1.5. Further Problem Examples

Let us start to solve the problems, these very easy problems that there is a square gate of dimensions of 1.5 meter into 1.5 meters. One of the vertical sides of a fully filled water tank has one side on the free surface. It is hinged on the lower horizontal sides. Here it is hinged it and is held in a position by force applied on the vertical central line at a depth of 0.75 meter below the free surface. The magnitude of this force is?

This problem describes a square gate in a water tank. To find the magnitude of the force holding the gate, we need to calculate the hydrostatic pressure acting on the gate at different depths. The pressure at any depth in a fluid is given by the formula P = ρgh, where ρ is the fluid's density, g is the acceleration due to gravity, and h is the depth. Since the gate is 1.5 m tall and submerged 0.75 m below the free surface, we can find the average pressure acting on the gate's surface. By calculating the force exerted by this pressure over the area of the gate, we can determine the force applied at the hinge.

Think of a large door in a swimming pool. The deeper you go, the heavier the water pushing against the door becomes. Just like the swimming pool door needs a strong handle (or force) to keep it closed, the square gate also needs force at the hinge to stay in place against the water pressure.

Now the pressures at the water surface levels will be the atmospheric pressures, will be the atmospheric pressures and we can consider this atmospheric pressure is close to zero okay that as compared to the pressure distributions. This the correct the pressure distribution varies linearly with respect to the depth.

In a fluid at rest, pressure increases linearly with depth. This means that the deeper you go in the fluid, the higher the pressure you will encounter. At the water surface, the pressure is considered as atmospheric pressure (often treated as zero for calculation purposes), and as you go lower into the fluid, pressure builds up because of the weight of the fluid above. This principle allows us to predict how much pressure will be exerted at a certain depth and can be calculated using P = ρgh.

Imagine diving into a lake. The deeper you dive, the more you feel the weight of water above you pressing down. This feeling is the increasing water pressure, just as the problem describes.

As you know in a triangular pressure distribution diagrams, the resultant force acts at a one third distance from the bottoms or two thirds distance from the free surface. So one third distance locations that what will be act it the fluid F we know it what is the distance from this. So we just take a moment about the hinge to compute what will be the F, the force into distance, force into distance.

When a force acts on a submerged surface, its effect can be analyzed by considering moments around a pivot point—in this case, the hinge. The resultant force due to the fluid pressure acts at a particular distance from the hinge, specifically one-third of the height of the gate from the base. To find out how much force is needed to keep the gate from moving, we can set the moments created by the forces acting on either side of the hinge to be equal. This method uses the formula: moment = force × distance from the pivot.

Think of a seesaw. If one side is heavier, it will tip. The force (the weight of a person) needs to be balanced by the moment it creates at the pivot (the hinge). If you shift farther from the hinge, you don't need as much weight to balance out the seesaw, much like finding the force on the gate.

Now let us we another GATE question we can solve it with as it is a diagrams given here. There is a semi-circular 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?

In this problem, we need to analyze a semi-circular gate submerged to a depth of two meters in water. Similar to the square gate problem, we must identify both vertical and horizontal forces acting on it due to hydrostatic pressure. The pressure can be calculated using the fluid's density and height, then applied to determine the forces acting on the curved gate surface. By summing these forces and considering the moments around the hinge, we can figure out the applied force needed for equilibrium.

Imagine a large beach ball held beneath the water; like the ball, the semi-circular gate feels different pressures pushing on it depending on how deep it is underwater. If you wanted to keep the ball from floating up or sinking, you’d need to apply a force—similar to how we calculate the force on the gate to keep it in place.

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