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Displacement of COM:Application

A glass capil...

Question

A glass capillary tube is of the shape of 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).

\frac{2 S}{b ρ g} cos (θ - α)

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\frac{2 S}{b ρ g} cos (θ + α)

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\frac{2 S}{b ρ g} cos (θ - \frac{α}{2})

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\frac{2 S}{b ρ g} cos (θ + \frac{α}{2})

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Solution

The correct option is D $\frac{2 S}{b ρ g} cos (θ + \frac{α}{2})$

Let $r$ be the radius of curvature of the meniscus.

$\Rightarrow b = r cos (θ + \frac{α}{2})$

Excess pressure on concave side of the meniscus

$Δ P = \frac{2 S}{r}$

$\Rightarrow (P_{0} + h ρ g) - P_{0} = \frac{2 S}{r}$

$\Rightarrow h ρ g = \frac{2 S cos (θ + \frac{α}{2})}{b}$

$\Rightarrow h = \frac{2 S}{b ρ g} cos (θ + \frac{α}{2})$

Hence, option $(A)$ is correct.

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