The frequency of a sonometer wire is $f$. The frequency becomes $f/2$ when the mass producing the tension is…

The frequency of a sonometer wire is $f$. The frequency becomes $f/2$ when the mass producing the tension is completely immersed in water and on immersing the mass in a certain liquid, frequency becomes $f/3$. The relative density of the liquid is
  1. (a) 1.32
  2. (b) 1.03
  3. (c) 1.41
  4. (d) 1.18

Solution

Frequency $\propto \sqrt{T}$ $\frac{f}{f / 2} = \sqrt{\frac{T}{T - F_w}}$ ($F_w = \text{upthrust in water}$) $\therefore 1 - \frac{F_w}{T} = \frac{1}{4}$ or $\frac{F_w}{T} = \frac{3}{4}$ ...(i) In second case, $\frac{f}{f / 3} = \sqrt{\frac{T}{T - F_l}}$ ($F_l = \text{upthrust in liquid}$) $\Rightarrow 1 - \frac{F_l}{T} = \frac{1}{9}$ $\therefore \frac{F_l}{T} = \frac{8}{9}$ ...(ii) From Eqs. (i) and (ii), we get $\frac{F_l}{F_w} = \frac{8}{9} \times \frac{4}{3} = 1.18$ Now, upthrust is in the ratio of relative density.

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