10.3. Derivations of Hydrostatic Pressures

Now if you look it that, let us coming to the derivations of the hydrostatic pressures on a submerged surface. That means surface is submerged in a liquid. So we have the atmospheric pressure acting on that. We have the free surface and I am just putting a surface which is the x and y projections in these directions is it look it from this point you will have an inclined surface which is having this x and y.

In this introduction, we are focusing on hydrostatic pressure, which is the pressure exerted on a submerged surface. When a surface is submerged in a liquid, atmospheric pressure affects it, combined with the pressure from the liquid above it. In our scenario, we visualize it with a surface that is inclined, and we need to analyze the force acting on it due to these pressures.

Think of a person diving into a swimming pool. As they go deeper, they feel more pressure on their body from the water above them. The deeper they go, the greater the pressure, similar to how the submerged surfaces experience hydrostatic pressure.

If these two things we know it then we can design these things that what type of material to be used, what type of lifting arrangements will be there, if this is as equal to a get conditions. The submerged plane can be considered as a get conditions. So total force caused by retaining liquid that what we are looking and directions and lineup for actions as the directions will be always perpendicular to the surface that what the directions and the line of actions what of the force is the center of pressure.

To understand how hydrostatic forces act on submerged surfaces, we focus on small elements of the surface area. Each element experiences a force due to the water pressure, which acts perpendicularly to the surface. The total force on the submerged surface can be determined by summing up the contributions from all these small elements. This also helps us understand how to design structures, like gates or walls, that need to resist this force.

Imagine a large sheet of paper being pressed down on by a heavy book. Each small part of the paper experiences a force from the book's weight. In a similar way, each small element of a submerged surface feels the pressure from the water above it.

If I consider a small element okay, let me consider a small element. In this small element of the dA if the pressure is P Then P into dA will be the force, pressure into area will be force.

Here, we are taking a more mathematical approach by introducing integrals. For each small area element (dA) on our submerged surface, the force can be calculated as the pressure (P) multiplied by the area. To find the total force acting on the entire submerged surface, we would integrate these forces across the entire area.

Think of using a sponge to absorb water. Each tiny part of the sponge absorbs a certain amount of water based on the pressure of the water it's in. To find out how much water the entire sponge absorbs, you need to consider all those tiny parts together—just like using integration to find total force.

That means if I just put all these values what I am getting this part, which is the Pc multiply the area that what will come is the force.

Once we have integrated and found the pressure at the centroid of the submerged surface, we can calculate the total hydrostatic force. This is done by multiplying the pressure at that centroid (Pc) by the area of the submerged surface. This gives us a clear numerical value for the force acting on that surface.

Picture a water balloon. When you press on it, the pressure at the center of the balloon will tell you how much force you're exerting. Just like that, knowing the pressure at the centroid of an area lets us calculate the total force acting on it.

Now let us compute to that we need to compute center of pressure. That means, the point of action of total hydrostatic force on this submerged surface.

In the analysis of hydrostatic forces, understanding where this total force acts is crucial. This point is known as the center of pressure. It is determined by considering the moments produced by the force about a reference point (like an axis) and can be calculated using principles similar to those used in solid mechanics.

If you think of a seesaw, the point where the forces are balanced is like the center of pressure. Just as the seesaw tilts towards the heavier side, the center of pressure helps us understand how the structure will behave under hydrostatic forces.

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