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  4. A glass capillary tube is of the shape of a truncated cone with an apex angle α so that its two ends have cross sections of different radii. When dipped in water vertically, water rises in it to a height h, where the radius of its cross section is b. If the surface tension of water is S, its density is ρ, and its contact angle with glass is θ , the value of h will be (g is the acceleration due to gravity)

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Q. A glass capillary tube is of the shape of a truncated cone with an apex angle $\alpha$ so that its two ends have cross sections of different radii. When dipped in water vertically, water rises in it to a height $h$, where the radius of its cross section is $b$. If the surface tension of water is $S$, its density is $\rho$, and its contact angle with glass is $\theta$ , the value of $h$ will be ($g$ is the acceleration due to gravity)

Mechanical Properties of Fluids

Solution:

Let $r =$ radius of curvature of meniscus
$b=r cos \left(\theta+\frac{\alpha}{2}\right)$
Excess pressure on concave side orr meniscus $= \frac{2S}{r}$
$\Rightarrow \left(P_{0}+h\rho g\right)-P_{0}=\frac{2S}{r} $
$\Rightarrow h\rho g=\frac{2S\, cos\left(\theta+\frac{\alpha}{2}\right)}{b}$
or $h=\frac{2S}{b\rho g}cos \left(\theta+\frac{\alpha}{2}\right)$
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