Question

A uniform capillary tube of inner radius $r$ is dipped vertically into a beaker filled with water. The water rises to a height $h$ in the capillary tube above the water surface in the beaker. The surface tension of water is $\sigma$. The angle of contact between water and the wall of the capillary tube is $\theta$. Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

• A. For a given material of the capillary tube, $h$ decreases with increase in $r$
• B. For a given material of the capillary tube, $h$ is independent of $\sigma$
• C. If this experiment is performed in a lift going up with a constant acceleration, then $h$ decreases
• D. $h$ is proportional to contact angle $\theta$

Answer 1

Answer: (A), (C)

$\frac{2\sigma}{R} = \rho gh $ [R $\to$ Radius of meniscus]
$h = \frac{2 \sigma}{R \rho g} R = \frac{r}{\cos \theta} $ [r $\to$ radius of capillary ; $\theta \, \, \to $ contact angle]
$h = \frac{2\sigma\cos\theta}{r \rho g}$
(A) For given material, $\theta \, \to $ constant
$ h \propto \frac{1}{r}$
(B) h depend on $\sigma$
(C) If lift is going up with constant acceleration,
$ g_{eff} = \left(g +a \right) $
$h = \frac{2 \sigma \cos\theta}{r \rho\left(g +a\right)} $ It means h decreases
(D) h is proportional to cos $\theta$ Not $\theta $