Question

The pressure in the water pipe at the ground level of a building is 120000Pa, whereas, the pressure on a third floor is $30000Pa$. What is the height of the third floor?
(Take$g = 10\,m{s^{ - 2}}$, density of water$ = 1000\,kg{m^{ - 3}}$).

Answer 1

Hint: Firstly we will calculate the height of the ground floor and third floor from the tank using the suitable formula and then, we will calculate the actual height of the building. Just remember, the level of floors from the ground floor, assuming the level of the ground floor to be 1.

Formula used:
We will use the formula of variation of pressure with depth, which is given by
$P = h\rho g$
Where $P$ is the pressure, $h$ is the height, $\rho $ is the density, and $g$ is the gravity.

Complete step by step solution:
Let us first write the given terms in the question
${P_1} = 120000\,Pa$(Pressure in water pipe on the ground floor)
${P_2} = 30000\,Pa$(Pressure in water pipe on the third floor)
$\rho = 1000\,kg{m^{ - 3}}$(Density of water)
$g = 10\,m{s^{ - 2}}$(Gravity)
Now, for calculating the height of the tank, we will use the formula which is given by
${P_1} = h\rho g$
$120000\,Pa = h \times 1000 \times 10$
$\therefore \,\,h = \dfrac{{120000}}{{1000 \times 10}}$
$ \Rightarrow \,h = 12m$
Therefore, the height of the ground floor from the tank is $12\,m$.
Therefore, the building is $12\,m$ tall.
Now, we will calculate the height of the third floor from the tank as
${P_2} = h\rho g$
$30000 = h \times 1000 \times 10$
$\therefore \,\,h = \dfrac{{30000}}{{1000 \times 10}}$
$ \Rightarrow \,h = 3m$
Therefore, the height of the tank from the third floor is $3m$.
Now, assume that the ground level is $1$, so, the level of the third floor will be $4$.
Now, the height of the third floor from the ground floor can be calculated as
$height\,of\,third\,floor = \dfrac{{height\,of\,the\,building}}{{level\,of\,ground\,floor}} \times \,level\,of\,third\,floor$
$h = \dfrac{{12}}{4} \times 3$
$ \Rightarrow \,h = 3 \times 3$
$\therefore \,h = 9m$

Therefore, the height of the third floor is$9m$.

Additional Information:
You might have observed that while on a plane flight your ears may pop or ears may ache during a deep dive in a swimming pool, this is because of the effect on depth of pressure of a fluid. Now, on the Earth’s surface, the pressure exerted on you will be the result of the pressure of the air above you. This pressure of air will be reduced as you climb up in an altitude because the weight of air in the high altitude reduces.

Also, in the case of water, when we will go underwater, the pressure of water increases with an increase in depth. Also, the pressure on you will be the result of both the pressure of water and the pressure of the atmosphere above you.

Note: As we all know that liquids are incompressible, therefore, the equation used above i.e. $P = h\rho g$ can be used to great depth.
But, in the case of gases, which are compressible, we can apply this equation only when the density change is small over the depth considered.