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(a) The pressure P at depth y from the water surface is:
P = pgy
where p is the density of water, g is the acceleration due to gravity.
Hence, force dF at an elemental area (dy).w is:
dF = P.[w.(dy)] where w is the width of the gate
Therefore, total force F. = integral { P.w.(dy)} with limits of integration from 0 to D (the depth of the water).
F = integral { (w.pgy).dy} integrate from 0 to D
i.e. F = (1/2)wpg(D^2)

Hence, lateral force on gate = (1/2).(1.6).(1000).(9.81).(0.8^2) N = 5023 N

(b) Mean pressure on gate, P = F/(w.D) = [ (1/2)wpg(D^2)]/(wD) = (1/2)pgD

Let H be the centre of pressure, measure from the surface of water.
Moment about the top of gate = force acting on gate x H = [P.(w.D)].H
= (1/2).(wpgD^2).H

Consider pressure at depth y, moment of pressure at this depth about top of gate
(pgy)(dy.w).y
Total moment = integral {(wpgy^2. (dy)} with limits of integration from 0 to D
= (wpg/3)D^3

Therefore, (1/2).(wpgD^2).H = (wpg/3)D^3
H/2 = D/3
i.e. H = 2D/3 = 2 x 0.8/3 m = 0.533 m

(c) Moment acting about the base
= 5023 x (D/3) = 5023 x (0.8/3) N.m = 1339 N.m