A uniform capillary tube of inner radius r is dipped vertically i… - Vidyadip

MCQ

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, $\mathrm{h}$ decreases with increase in $\mathrm{r}$.

$(B)$ For a given material of the capillary tube, $\mathrm{h}$ is independent of $\sigma$

$(C)$ If this experiment is performed in a lift going up with a constant acceleration, then $\mathrm{h}$ decreases

$(D)$ $\mathrm{h}$ is proportional to contact angel $\theta$

A
$A,B$
✓
$A,C$
C
$A,D$
D
$A,B,C$

✓

Answer

Correct option: B.

$A,C$

b
$\frac{2 \sigma}{\mathrm{R}}=\rho g \mathrm{~h}$

where $\mathrm{R}$ is the radius of meniscus.

$h=\frac{2 \sigma}{\rho g R}$

$\mathrm{R}=\frac{\mathrm{r}}{\cos \theta}$, where $\mathrm{r}$ is the radius of capillary and $\theta$ is the angle of contact.

$h=\frac{2 \sigma \cos \theta}{\rho g r}$

$(A)$ For a given material, $\theta$ is constant.

Therefore, $\mathrm{h} \propto \frac{1}{\mathrm{r}}$

$(B)$ $\mathrm{h}$ depends on $\sigma$ as $\mathrm{h} \propto \sigma$.

$(C)$ If lift is going up with constant acceleration a,

$\mathrm{g}_{\text {eff }}=\mathrm{g}+\mathrm{a}$

$\mathrm{h}=\frac{2 \sigma \cos \theta}{\rho(\mathrm{g}+\mathrm{a}) \mathrm{r}} ; \mathrm{h} \text { decreases }$

$(D)$ $\mathrm{h}$ is proportional to $\cos \theta$.

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