I have a question on fluids.

Question

Water in a section 1.2m wide, 2m high. retained by sluice gate that can pivot about its base.
There is a retaining cable at the top of the gate. Determine tension in the retaining cable at the top of the gate when the depth of water is 1.8m. Density of water is 1020 kg/m3.

Is this to do with hydrostatic pressure?

So work out area of section: 1.2m x 2m = 2.4 m2

Then F = pAgh
= 1020 x 2.4 x 9.81 x 1.8
=43.23 kPa or NM

Answer 1

We have an increasing hydrostatic pressure varing from zero at water surface to the depth of the water, which exerts a varying force on the gate.
The tension on the cable can be obtained by taking moments about the hinge of the gate (at the bottom).

Let
H=height of gate (where the cable is attached) = 2 m
h=height of water=1.8 m
ρ=density of water = 1020 kg/m³
w=width of gate = 1.2 m
T=tension in cable (N)

Hydrostatic pressure at x m from base
= (h-x)*ρg
hydrostatic force over width of gate
F(x) = w(h-x)ρg

Take moments about the base of the gate
due to cable = H×T N-m
due to hydrostatic pressure
= ∫ F(x)×x dx from 0 to h.

Equate the two moments and solve for T.
I get a touch less than 10 kN.

Answer 2

corrections.
I forgot to multiply F(x) by x in the integral.
So T should read less than 6 kN instead of 10.