First slide

Surface tension

Question

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

Moderate

A

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

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

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

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

Solution

$h = \frac{2 T \cos θ}{ρgr}$
as r, h, T are same
$\Rightarrow \frac{\cos θ}{ρ} = constant$
$\Rightarrow \frac{\cos θ_{1}}{ρ_{1}} = \frac{\cos θ_{2}}{ρ_{2}} = \frac{\cos θ_{3}}{ρ_{3}}$
as $ρ_{1} > ρ_{2} > ρ_{3}$
$\Rightarrow {cosθ}_{1} > {cosθ}_{2} > {cosθ}_{3}$
$\Rightarrow θ_{1} < θ_{2} < θ_{3}$
As water rises $\Rightarrow$ $θ must be acute$
So, $0 \leq θ_{1} < θ_{2} < θ_{3} < \frac{π}{2}$

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