Three liquids of densities and (with ), having the same value of surface tension , rise to the same height in three identical capillaries. The angles of contact and obey [2016]

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Q.
Three liquids of densities $ρ_{1}, ρ_{2}$ and $ρ_{3}$ (with $ρ_{1} > ρ_{2} > ρ_{3}$ ), having the same value of surface tension $T$ , rise to the same height in three identical capillaries. The angles of contact $θ_{1}, θ_{2}$ and $θ_{3}$ obey [2016]

1 $\frac{π}{2} > θ_{1} > θ_{2} > θ_{3} \geq 0$

2 $0 \leq θ_{1} < θ_{2} < θ_{3} < \frac{π}{2}$

3 $\frac{π}{2} < θ_{1} < θ_{2} < θ_{3} < π$

4 $π > θ_{1} > θ_{2} > θ_{3} > \frac{π}{2}$

Ans.
(2)

Capillary rise, $h = \frac{2 T \cos θ}{r ρ g}$

For a given value of $T$ and $r,$ $h \propto \frac{\cos θ}{ρ}$

Also, $h_{1} = h_{2} = h_{3}$ or $\frac{\cos θ_{1}}{ρ_{1}} = \frac{\cos θ_{2}}{ρ_{2}} = \frac{\cos θ_{3}}{ρ_{3}}$

Since, $ρ_{1} > ρ_{2} > ρ_{3}$ , so $\cos θ_{1} > \cos θ_{2} > \cos θ_{3}$

For $0 \leq θ < \frac{π}{2},$ $θ_{1} < θ_{2} < θ_{3}$

Hence, $0 \leq θ_{1} < θ_{2} < θ_{3} < \frac{π}{2}$

Solution

Q.
Three liquids of densities $ρ_{1}, ρ_{2}$ and $ρ_{3}$ (with $ρ_{1} > ρ_{2} > ρ_{3}$ ), having the same value of surface tension $T$ , rise to the same height in three identical capillaries. The angles of contact $θ_{1}, θ_{2}$ and $θ_{3}$ obey [2016]

1 $\frac{π}{2} > θ_{1} > θ_{2} > θ_{3} \geq 0$

2 $0 \leq θ_{1} < θ_{2} < θ_{3} < \frac{π}{2}$

3 $\frac{π}{2} < θ_{1} < θ_{2} < θ_{3} < π$

4 $π > θ_{1} > θ_{2} > θ_{3} > \frac{π}{2}$

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Solution

Q. Three liquids of densities ρ1,ρ2 and ρ3 (with ρ1>ρ2>ρ3), having the same value of surface tension T, rise to the same height in three identical capillaries. The angles of contact θ1,θ2 and θ3 obey [2016]

1 π2>θ1>θ2>θ3≥0

2 0≤θ1<θ2<θ3<π2

3 π2<θ1<θ2<θ3<π

4 π>θ1>θ2>θ3>π2

Ans. (2) Capillary rise, h=2Tcosθrρg For a given value of T and r, h∝cosθρ Also, h1=h2=h3 or cosθ1ρ1=cosθ2ρ2=cosθ3ρ3 Since, ρ1>ρ2>ρ3, so cosθ1>cosθ2>cosθ3 For 0≤θ<π2, θ1<θ2<θ3 Hence, 0≤θ1<θ2<θ3<π2

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Q.
Three liquids of densities $ρ_{1}, ρ_{2}$ and $ρ_{3}$ (with $ρ_{1} > ρ_{2} > ρ_{3}$ ), having the same value of surface tension $T$ , rise to the same height in three identical capillaries. The angles of contact $θ_{1}, θ_{2}$ and $θ_{3}$ obey [2016]

1 $\frac{π}{2} > θ_{1} > θ_{2} > θ_{3} \geq 0$

2 $0 \leq θ_{1} < θ_{2} < θ_{3} < \frac{π}{2}$

3 $\frac{π}{2} < θ_{1} < θ_{2} < θ_{3} < π$

4 $π > θ_{1} > θ_{2} > θ_{3} > \frac{π}{2}$