Question

A U -tube contains water and methylated spirit separated by mercury. The mercury columns in the two arms are in Level with 10.0cm of water in one arm 12.5 cm of spirit in the other. What is the Specific gravity of spirit?

Answer 1

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

Complete step by step answer:
Refer to the below figure for a proper explanation of a U tube manometer shown below.

     

We know that the pressure at the same line or base level is always the same. Hence the pressure ${P_1}$ at A due to the water column and the pressure ${P_2}$ at B due to the spirit column is always the same. Hence,
$\begin{array}{c}
  \Rightarrow {P_0} + HDg = {P_0} + hdg
\end{array}$
Here, ${P_0}$ is the atmospheric pressure, H=10 cm=0.1m is the height of the water column, h=12.5 cm=0.125m is the height of the mercury column, and g is the acceleration due to gravity.
$\begin{array}{l}
 \Rightarrow HD = hd\\
 \Rightarrow \dfrac{d}{D} = \dfrac{H}{h}\\
 \Rightarrow \dfrac{d}{D} = \dfrac{{0.1\;{\rm{m}}}}{{0.125\;{\rm{m}}}}\\
 \Rightarrow \dfrac{d}{D} = 0.8
\end{array}$
Since the specific gravity is defined as the ratio of the density of a liquid to the density of water, therefore the specific gravity of spirit is 0.8.

Additional Information:We know that pressure is a scalar quantity defined as the force applied per unit area, and the force should be perpendicular to the area. But fluid pressure is the measure of force per unit area of an object in a closed container or a fluid. According to Pascal’s law, the pressure applied on fluid will be transmitted throughout the fluid and to the walls of the container without any change in the magnitude of the pressure.

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.