1.2. Real life example of moment

So, now we proceed with one of the real life examples of moment. So, the question is, an elliptical gate covers the end of a pipe 4 meters in diameter. If the gate is hinged at the top, what normal force F applied at the bottom of the gate is required to open the gate when the water is 8 meters deep above the top of the pipe, and the pipe is open to atmosphere. Neglect the weight of the gate.

In this example, we have an elliptical gate positioned at the end of a pipe. The objective is to calculate how much force is needed to open this gate when submerged under a certain depth of water. The key factors include the dimensions of the gate, the depth of the water above it, and the atmospheric pressure acting on the gate.

Think of a heavy door submerged in water. Just like you need to push harder to open the door underwater compared to on land due to water pressure, here, we need to determine the exact force needed to open the gate due to the water pressure above it.

The resultant force is going to be the pressure at the centroid into area. The area of the ellipse is pi into a b. For h c, h c is 10 meters: 8 meters (water depth) plus 2 meters (depth to the centroid). So, F R is going to be ρ g h c π a b.

To find the resultant force exerting on the gate, first, we need to determine the pressure acting at the centroid of the gate. This pressure multiplied by the area of the gate’s elliptical shape gives us the resultant force. The effective height (h_c) is derived from the water depth plus the depth from the gate's top to its centroid.

Consider a large balloon underwater. The deeper you go, the greater the water pressure on the balloon's surface. Similarly, we're calculating how the water depth influences the force exerted on our elliptical gate.

We need to find out y R and x R. y R is given by y c + a squared / 4 h c sin theta. Since the ellipse is symmetrical about its centroid, x R will be 0.

Next, we calculate where the resultant force acts. This is important because it affects how the force is applied when trying to open the gate. The formula takes into account the distance from the centroid to the hinge as well as the shape of the ellipse, ensuring we find the precise line of action for the resultant force.

Imagine a seesaw with a heavy load on one end. The balance point or pivot determines how easily you can tip it. In our case, finding the location of the resultant force helps understand how hard we need to push to open the gate.

The most important part is the force required to open the gate. The moment about the hinge is calculated, using F l = R cp, substituting values yields a required force.

To determine how much force needs to be applied to open the gate, we calculate the moments around the hinge. Leveraging the distance to where the resultant force acts, we can isolate and solve for the force applied at the bottom of the gate. This tells us how much effort is needed to overcome the water pressure.

Imagine trying to push a heavy sliding door open. The further you apply force from the hinge, the easier it is to move the door. Similarly, by calculating the moment and applying this logic, we can find the exact force to open our gate.

Thus, the force required to open the gate is approximately 809 kilo Newton.

After completing our calculations, we find that the force needed to open the gate is significant due to the pressure from the water. This outcome emphasizes the importance of understanding fluid mechanics and forces in practical applications, especially in hydraulic engineering.

Just like enormous machinery often requires considerable force to operate against resistance, the calculations reveal the tough reality of managing large water bodies and why hydraulic systems are carefully designed to handle such forces.

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