1.5.1. GATE Question - Semi-Cylindrical Gate

Now let us we another GATE questions we can solve it with as it is a diagrams given here. There is a semi circles cylindrical gate is there and force F is acting here and there is a hinge at this point. So we need to compute it if this is a free surface. This is the two meter depth of the water is there. What could be the force is required if the hinge at this point. This is what the questions.

In this problem, we analyze a semi-cylindrical gate submerged in water. The gate is hinged at a point, which means it can rotate about that hinge. The depth of the water above the gate is 2 meters. Our goal is to determine the force needed to keep the gate in equilibrium. This involves understanding the forces acting on the gate due to the water pressure as well as the applied force F.

Imagine a hinged trapdoor at the bottom of a swimming pool. If you wanted to keep the door closed while water presses against it, you'd need to apply a certain force to counteract the water pressure pushing on it. This scenario is similar to our semi-cylindrical gate problem.

Flow classification: Fluid is homogeneous, Density is constant, Fluid is static. So there are the two force component will come it. One is a horizontal force and other is vertical force. And another F which is applied to here. So we will try to find out what could be this force component to compute, so that this will not rotate it. So we need to compute what is F is acting on this if there is a two force component, one is a horizontal force because the fluid at the rest.

For the semi-cylindrical gate, we identify two key components of force acting on it: the horizontal force from the water pressure and the vertical force due to the weight of the water. These forces must be balanced for the gate to remain steady and in equilibrium. The horizontal force tends to push the gate outward, while the vertical force affects how deep the gate can be submerged without rotating.

Think about how a large billboard stands against the wind. The wind exerts horizontal pressure on the billboard, trying to push it down, while the vertical force would be gravity pulling it down. Just as the billboard needs support to stay upright, the gate needs forces acting on it to maintain its position.

Now to compute horizontal forces, what is going to act it. So it is very easy that the places what will get it that what will be the force into vertical projections of the curved surface area. That is what we derived earlier. So we can compute it what will the force will act on this as the surface the pressure distributions where is like this, we can find out the pressures at the centroid locations. Horizontal force = $\rho g h A \cos(\theta)$ (where A is the area of vertical projection).

To calculate the horizontal force acting on the gate, we look at the pressure distribution over its surface. The pressure at any point increases with depth, and the total horizontal force can be found by integrating over the area of the gate that’s submerged. We use the equation presented where A represents the area projected vertically, and cosine of the angle adjusts for the orientation of the force acting on the area.

Consider a kite in the wind. The wind applies force to the kite at different angles, much like water pressure on the gate. The more area the kite presents to the wind, the greater the force exerted on it. Similarly, the submerged area of the gate determines how much water pressure acts on it.

Similar way if I need to compute how much vertical force is acting on this. So we can find out if this is my free surface. This is the area, the shaded part will give a vertical force component of F. The vertical force on portion BC = weight of liquid contained in the shaded area acting vertically upward = $\rho g \cdot h \cdot A$.

The vertical force acting on the gate is derived from the weight of the fluid above that gate. We calculate the vertical force by looking at the height of the water acting on the area of the gate that is submerged, expressed using the density of the water, gravity, and the area of the impacted surface.

Think about how a boat floats on water. The weight of the water displaced by the boat pushes upward with a force equal to the weight of the water. In the case of our gate, the weight of the water above it pushes down, exerting a vertical force that we must consider.

F for equilibrium taking moment about hinge 'O'. By taking moments about hinge, we establish that the sum of moments due to the forces acting must equal zero for equilibrium. The equation for moments gives us $F \cdot r_1 = F_h \cdot r_2$, balancing the horizontal and vertical moments.

For the gate to be stable and not rotate, we need to ensure that the clockwise moments equal the counterclockwise moments about the hinge point. By calculating the distances from the hinge to where each force is applied and multiplying these distances by their respective forces, we can set up an equation to solve for the unknown force F. This step is crucial for finding the required force to maintain the gate’s balance.

Imagine trying to balance a seesaw. If one side is heavier, you can adjust things (like moving a person closer to the hinge) to balance it out. In our problem, we’re doing something similar – balancing forces acting from different points to keep the gate from swinging open or closed.

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