11.3.1. Forces in a Fluid at Rest

As I told it when the fluid is the rest we have the two force component. One is force due to the pressure and the second is the gravity force. There is no velocity components, there is no shear stress component.

When a fluid is at rest, it experiences two main forces: the force due to pressure acting in all directions and the gravitational force acting downward. Since the fluid isn't moving, there aren't any velocity or shear stress components to consider, simplifying our analysis of the fluid's behavior.

Think of a swimming pool filled with water. When no one is splashing around and the water is perfectly still, the only forces acting on the water at any given point are the weight pushing down due to gravity and the pressure from the surrounding water pushing up. It's like a calm lake on a sunny day.

If you have to determine what could be the forces on a curved surface? What the problems we face it. If you have a let me take this example. So you have the free surface and you have a curved surface and you have the liquid which is having the density let be the rho.

To analyze the forces acting on a curved surface submerged in a fluid, we must consider that pressure acts normally (perpendicularly) at each point of the surface. As pressure varies with depth, we break the curved surface into smaller flat segments. By integrating these forces across the surface, we can calculate the total force exerted on the curved surface.

Imagine submerging a basketball in a swimming pool. The water applies pressure on the ball both from the top and the sides, and this pressure increases the deeper the ball goes. By understanding how this pressure works at various points on the ball, engineers can design stronger balls that can withstand this force even at deeper waters.

So that is the reasons if you take a control volumes, it is easy to draw the pressure diagrams and from that pressure diagrams, if you equate with the gravity force, you can solve the problems.

Creating pressure diagrams allows us to visualize how pressure changes across the surface of a submerged structure. By equating these pressures with gravitational forces acting on the fluid, we can solve for unknown forces. This method is helpful in analyzing how the structure will respond to the pressures it encounters.

Think about a snorkeling experience. When you're underwater, you can feel the pressure of the water on your body. If you were to draw a diagram showing how the pressure changes with depth while you're underwater, you would see that it increases the further down you go. This knowledge helps engineers design diving equipment that can safely withstand these pressures.

The problems come it that we can simplify that ones like we can resolve the curved surface into a two-part horizontal and vertical components.

To make calculations easier, we can resolve the total force acting on a curved surface into horizontal and vertical components. By doing this, we can separately analyze the upward buoyant forces and downward gravitational forces, simplifying our problem-solving process.

Imagine trying to lift a large balloon underwater. The balloon has an upward force due to buoyancy and a downward force due to its weight. By breaking these forces down into vertical (up and down) and horizontal (side to side) components, you can figure out how much assistance you may need to lift it out of the water.

As you know it the famous scientist Archimedes what invented this buoyancy concept when he was in a bathtub experienced the lighter weight because of the buoyancy forces.

Archimedes' principle states that any object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. This principle helps us understand why objects feel lighter in water than in air, as the water pushes up against the object, counteracting some of its weight.

Consider an ice cube in a glass of water. The cube floats because it displaces a volume of water equal to its weight. As it floats, the water level rises to accommodate the weight of the cube, showcasing Archimedes' principle in action.

Now if you look it another point is we also look it that the center of buoyancy. That means the line of actions of the buoyant force is the center of buoyancy.

The center of buoyancy is the point through which the buoyant force acts, and it is located at the centroid of the displaced fluid volume. Understanding where this point is relative to the center of gravity of the object is crucial for stability. A stable floating object has its center of buoyancy above its center of gravity, ensuring it right itself when tilted.

Think of how a cork floats. The cork naturally turns upright in water because its center of buoyancy is above its center of gravity. If you push it under the water, it will want to come back up – this restoration is all about the relationship between the center of buoyancy and gravity.

To summarize and to make it very easy to say that the Archimedes principles what it says that a body immersed in a fluid experience a vertical buoyant force.

To recap, Archimedes' principle asserts that an object immersed in a fluid will experience an upward buoyant force equal to the weight of the fluid displaced by the object. This principle is fundamental in understanding why we float or sink in water.

A good example is when you’re at the beach. When you walk into the water, you feel your weight decrease. This feeling is the buoyant force from the water pushing up against you, demonstrating Archimedes' principle.

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