Solving Fluid Mechanics Question: Forces on Plane Areas

[philharg]

Ok I have a question that I am stuck on which is about forces on plane areas. I can do the questions ok where the water acts aboce the gate but in this case the water acts below it if you get what i mean. Can someone please give an explanation of the approach to this question and how you do it, I would be very grateful. Picture is below:

http://img503.imageshack.us/img503/2701/dscf0244yo0.jpg

 

[Astronuc]

The gate has a distributed force, which is a function of distance (height) from the pivot (hinge). The local force is due to the hydrostatic pressure of the fluid, which is a function (mgh) of the height (depth) of water above that point. At the top of the gate, the pressure is mgH, and at the bottom the water pressure is mg(H+4), and it varies linearly in between.

The problem requires a balance of moments. The pressure varies along the face of the plate normal to the surface and parallel to force P.

The moment of P must equal the moment of the force generated by the water pressure on the area of the gate.

Let x be the distance from the hinge, and the pressure varies as p(x sin\theta), where \theta the angle between the plane of the gate and the horizontal. The force is p*A and assuming unit width, an increment of area is given by 1*dx, so the local force at x is f(x) = p(x)dx, the moment of the force if x*f(x).