1.3. Calculating pressure and force

The resultant force is going to be the pressure at the centroid into area that we have seen in the derivation in last lecture. What is the area? Area of the ellipse that we have seen is pi into a b.

In fluid mechanics, pressure is defined as the force exerted per unit area. In this context, when calculating the resultant force on a surface submerged in fluid (like a gate), we use the pressure at the centroid of that surface and multiply it by the area of the surface. Here, the area of an ellipse (which is the shape of the gate) is calculated using the formula πab, where 'a' is the semi-major axis and 'b' is the semi-minor axis. This gives us a way to quantify the force acting on the gate due to the fluid pressure.

Imagine pressing your hand against a beach ball. The pressure you feel on your hand depends on how hard you press (force) and how big the hand area (size of the contact). If you press on a smaller area with the same force, the pressure increases. Similarly, for this gate, the fluid exerts pressure on it depending on how deep the fluid is and the area of the gate.

So, h c is 10 meters actually. Why? 8 meters plus 2 because of the ellipse, depth to this centroid, very clear.

The depth to the centroid of the elliptical gate is crucial as it determines how much pressure is exerted by the water above it. Here, the total depth (h_c) is calculated by adding the depth of water above the gate (8 meters) to the additional depth of the centroid of the ellipse below the top surface of the pipe (2 meters), resulting in a total of 10 meters. This depth will be used in further calculations to determine the resultant force.

Think of it like measuring how deep a swimming pool is at different points. The deeper you go (like where the centroid is), the more water pressure you feel pressing down. Just like understanding the depth of the centroid helps us calculate how much pressure is acting on the gate.

So, F R is going to be ρ g h c pi a b. So, ρ is 1000 kilogram per meter cube, g is 9.8...

The resultant force (F_R) acting on the gate is calculated using the formula F_R = ρ * g * h_c * π * a * b. In this equation, ρ is the density of the fluid (water, in this case), g is the acceleration due to gravity, h_c is the depth to the centroid, and a and b are the semi-major and semi-minor axes of the ellipse. Each of these variables contributes to the total force acting against the gate.

Consider how a diver feels pressure in the water. The deeper they go, the heavier the water feels because there's more water above them. Similarly, in this case, the deeper the centroid of the gate is in water, the greater the pressure, and thus the greater the total force that needs to be calculated.

Now, the most important part is force that is required to open the gate...

To find the force required to open the gate, we need to consider the moments about the hinge. The force (F) applied at the bottom of the gate must create a moment around the hinge equal to the moment created by the resultant force (F_R) acting at its line of action. The equation used here allows us to calculate the force required based on the lengths involved (from the hinge to the point where the force is applied and where the resultant force acts). This calculation ensures that the gate can be opened against the pressure from the water.

Think of trying to lift a heavy door. You need a certain amount of force to pull it open, but the farther you pull from the hinge (using a longer arm), the easier it is to open. This concept of moments is similar when it comes to calculating how much force is needed to open the gate against the water pressure.

So, we also need to, you know, to wind up the, you know, the forces on the plane surfaces...

For our gate scenario, we need to consider both vertical and horizontal components of force. The vertical force is equivalent to the weight of the liquid directly above the curved surface and extends up to the pressure reference point (normally the free surface). The horizontal force, on the other hand, is computed based on the pressure at the centroid of the submerged surface multiplied by the area. Most notably, these forces balance each other out to keep the gate in equilibrium with the fluid.

Imagine you have a water balloon; the pressure on the sides (horizontal) keeps it within your hands, while the weight of the water inside pushes down (vertical). Similarly, both forces act on the gate to maintain balance as it resides in the water.

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