Buoyancy and Floating Stability

There is no law of buoyancy separate from hydrostatics. The bottom face of a submerged block is deeper than its top, so it is pushed harder — upward. The side faces cancel in pairs. What remains is a net upward force.

1. 1The pressure difference between top and bottom is Δp = ρg × (block height).
2. 2Multiply by the face area: F = Δp·A = ρg × height × A.
3. 3Height × A is the block's volume, so F = ρgV.
4. 4ρgV is exactly the weight of the fluid that volume displaced.

Nothing about the body's material appears anywhere in that derivation. Only the volume it occupies. Any shape works, because any shape is a stack of thin blocks and the argument survives the sum.

A floating body displaces exactly its own weight of fluid, settling at whatever draught makes those equal. Whether that is achievable at all depends on one comparison.

"Steel is denser than water" is true and irrelevant. A ship's average density — hull plus enclosed air — is what counts. Hole the hull, the air is replaced by water, the average crosses the threshold, and the outcome reverses.

Floating and being stable are separate questions, and a body can pass the first while failing the second. Two points decide it.

Centre of gravity G

Where the body's own weight acts. Fixed within the body, and it moves only if the loading changes.

Centre of buoyancy B

The centroid of the *submerged volume*. It moves when the body heels, because the underwater shape changes.

Metacentre M

Where the vertical through the shifted B crosses the body's original centreline.

Metacentric height GM

The distance from G to M. Positive means stable; it is the number naval architects quote.

1. 1Upright, G and B lie on the same vertical line and there is no moment.
2. 2Heel the body: the submerged shape changes, so B migrates toward the immersed side.
3. 3G stays where it was, so weight and buoyancy now form a couple.
4. 4If M is above G, the couple is restoring and the body rights itself.
5. 5If M is below G, the same couple is overturning and the heel accelerates.

GM also changes during a voyage. Burning fuel from low tanks raises G and reduces stability, which is why ballast is actively pumped rather than set once at departure.

GM is only the initial slope of the righting-lever curve and describes the first few degrees. Further over, the lever GZ rises, peaks, and falls back to zero at the *angle of vanishing stability* — beyond which the couple capsizes rather than rights.

Loose water in a part-full tank runs to the low side as the vessel heels, effectively raising G and shrinking the whole curve. This free-surface effect is why tanks are subdivided by longitudinal bulkheads, and why a part-full tank is more dangerous than a full one.