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 $\theta$