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)

The correct options are
A For a given material of the capillary tube, $h$ decreases with increase in $r$
C If this experiment is performed in a lift going up with a constant acceleration, then $h$ decreases.

$\frac{2\sigma }{R}=\rho gh$ [$R\to$ 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) $h$ depends on $\sigma$

(C) If lift is going up with constant acceleration,
$g_{eff}=(g+a)$

$h=\frac{2\sigma \cos \theta }{r\rho (g+a)}$ which means $h$ decreases

(D) $h$ is proportional to $\cos \theta$, NOT $\theta$