Q145 / 150 · MHT CET 2024 (09 May Shift 2) · Physics
A sonometer wire is in unison with a tuning fork of frequency ' $n$ ' when it is stretched by a weight of specific gravity ' $d$ '. When the weight is completely immersed in water, ' $x$ ' beats are produced per second, then
- A$\frac{\mathrm{n}}{\mathrm{n}-\mathrm{x}}=\frac{\mathrm{d}}{\mathrm{d}-1}$
- Correct answerB$\frac{n}{n-x}=\sqrt{\frac{d}{d-1}}$
- C$\frac{n-x}{n}=\frac{d-1}{d}$
- D$\frac{\mathrm{n}-\mathrm{x}}{\mathrm{n}}=\sqrt{\frac{\mathrm{d}}{\mathrm{d}-1}}$
Solution
For Sonometer,
We know, relative density $\sigma=\frac{\rho_{\text {load }}}{\rho_{\text {water }}}$
Also, $\mathrm{n} \propto \sqrt{\mathrm{T}}$
Tension in air $\mathrm{T}_{\text {air }}=\rho_{\text {load }} \cdot \mathrm{V} \cdot \mathrm{g}$
Tension in water $\mathrm{T}_{\text {water }}=\left(\rho_{\text {load }}-\rho_{\text {water }}\right) \cdot \mathrm{V} \cdot \mathrm{g}$
$=\rho_{\text {water }} \cdot(\sigma-1) \cdot \mathrm{V} \cdot \dot{\mathrm{~g}}$
$\therefore \quad \frac{\mathrm{n}_{\text {load in air }}}{\mathrm{n}_{\text {load inmersed in water }}}=\frac{\sqrt{\mathrm{T}_{\text {air }}}}{\sqrt{T_{\text {water }}}}$
On substituting the respective values,
$\frac{\mathrm{n}_{\text {load in air }}}{\mathrm{n}_{\text {load immersed in water }}}=\frac{\sqrt{\rho_{\text {load }} \cdot \mathrm{V} \cdot \mathrm{g}}}{\sqrt{\rho_{\text {water }} \cdot(\sigma-1) \cdot \mathrm{V} \cdot \mathrm{g}}}$
$=\sqrt{\frac{\sigma_{\text {load }}}{\sigma_{\text {load }}-1}}...(i)$
where, $\sigma=$ relative density or specific gravity of load
Given, $\mathrm{n}_{\text {load }}=$ frequency
$\therefore \quad$ ' $x$ ' beats are produced when weight is completely immersed in water
$\mathrm{n}_{\text {load }}$ in water $=\mathrm{n}-\mathrm{x}$
Given, Specific gravity of load $\left(\sigma_{\text {load }}\right)=d$
$\Rightarrow \frac{\mathrm{n}}{\mathrm{n}-\mathrm{x}}=\sqrt{\frac{\mathrm{d}}{\mathrm{~d}-1}}$
...[From(i)]
Hence, the correct answer is option: 'B'.