Question

The pressure in the water pipe at the ground floor of a building is 120000 Pa, whereas the pressure on the third floor is 30000 Pa. What is the height of the third floor?
[Take $g = 10m{s^{ - 2}}$, density of water = $1000kg{m^{ - 3}}$].
A. 9 m
B. 10 m
C. 11 m
D. 12 m

Answer 1

Hint: Here, it is important to understand the definition of pressure. Pressure is equal to the force applied per unit area.

Pressure, $P = \dfrac{F}{A}$
where F = force in newtons and A = area in ${m^2}$
The unit of pressure is pascal (Pa).

Complete step-by-step answer:
In a fluid column, the pressure is applied uniformly on all the directions of the walls of the container. This fluid pressure depends upon the depth. As the depth increases, the pressure applied by the fluid column increases.
We can say that pressure at a point in the fluid is due to the weight of the fluid, W at that point.
So, the pressure at that point, $P = \dfrac{W}{A}$
The weight of the fluid, $W = mg$ where m=mass of the fluid in kg and g = acceleration due to gravity, given here -$10m{s^{ - 2}}$
Also, the mass of the fluid is equal to the product of its density and the volume.
\[
  \therefore P = \dfrac{{mg}}{A} \\
   \to P = \dfrac{{V\rho g}}{A} \\
 \]
where $\rho $ = density in $kg{m^{ - 3}}$
The volume of the fluid column is equal to the product of the area of the fluid column and the height of the column. $\therefore V = A \times h \to h = \dfrac{V}{A}$
Substituting, we get –
\[
  P = \dfrac{{V\rho g}}{A} \\
   \to P = h\rho g \\
 \]
Now, given $g = 10m{s^{ - 2}}$, density of water, $\rho $ = $1000kg{m^{ - 3}}$.
Pressure at the ground floor, ${P_g} = {h_g}\rho g$
Substituting, we get –
$
  120000 = {h_g} \times 1000 \times 10 \\
$
Solving,
$
  {h_g} = \dfrac{{120000}}{{10000}} = 12m \\
 $
Pressure at the third floor, ${P_3} = {h_3}\rho g$
Substituting, we get –
 $
  30000 = {h_3} \times 1000 \times 10 \\
$
Solving,
$
  {h_3} = \dfrac{{30000}}{{10000}} = 3m \\
 $
The height of the third floor, $H = {h_g} - {h_3} = 12 - 3 = 9m$

Hence, the correct option is Option A.

Note: Here, there is a point mentioned that the pressure is the same throughout the container. This is actually a scientific law called Pascal’s law of pressure, which states that that pressure applied to an enclosed fluid will be transmitted without a change in magnitude to every point of the fluid and to the walls of the container. This means that the pressure applied in all the directions will be uniform unlike the solids.