What Is The Center Of Floating?

Any floating vessel experiences static and dynamic motions about the longitudinal, transverse, and vertical axes (commonly referred to as the x, y, and z axes).

According to the fundamentals of naval engineering, when a vessel tilts or heels about its longitudinal or x-axis due to external or internal factors, it falls under the rubric of vertical, transverse, and static stability.

Similarly, all issues related to the dynamic or kinetic behavior of a vessel about the x-axis, or longitudinal axis, fall under the rubric of marine balance. The rolling motion of a vessel is the primary focus of all marine balance studies.

All studies related to the behavior of a vessel about the z-axis, or vertical axis, can be categorized as maneuverability studies.

Similarly, all issues related to the static and dynamic behavior of a vessel about the y-axis, or transverse axis, fall under the rubric of longitudinal stability, both static and dynamic.

Rolling, the significant change in a vessel’s draft along its length, falls under the rubric of static longitudinal stability. On the other hand, the heeling motion of a vessel falls under the rubric of dynamic stability.

A vessel’s center of buoyancy is fundamentally linked to its longitudinal stability and response. How are the two related?

To do this, we need to understand the concept of water level.

Water level can be simply defined as the area created or displaced by a floating body at the waterline. For a ship floating at a given draft x above the baseline, it creates a specific water level at the waterline, which is best understood from a top or bottom view. Water level can vary greatly depending on the ship’s design and draft.

The geometric center of gravity of the water level is called the center of buoyancy.

Revisiting Simple Lever Theory

But the importance of the center of buoyancy goes far beyond its geometrical center of gravity. It is essentially the point at which the physics of equilibrium and heeling (dynamic stability) are fully defined.

The transverse axis that determines a ship’s response under stability or longitudinal behavior is considered to pass through this center of buoyancy.

Thus, the center of buoyancy can be understood as the specific point that moves or oscillates longitudinally (or, more commonly, along the x-z plane) through the transverse line of a floating ship.

The physics of the center of buoyancy is best explained using the example of a seesaw.

When two people of different weights sit on a seesaw, the side with the heavier person will fall. Seesaws rely directly on the simple principle of a lever. The fulcrum is the hinge, or the point around which a crane beam rotates. When any type of load or force is applied to either end of the beam, a torque is generated about that point.

In the typical seesaw found in parks, this fulcrum is located at the center of the beam. Therefore, the moment arms at both ends are equal.

So, what is torque? It’s force multiplied by distance. Since the distance is constant in both cases, the torque generated (i.e., the difference between the opposing torques acting on the ends) depends only on mass. Therefore, the heavier person will inevitably fall first because they are applying a greater load on one end of the lever, thus generating a greater torque on that side.

Fulcrum

However, in most real-world applications of levers as simple machines, the location of the fulcrum on the beam is crucial. This is because the torque applied in a simple lever is a result of any applied load. Consider a simply supported beam of length L.

Assume that the fulcrum (or pivot point) is located 0.8L from each end of the hinge (for example, end A).

The distance between the hinge and the other end (for example, end B) is 0.2L.

Now, consider an object of mass W placed at end B. What is the induced torque?

Mathematically, it can be expressed as 0.2L × W. To lift this load, or more physically, to counteract the induced torque, we must find an equal and opposite torque from first principles. Using a simple equilibrium equation, or the equation for equal equilibrium torques, we can find:

0.2L × W = 0.8L × w’

The left side of the equation represents the torque generated by the load, while the right side represents the balancing torque required to maintain mechanical equilibrium. As we all know, 0.8L and 0.2L represent the distances from ends A and B, respectively, to the fulcrum or pivot point, with weight W placed at end B. The right side of the equation, w’, represents the load or force required to generate the balancing torque or counteracting torque. Solving for w’, we find that it equals W/4, or one-fourth the weight placed at the other end, B.

Thus, the load or force required at the other end is much smaller than the weight initially placed to lift it, because the difference in torque or lever arm achieves the desired balance.

This is known as the mechanical advantage of a lever system. From simple everyday uses to complex mechanical devices, this system has helped humans lift heavy objects worldwide.

In the seesaw example above, because the moment arms are equal—that is, the distances between the load application points and the point of action (the fulcrum or hinge) are equal at both ends—weight is the sole determinant of the resulting total torque.

The system is not in equilibrium, and the total moment acting on the heavier side is the product of the difference between the two weights and the equal moment arms.

Detailed Description of the Center of Buoyancy

In the context of floating structures or objects, the center of buoyancy is the fulcrum or hinge. Suppose that a weight W moves a distance (d) from a point. This generates a moment W × d.

According to the physics of floating objects, any imbalance in force or load manifests itself as a change in draft. In this case, as the weight shifts aft, a torque is generated in that direction, causing the stern of the ship to sink more than the bow. This is called “stern trim.” Conversely, when the bow draft increases due to increased sinking, this is called “bow trim.”

Due to the weight shift, the longitudinal center of gravity (CGL) also shifts slightly in the direction of the applied load, in this case, aft (from G to G1).

As the stern sinks, the center of buoyancy (CB) also shifts aft (from B to B1). From the nature of this effect, we can see that as part of the stern sinks, the bow rises.

According to the physical law of buoyancy equilibrium required for a ship to remain afloat, the submerged portion of the ship is equal in volume to the floating portion. Static equilibrium is achieved when the new center of buoyancy, G1, and the center of gravity, B1, are vertically aligned. Assume two perpendicular projections pass through points B and G, and points B1 and G1, respectively. They intersect at point M. This point is called the longitudinal center of equilibrium and is stable in a given equilibrium state.

Starting from the moment of displacement torque due to a given weight, w, we can derive:

GG1 × W = w × d

W is the ship’s displacement, and GG1 is the displacement of its longitudinal center of gravity. Therefore, the moment caused by the displacement of weight w is the ship’s equilibrium moment, expressed as W × GG1.

We will not delve into the more detailed aspects of equilibrium and stability, as these have been explained in detail in other articles.

Now, let’s return to the essence of this compensation effect. We note that in this case, the waterline forms an angle with the waterline in the previous equilibrium state (when the ship was uniformly floating). As shown in the diagram, these two waterlines intersect at a specific point, F. This point is also the center of buoyancy. We also note that, as previously mentioned, point F represents the intersection of the submerged and surface portions of the hull.

In this case, in static equilibrium, the aft displacement of the weight generates a counterclockwise torque that opposes the induced torque. This torque is generated by the buoyancy force, which is equal in magnitude and opposite in direction to the displacement of the submerged portion.

Thus, in our example, the counterclockwise torque induced by the weight displacement is balanced by the clockwise torque generated by the buoyancy force (B).

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What Is The Center Of Floating?