6.1. Rectangle Properties

We have to see what are the forces on plane areas, that is horizontal surface. So, if you see, this is a figure that shows a horizontal surface at depth h. So, this h is the vertical distance to the free surface and this we what is the P here. And what is the resultant force at the bottom?

In this section, we explore how fluid exerts pressure on horizontal surfaces. The pressure increases with depth, and the depth is represented by h, which is the vertical distance from the surface of the fluid to the point on the surface in question. The resultant force can be calculated by integrating the pressure over the area of the surface. This is crucial for understanding how fluids interact with surfaces, especially in applications like tanks and dams.

Imagine a swimming pool. The deeper you go into the pool, the heavier the water above presses down on you. If you are at the bottom, you feel more pressure from the water than if you were sitting at the surface. This is similar to how forces on plane surfaces are calculated.

The force resultant force is going to be the integration of pressure into area. So, p is constant so it comes out and that becomes pressure into area pA, where p is rho gh, okay, this is the gauge pressure.

Here, the formula for calculating the resultant force is established. Since pressure (p) at a specific depth in a fluid is given by the formula p = ρgh, where ρ is the fluid density, g is the acceleration due to gravity, and h is the depth, we can compute the resultant force (F) on a plane area by multiplying this pressure by the area (A). This gives us F = pA.

Think of filling a balloon with water. The pressure inside increases as you fill it more. If you imagine the bottom of the balloon as an area A, the force pushing downwards on that area can be thought of as the weight of the water inside (pressure times area).

Another important thing is, we have to learn and revise again, what are the forces on the plane areas or the inclined surface. The force will start acting perpendicular to the plane, right.

Forces acting on a plane, whether horizontal or inclined, always act perpendicular (normal) to that surface. This is crucial to understanding how fluids exert pressure. When calculating forces on inclined planes, the pressure varies with depth, making it necessary to compute the resultant force considering the varying pressure along the surface.

Picture a door being pushed open by wind. The force of the wind acts straight against the surface of the door, perpendicular to it. Similarly, when fluid exerts its force on any surface, it does so perpendicularly to that surface.

The point through which the resultant force acts is called the center of pressure. This is not at the centroid. The reason is because the pressure is increasing with depth.

The center of pressure is a vital concept in fluid mechanics. It is the specific point at which the total hydrostatic force can be considered to act. Unlike simple geometric centers (centroids), the center of pressure shifts lower as pressure increases with depth. Consequently, understanding this concept is important for designing structures like dams and ships.

When you dive into a pool, the point where the water pushes up against your body is not exactly in the center of your mass because pressure increases with depth. The water beneath you exerts a stronger force where you're deeper, so the net force, or resultant, is felt lower, similar to how the center of pressure works.

Coordinate y R can be determined by summation of moment around the x-axis. So, y R into F R will be equal to integral y dF.

To find the location of the center of pressure, we can compute the moments about a designated axis. The moments created by each differential force (dF) need to balance with the moment created by the resultant force (FR). By calculating these moments and solving for yR, we get a precise location of where that resultant force acts relative to the surface.

Think of a seesaw. If you want to know where to sit to balance it, you would need to consider where the forces (people) are acting. You compute your position based on their weights and distances. Similarly, calculating yR involves balancing the moments created by the forces on the surface.

We also need to find out X R, the moment calculation around the y-axis will yield that X R can be found using the product of inertia.

To find the x-coordinate for the center of pressure, we again use the moment method but this time around the y-axis. The results from these calculations end up involving the product of inertia, which allows us to determine how the resultant force is distributed along both the x and y axes. Such calculations are fundamental for stability analysis in fluid mechanics.

Similar to the way sports teams analyze players' movement on a field. Coaches evaluate positions where players are and how they should adjust to balance the strategy (forces) on the field. Here, we adjust xR to ensure fluid forces act effectively.

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