$ ' is '…

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
  1. $\mathrm{n}\left[\frac{\rho-\sigma}{\rho}\right]^{\frac{1}{2}}$
  2. $\mathrm{n}\left[\frac{\rho-\sigma}{\sigma}\right]^{\frac{1}{2}}$
  3. $\mathrm{n}\left[\frac{\rho}{\rho-\sigma}\right]^{\frac{1}{2}}$
  4. $n\left[\frac{\sigma}{\rho-\sigma}\right]^{\frac{1}{2}}$

Solution

$\begin{aligned} & \mathrm{n} \propto \sqrt{\mathrm{T}} \\ & \mathrm{T}=\mathrm{mg}=\rho \mathrm{Vg} \\ \therefore \quad \mathrm{n} & \propto \sqrt{\rho \mathrm{Vg}} \end{aligned}$ After immersion in the liquid, $\begin{array}{ll} \therefore & \frac{\mathrm{n}_2}{\mathrm{n}} \propto \frac{\sqrt{\mathrm{V}(\rho-\sigma) \mathrm{g}}}{\sqrt{\mathrm{V} \rho \mathrm{g}}} \\ \therefore & \mathrm{n}_2=\mathrm{n}\left[\frac{\rho-\sigma}{\rho}\right]^{\frac{1}{2}} \end{array}$

Asked in: MHT CET 2023 (13 May Shift 1)

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