Question

A U-tube contains water and methylated spirit separated by mercury. The mercury columns in the two arms are in level with $10.0\;{\text{cm}}$ of water in one arm and $12.5\;{\text{cm}}$of spirit in the other. What is the specific gravity of spirit?

Answer 1

Finding the ratio of the density of the spirit and water columns is important. Hence the specific gravity is the ratio of density with density of the reference material. Here the reference is the water and specific gravity of spirit is to be determined.

Complete step-by-step solution:
Given the height of the water column is ${h_1} = 10\;{\text{cm = 0}}{\text{.1}}\;{\text{m}}$
Height of the spirit column is ${h_2} = 12.5\;{\text{cm = 0}}{\text{.125}}\;{\text{m}}$
The expression for the pressure at a point inside the tube is given as,
$P = {P_0} + \rho hg$
Where, ${P_0}$ is the atmospheric pressure, $\rho$ is the density, $g$ is the acceleration due to gravity and $h$ is the height of the column.
Therefore the pressure at a point A inside the water column is given as,
${P_1} = {P_0} + {\rho _1}{h_1}g$
Where, ${\rho _1}$is the density of water.
And the pressure at a point B inside the spirit column is given as,
${P_2} = {P_0} + {\rho _2}{h_2}g$
Where, ${\rho _2}$is the density of spirit.
The pressure at all points inside the tube will be the same. Therefore the pressure at A and pressure at B will be the same.
${P_0} + {\rho _1}{h_1}g = {P_0} + {\rho _2}{h_2}g \\\ {\rho _1}{h_1}g = {\rho _2}{h_2}g \\\ {\rho _1}{h_1} = {\rho _2}{h_2} \\\ \dfrac{{{\rho _2}}}{{{\rho _1}}} = \dfrac{{{h_1}}}{{{h_2}}} \\\$
Substitute the values in the above expression.
\dfrac{{{\rho _2}}}{{{\rho _1}}} = \dfrac{{{h_1}}}{{{h_2}}} \\\ = \dfrac{{{\text{0}}{\text{.1}}\;{\text{m}}}}{{{\text{0}}{\text{.125}}\;{\text{m}}}} \\\ = 0.8 \\\
Thus the specific gravity of spirit is $0.8$.

Note:- We have to note that the pressure at all points inside the column will be the same. And the pressure depends on the height and density of the liquid. Also while calculating the specific gravity the reference material will be water always.