A fluid in contact with any surface exerts a
pressure and hence forces on it. Consider a
rectangular vessel of base `l xx b` filled with fluid up to
height H as shown.

The forces exerted by fluid at base

`F_b = ` Pressure (at base) `xx` Base Area

`F_b = Hrhogxx(lb)`

`I b H = ` volume of fluid

`F_b = rho g V =` Weight of fluid in vessel

At base, pressure is uniform everywhere and is
equal to weight of fluid.

Pressure at vertical wall of vessel is not uniform but
increases linearly with depth from the free surface.
Consider a small rectangular element of width (b)
and thickness (dh) at a depth h from the free
surface.

Pressure due to fluid at this position is

`P = h rho g`

The force at this element

`dF = PdA = h rho g (b dh)`

Total force on wall is `F = int _(h=0)^(h=H) h rho g(b dh) = 1/2 rho g b H^2`

The total force per unit width of vertical wall

`= F/b = 1/2 rho g H^2 `

The centre of force (he ) is the point of application
of total force from the free surface is given by

`h_c = 1/F int_0^H hdF = 1/F int_0^H h^2 rho g b (dh)` ,

`h_c =1/F (rho g b) int_0^H h^2 dh = 1/F(rho g b)H^3/3`

`h_c = ((rho gb H^3)/3)/((rho gb H^2)/2) = 2/3 H`

`h_c =2/3 H`

Center of force is at two-third of the total depth.