Question

Deduce an expression for the pressure at a depth inside a liquid.

Answer 1

Let us consider a container which contains a liquid of density d.
consider a cylindrical column of height h from the surface of liquid of cross section area A.
the volume of liquid column
V=Ax h
mass = density x volume
m= Axhxd
weight w= mxg
w=mg
= Axhxdxg
there are three forces acting on water column.
1. weight, vertically down.
2. force on top surface due to atmospheric pressure ( P°A), acting vertically Down.
3. Force on bottom surface if column due to static pressure of liquid ( PA), acting vertically up
from Newton's law
PA= P° A +w
PA = P° A + hdgA
P= P° + hdg
note : P° read it as p not.

Answer 2

Pressure = Force/ Area
Force in a fluid = Thrust= weight of the liquid = (volume of liquid × density × Acceleration due to gravity.)
volume= v (Area,A × height,h)
density= d
gravitational pull= a
Thrust = A × h × d × a

Pressure= Thrust / area
               =A × h × d × a
                       A
Therefore the pressure = h × d × a        (height of the column of the liquid                                                    ×density of the liquid × acceleration due to                                                                  gravity)
Pressure in liquid = h×d×a
However when the liquid is exposed to the atmosphere, you have to factor in the pressure due to the column of the air.

Pressure in liquid =  atmospheric pressure + (h×d×a)