Centre Gravity Buoyancy

Overview

Engineering reference data for Centre Gravity Buoyancy in mechanics.

Key Formulas

Newton's Second Law

$F = ma$

Force = mass × acceleration.

Work

$W = Fd\cos\theta$

Work = force × displacement × cos(angle).

Kinetic Energy

$E_k = \frac{1}{2}mv^2$

Energy of motion.

Potential Energy

$E_p = mgh$

Gravitational potential energy.

Variables

Stability Principles

The interaction between the center of gravity (G) and the center of buoyancy (B) determines a vessel's stability. This relationship defines three critical states:

1. Upright Position (Stable)

• In the upright, static equilibrium condition, G and B lie on the same vertical line.
• For most conventional hulls, B is located below G. This arrangement is termed meta-stable.
• The net buoyancy force and the gravitational force are equal, opposite, and collinear, resulting in zero net moment.

2. Small Angle of Heel (Stable Equilibrium)

• When the hull tilts by a small angle (heel), G remains fixed relative to the hull (assuming no cargo shift).
• B moves laterally to align with the new centroid of the submerged hull volume.
• The horizontal separation between the vertical lines of action of the weight (through G) and the buoyancy force (through B) creates a righting moment.
• This moment acts to rotate the hull back toward the upright position, characterizing positive stability.

3. Critical Heel Angle (Capsizing Instability)

• As the heel angle increases beyond a critical threshold, the geometry of the submerged hull changes such that the lateral shift of B can no longer outpace the horizontal projection of G.
• At this point, the lines of action of the weight and buoyancy forces converge and cross.
• The resulting moment reverses direction, acting to increase the heel angle rather than correct it. This leads to capsizing.

Key Engineering Insight

Stability is not merely a function of the positions of G and B at rest, but of how the center of buoyancy shifts dynamically in response to the vessel's attitude. The metacentric height (GM) is the primary quantitative metric used to quantify initial stability.

Righting Moment and Stability

The righting moment (GZ) is the torque that restores a vessel to upright when heeled. It is the product of the buoyancy force and the perpendicular distance (GZ) between the lines of action of gravity (G) and buoyancy (B').

Righting Moment Formula:

$$\text{Righting Moment} = \Delta \cdot g \cdot GZ$$

Where:

• $\Delta$ is the displacement (mass of the vessel)
• $g$ is acceleration due to gravity
• $GZ$ is the righting arm

The righting arm $GZ$ is a function of the heel angle ($\theta$) and the vessel's geometry. For small angles of heel, it can be approximated using the metacentric height ($GM$):

$$GZ \approx GM \cdot \sin(\theta)$$

Capsize Condition

A vessel will capsize when the righting moment becomes zero or negative. This occurs when the center of buoyancy (B') moves such that the line of action of buoyancy force passes through (or beyond) the center of gravity (G). At this critical angle, known as the angle of vanishing stability, the vessel has no inherent tendency to return upright.

Example: For a vessel with a displacement ($\Delta$) of 10,000 tonnes and a metacentric height ($GM$) of 1.5 m, the righting moment at a 5° heel angle is:

$$\text{Righting Moment} \approx 10{,}000{,}000\ \text{kg} \times 9.81\ \text{m/s}^2 \times (1.5\ \text{m} \times \sin(5^\circ)) \approx 12.8 \times 10^6\ \text{N·m}$$

Stability Margins

The text describes a critical concept in naval architecture: metacentric height (GM). This value quantifies the initial stability of a vessel.

When a ship is tilted by a small angle, the center of buoyancy (B) shifts horizontally. The vertical line from the new center of buoyancy intersects the original vertical centerline at a point called the metacenter (M). The distance between the center of gravity (G) and the metacenter (M) is the metacentric height (GM).

A positive GM means the metacenter is above the center of gravity, creating a righting moment that tends to return the ship to upright—the condition described as "meta-stable." A negative GM indicates that the center of gravity is above the metacenter, creating an upsetting moment that promotes capsizing—the condition where forces "start to create a moment that work in the same direction."

The magnitude of GM directly influences the vessel's range of stability and how strongly it resists heeling.

References

• Original Source