Question

In a U-tube experiment, a column AB of water is balanced by a column CD of paraffin. The relative density of paraffin is:

$\begin{array}{l}A.{\displaystyle \frac{h_2}{h_1}}\\ B.{\displaystyle \frac{h_1}{h_2}}\\ C.{\displaystyle \frac{h_2-h_1}{h_1}}\\ D.{\displaystyle \frac{h_2}{h_1+h_2}}\end{array}$

Answer 1

Hint: This question tests us of our knowledge of fluid dynamics. Here we have to use the condition of the U-tube and the condition of a balanced U-tube. And then we will use that condition to find the relative density of paraffin.

Complete step by step answer:For a balanced U-Tube the pressure in both the columns is equal. Therefore, the pressure acting due to the liquids in each column is given as:
$P=P_0+\rho gh$, where $\rho$ is the density of the liquid, g is the acceleration due to gravity and h is the height of the liquid column and $P_0$ is the atmospheric pressure.
Therefore, the pressure in column one is:
$P_{CD}=P_0+{\rho }_{p}g{h}_1$
And the pressure in the second column is:
$P_{AB}=P_0+{\rho }_{w}g{h}_2$
And for the condition of balanced U-Tube these two pressure values would be equal.
Equating the above two equations, we will get:
$\begin{array}{l}{P}_{AB}=P_{CD}\\ P_0+{\rho }_{w}g{h}_2=P_0+{\rho }_{Paraffin}g{h}_1\end{array}$
Cancelling the atmospheric pressure $P_0$ from the above expression, we get:
${\rho }_{w}g{h}_2={\rho }_{Paraffin}g{h}_1$
Now we have a relation involving the density of the paraffin, water and the heights of the water columns.
Now if we consider the formula of relative density of paraffin it is given below:
$$relativedensity={\displaystyle \frac{{\rho }_{paraffin}}{{\rho }_w}}$$
Therefore, if we arrange the relation which we derived we can find the relative density of paraffin:
$\begin{array}{l}{\rho }_{w}g{h}_2={\rho }_{Paraffin}g{h}_1\\ {\displaystyle \frac{{\rho }_{Paraffin}}{{\rho }_w}}={\displaystyle \frac{g{h}_2}{g{h}_1}}\\ {\displaystyle \frac{{\rho }_{Paraffin}}{{\rho }_w}}={\displaystyle \frac{h_2}{h_1}}\\ relativedensity={\displaystyle \frac{h_2}{h_1}}\end{array}$
Hence, the relative density is ${\displaystyle \frac{h_2}{h_1}}$ and the correct option from the given options is (A.)

Note:In questions like these we have to apply concepts of relative quantities which is the ratio of the quantity with the standard quantity. Also, when calculating pressure in a U-tube one should always add the factor of atmospheric pressure.