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An open capillary tube contains a drop of a liquid. When the tube is in its vertical position, the drop forms a column with a length of $3 \mathrm{~cm}$ . The internal diameter of the capillary tube is $0.8 \mathrm{~mm}$ . Determine the radii of curvature of the upper and lower meniscus in each case. Consider the wetting to be complete. The surface tension of the liquid is $0.08 \frac{\mathrm{N}}{\mathrm{m}}$ . Assume gravitational acceleration to be $9.8 \frac{\mathrm{m}}{\mathrm{s}^2}.$
Option: 1
$R_1=0.4 \mathrm{~mm}, R_2=0.4 \mathrm{~mm}$

Option: 2
$R_1=0.4 \mathrm{~mm}, R_2=1.5 \mathrm{~mm}$

Option: 3
$R_1=0.8 \mathrm{~mm}, R_2=0.4 \mathrm{~mm}$

Option: 4
$R_1=0.05 \mathrm{~mm}, R_2=0.4 \mathrm{~mm}$

Answers (1)

best_answer

The radius of curvature of the upper meniscus is equal to the radius of the capillary tube, which is $0.4 \mathrm{~mm}.$

The pressure due to surface tension acting in the upward direction on the upper meniscus is given by:

$P_1=\frac{2 T}{r}$

Where $\mathrm{T}$ is the surface tension of the liquid.

Substituting the given values, we get:

$P_1=2 \times \frac{0.08}{0.0004}$

$P_1=400 \frac{\mathrm{N}}{\mathrm{m}^2}$

The hydrostatic pressure acting downward on the lower meniscus is given by:

$P_2=h \rho g$

Assuming the density of the liquid to be $1000 \frac{\mathrm{kg}}{\mathrm{m}^3} \text {. }$

Substituting the given values, we get:

$P_2=\frac{(0.03 \times 1000 \times 9.8) N}{m^2}$

$P_2=294 \frac{\mathrm{N}}{\mathrm{m}^2}$

Since $P_1>P_2$ the resulting pressure is directed upward, and for equilibrium, the pressure due to the lower meniscus should be directed downward, which makes the lower meniscus concave downwards. Therefore, the radius of curvature of the lower meniscus is given by:

$P_1-P_2=\frac{2 T}{R_2}$

Substituting the given values, we get:

$\begin{aligned} & 400-294=2 \times 0.08 / R_2 \\ & 106=\frac{0.16}{R_2} \end{aligned}$

$R_2=1.5 \mathrm{~mm}$

Therefore, the radius of curvature of the upper meniscus (concave upwards) is $0.4 \mathrm{~mm}$ , and the radius of curvature of the lower meniscus (concave downwards) is $1.5 \mathrm{~mm}$ , which is option $\mathrm{B}$ .

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