Question

A U-tube is partially filled with water. Oil which does not mix with water is next poured into one side until water rises by $25cm$ on the other side. If the density of oil is 0.8 the oil level will stand higher than the water level by:
(A) $6.25cm$
(B) $12.50cm$
(C) $31.75cm$
(D) $62.50cm$

Answer 1

As we know that the density of water is 1 and we have the density of oil given already. Then we will first assume the amount of water in the given tube to find the difference in height of water. And then we will use the pressure formula in which the pressure of oil is equal to the pressure of water. And we can conclude the height of oil. Then we can easily calculate by how much the oil level will stand higher than the water level.

Complete Step By Step Answer:
Given that-
Density of oil, ${\rho _{oil}} = 0.8$
and, density of water, ${\rho _{water}} = 1$
When water goes down by $x$ amount on one side of the U-tube, then it moves up by $x$ amount on the other tube.
As a result, the overall difference in water height on both sides is equal to $2x$ .
So, the difference in height of water $= 2 \times 25cm = 50cm$
By using the formula:
$\because {P_A} = {P_B}$
where, ${P_A}$ is the pressure for oil
and, ${P_B}$ is the pressure for water.
$\Rightarrow {P_0} + {\rho _{oil}}gh = {P_0} + {\rho _{water}}gh$
As, we know the height of water,
$\Rightarrow (0.8)gh = 1g(50) \\\ \Rightarrow h = 62.5cm \\\$
Therefore, the oil stands higher than water by amount:
$\therefore \Delta h = h - 50 = 62.5cm - 50cm = 12.5cm$
Thus, If the density of oil is 0.8 the oil level will stand higher than the water level by $12.50cm$ ,
Hence, the correct option is (B) $12.50cm$ .

Note:
The molecules that make up oil are bigger than those that make up water, they are unable to pack as closely as water molecules. They are less dense and take up more room per unit area.
With room temperature, maize syrup has the highest density of all liquids, at around $1.4gm{\text{ }}per{\text{ }}cubic{\text{ }}centimetre$ , whereas water has a density of roughly $1gm{\text{ }}per{\text{ }}cubic{\text{ }}centimetre$ .