Can you find the solution for this question and can you writ

Can you find the solution for this question and can you write your answer neatly so that I can see and understand it easily.

4) (a) What are the component forces due to the water acting on a curved surface? Give the formula of the forces (b) For the inclined circular gate 2.0 m in diameter with water on one side, as shown in Figure Q4, determine: (i) e total resultant force acting on the gate, and (ii) the location of the center of pressure below the water surface.

Solution

a) total resultant force: hydrostatic force acting on the plate

F = p dA, where p is the pressure and A the area = rho *g* z dA

Sum over all the area is rho g AZ where Z is the centroid, A the area

distance below the surface is inthis case (3.5 + 1) sin 60 = 3.897 m

This resultant force acts at right angles to the plate thru the centre of pressure ( Not the centroid)

To find this point need to estimate it as follows:

This is found by taking moments about any point, say the top point

summing over each element, rho g y (sin 60 )dA*y

=to the moment thru the centre of gravity, giving

resultant force* Zsin 60, acting theu centre of pressure at distance Z_c

Equating the moments   Zc=   sum (y² dA)/ AZ

since we are taking moments about the top point, the second moment of inertia given bythe nu,erator term has to be estimated by the parallel axis theorem, giving

I = Ig+ Area*x²    

x=4.5,

Ig = pi R⁴/4

Zc = 1/(4*4.5) +4.5 = 4.556, at a distance 3.945 below surface

whereas the centroid is 3.944 below surface

The diff is due t o the term I/(AZ) = 1/(4Z) in this case