1.5.2. Hinge Moment Calculation

Now let us start to solve the problems, these very easy problems that there is a square gate of a 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 a hinged it and is held in a positions by force applied on the vertical the central line at a depth of 0.75 meter below the free surface.

In this problem setup, we have a square gate with dimensions 1.5m x 1.5m. This gate is part of a vertical side of a water tank that is completely full. One side of the gate is at the water's free surface, and it is hinged at the bottom. A force is applied at the center of the gate, located 0.75 meters below the water surface. The goal is to compute the magnitude of this force, which is important in understanding how the gate interacts with the fluid pressure.

Imagine a heavy door that is partially submerged in water. The pressure from the water pushes against the door, just like the pressure from the tank water acts on the gate. If you were to push the door to keep it closed, you would need to exert a force depending on how deep the water is, which parallels our calculation here.

The magnitude of this force is? So this is 1.5 meters and 0.75 meters below means 0.75 meter from the bottom, from the hinge. This is what 0.75 meters, total is 1.5 meters. So what could be the magnitude of this force? The problem is that we need to know hydrostatic pressure distributions. Then we can compute the force due to the hydrostatic distribution.

To find the force acting on the gate, we first need to understand hydrostatic pressure. Hydrostatic pressure increases linearly with depth due to the weight of the fluid above it. The pressure at any point in the fluid can be calculated using the formula P = ρgh, where ρ is the fluid's density, g is the acceleration due to gravity, and h is the depth from the fluid surface.

Think of diving into a swimming pool. As you go deeper, you feel the weight of the water pressing down on you more strongly. This increasing pressure is similar to how hydrostatic pressure increases with depth in our water tank.

Once I know what is the force is acting because the fluid is at the rest and where it acts the force that the locations. If I know the force, the center of pressures or the force where is acting it, if I know that to the force magnitudes or the at the locations where the force acts then I can take a moment at the hinge locations to compute what will be the force component.

The total hydrostatic force acting on the gate can be calculated as the average pressure multiplied by the area of the gate. The point where this force acts is known as the center of pressure. This point is not at the geometric center of the gate but depends on the pressure distribution. To maintain equilibrium, the moments about the hinge must balance, which is essential for calculating the applied force.

Imagine pushing open a door. The point where you apply your push is critical for how easily the door swings. Similarly, in our problem, knowing where the hydrostatic force acts helps us balance the forces to find how much force is needed to keep the gate closed.

Taking Moment about hinge (1.5/3) × F = F × 0.75. As you know in a triangular pressure distribution diagrams the resultant force acts at a one-third distance from the bottoms or two-third distance from the free surface.

In this step, we calculate the moment about the hinge to find the required force F. The hydrostatic force acts at a distance of one-third the height of the gate from the bottom. We set up the moment balance: the moment due to the hydrostatic force equals the moment due to the applied force. Solving this gives us the magnitude of the applied force necessary for equilibrium.

Think of using a seesaw. If you have a heavier partner on one side, you need to apply force on the other to keep it balanced. The moments on both sides must be equal for it to be level, just like how we balance the forces and moments on our hinged gate.

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