Fluid Mechanics

JEE Main level, single correct, medium

139 of 192

Question 139 of 192

The ratio of surface tensions of mercury and water is given to be 7.57.5, while the ratio of their densities is 13.613.6. Their contact angles, with glass, are close to 135∘135^{\circ} and 0∘0^{\circ}, respectively. If it is observed that mercury gets depressed by an amount hh in a capillary tube of radius r1{r}_{1}, while water rises by the same amount hh in a capillary tube of radius r2{r}_{2}, then the ratio r1r2\dfrac{{r}_{1}}{{r}_{2}} is close to

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Answer: (D) 25\dfrac{2}{5}

∣hHg∣=∣hwater∣|{h}_{Hg}|=|{h}_{water}|
2S1cosθ1ρ1r1g=2S2cosθ2ρ2r2g\dfrac{2{S}_{1}cos{\theta }_{1}}{{\rho }_{1}{r}_{1}g}=\dfrac{2{S}_{2}cos{\theta }_{2}}{{\rho }_{2}{r}_{2}g}
Therefore, the ratio of the radii of the respective tubes is,
r1r2=ρ2ρ1×S1cos⁡θ1S2cos⁡θ2\dfrac{{r}_{1}}{{r}_{2}}=\dfrac{{\rho }_{2}}{{\rho }_{1}}\times \dfrac{{S}_{1}\cos {\theta }_{1}}{{S}_{2}\cos {\theta }_{2}}
r1r2=113.6×7.5×12\dfrac{{r}_{1}}{{r}_{2}}=\dfrac{1}{13.6}\times 7.5\times \dfrac{1}{\sqrt{2}}
r1r2≈25\dfrac{{r}_{1}}{{r}_{2}}\approx \dfrac{2}{5}

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