11.5.3. Stable Equilibrium

Let us come back to the topics what we have been discussing that fluid at the rest. As I told it when the fluid is at rest we have the two force components. 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. 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.

In fluid mechanics, when we talk about floating objects, we often refer to their stability. Stability is defined by how an object responds when it is disturbed. For any floating body, there are two main forces acting on it: the buoyant force (upward force due to fluid displacement) and the weight of the object (downward force). In a stable floating condition, if you slightly tilt or disturb the object (say, a ship), the forces will work together to bring it back to its original position. Conversely, if these forces do not balance correctly, the object may capsize or become unstable.

Consider a toy sailboat on a pond. If the wind hits it and tilts the boat, if the buoyant force is greater than the weight, it will right itself because those forces push it back up. If you pushed it too far and the sail got wet (making it heavy), it might capsize, showing instability.

Let me introduce one point, which is we call the metacenters, okay. That is what is called the metacenter. What is that metacenters, okay? If I tilt a floating object to an angle of delta theta, then there will be a new waterline that comes. The center of buoyancy will change from B to Bprime as the waterline changes, and along that, you will have the buoyancy force that is going to act.

The metacenter is a crucial point for understanding the stability of floating bodies. When an object is tilted, its center of buoyancy shifts due to the change in the waterline. The metacenter is defined as the point where the vertical line through the center of buoyancy meets the original waterline after tilting. If this metacenter is above the center of gravity (CG) of the object, the object is considered to be in a stable equilibrium because the restoring forces will correct any tilting.

Imagine a seesaw (teeter-totter) where one side is heavier but positioned well. When someone pushes it down, if the pivot (analogous to the metacenter) is correctly placed, the seesaw will return to its resting position. If the pivot is too low (metacenter below the center of gravity), it may flip entirely, just as a boat would capsize.

So let me summarize that, that when you have a floating object to test its stability, we generate a small disturbance of tilted at a small angle Δθ. The theoretically, if the BM and BG (distance between buoyancy and metacentric height) are having the relationship like BM > BG or BM < BG we have a stable or unstable equilibrium.

There are generally three types of equilibrium: stable, unstable, and neutral. In stable equilibrium, if the object is disturbed, it returns to its original position, whereas in unstable equilibrium, any disturbance causes the object to tilt further from its original position, potentially leading to capsizing. Neutral equilibrium means the object stays in its new position after being disturbed. Understanding these distinctions helps when designing ships and other floating bodies to ensure safety.

Think about balancing a pencil on your finger. If you can balance it perfectly (neutral equilibrium), it doesn’t fall but it doesn't return to your finger if it’s nudged. If you push it slightly and it falls down (unstable equilibrium), you’ll need to reposition it. If you place it on something wide enough to not fall (stable equilibrium), it stays put even if bumped.

Now how to compute this the metacentric height? Let you have a floating object. What we will do is take the area of the moment of inertia of the waterline about the axis of tilting to find out how we can establish a relationship between the metacentric height.

To determine whether a floating object is stable, we calculate the metacentric height (GM). This involves understanding the geometry of the submerged part of the object. The moment of inertia, which accounts for how mass is distributed, helps in calculating how buoyancy interacts with the object’s weight when tilted. If the spacing from metacenter (M) to center of gravity (G) is large, the object is likely to be stable.

If you picture a tall building, it has a high center of gravity (maybe compared to a wide one-story building). The taller it is, the easier it is to topple. The same goes for boats: a wide boat with low CG will be more stable in the water than a tall, narrow boat. Those calculations help find the ideal design for safety.

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