Question

A U-tube is partially filled with water. Oil which does not mix with water is next poured into one side; unit water rises by $25cm$ on the other side. If the density of oil is $0.8g/cc$, then oil level will stand higher than the water level by
A. $6.25cm$
B. $12.50cm$
C. $18.75cm$
D. $25cm$

Answer 1

Oil and water have a heterogeneous mixture and hence they do not mix with each other and can be differentiated. When oil is added, water rises by 25cm, but the pressure at both the points would remain the same. Thus applying Bernoulli's principle, for equal pressures, we can obtain the height, as both the densities are given.

Complete step by step answer:
There are two types of mixtures; homogeneous and heterogeneous mixture. Homogeneous mixture is when the two liquids blend with each other, the solute and the solvent get mixed with each other and one cannot differentiate between the initial components of the mixture. Heterogeneous n mixture is when two liquids do not blend or mixed with each other, and their surfaces are separate. The components of such mixtures can be easily differentiated.

Here, in the U-tube, water is partially filled and then oil is added, hence we obtain a heterogeneous mixture. Now, when oil is poured into the tube, let’s suppose from the right side, then the water level on the left side rises up to 25cm. So let us suppose that oil will stand higher than water by x amount of level. Thus as 25cm water rises after the addition, then oil rises by x+25cm. The Bernoulli’s theorem is expressed as below:
$P + \dfrac{1}{2}\rho {v^2} + \rho gh = Constant$
Where P is the pressure,$\rho$is the density of the fluid, g is the acceleration due to gravity, v is the velocity of the fluid and h is the height of the fluid.

Now as we know from Bernoulli's principle, we know that the pressure on both sides would be equal. Thus;

{{P}_{(water)}}={{P}_{(oil)}} \\\ \Rightarrow {{\rho }_{(water)}}g\centerdot 25={{\rho }_{(oil)}}g\centerdot (x+25) \\\ \Rightarrow 25g=0.8g(x+25) \\\ \therefore x=6.25cm$$ Here, $${\rho _{(water)}}$$is the density of water,$${\rho _{(oil)}}$$ is the density of oil and g is the acceleration due to gravity. **Hence option A is the correct answer.** **Note:** Here, the pressure can only be equated on the vertical portions of the capillary tubes and not the horizontal part of the tube. Pressure would be the same only at the vertical portion and not horizontal. Also, here as the U shaped tube is thin, we have ignored the capillary action of the tube which causes surface tension.