Buoyancy. A floating hull displaces its weight: Δ = ρ∇. Here ∇ is found by integrating the hull's submerged cross-sections along the length. The centre of buoyancy B is the centroid of that volume; at each heel angle the waterline is re-solved so the volume stays equal to Δ.
Metacentric height. GM = KB + BM − KG, with BM = Iwp/∇ (second moment of the waterplane area about the centreline over displaced volume). For a box barge KB = T/2, BM = B²/(12T). GM > 0 means a small heel produces a restoring moment.
Righting arm. GZ(φ) = (yB−yG)cosφ + (zB−zG)sinφ. For small angles GZ ≈ GM sinφ; for wall-sided hulls GZ = sinφ(GM + ½BM tan²φ). The curve's peak, the angle where it returns to zero (angle of vanishing stability) and the area under it describe how much energy the ship can absorb before capsizing. Deck-edge immersion is where buoyancy above the deck is lost; reserve buoyancy of the deckhouse is ignored.
Moving weights. Raising a weight raises G (δKG = wΔz/Δ); moving it sideways shifts G transversely, so the ship heels until GZ cancels it. In the inclining experiment tanθ = w d/(Δ GM), which is how real ships' GM is measured.
Free surface. A slack tank of length l and breadth b adds a virtual rise GGfs = ρl l b³/(12 ρ∇) to G; it scales with b³, so splitting a tank in two halves the effect fourfold per tank.
Rolling. T = 2πk/√(g GM), k = gyradius (≈0.35–0.45 B including added mass). A stiff ship (large GM) rolls fast; a tender ship rolls slowly. Forcing by waves is resonant when the wave period matches T. Wave slope is attenuated with depth: α = kw a e−kwT/2.
Wind and turning. Wind heeling arm ½ρaCdA V² l cos²φ /(Δg); centrifugal heeling arm in a turn (v²/gR)(KG − T/2)cosφ. The equilibrium heel is where GZ equals the heeling arm.
Wave resistance. Froude number Fn = V/√(gL). Hull speed ≈ 1.34√Lft knots (Fn ≈ 0.40): transverse waves of wavelength 2πV²/g equal the hull length. Friction uses ITTC-57 CF = 0.075/(log10Re − 2)² with a 15% form factor. The wave part is a stylised bow/stern interference model (humps near Fn 0.5, 0.29, 0.21), not a validated hull-specific prediction.
Assumptions: even keel (no trim), rigid hull, weight above deck only affects G, quasi-static hydrostatics (no dynamic added mass except through k), deep water, linear wave-slope forcing. This is a physics teaching model of a generic unarmed displacement hull, not a design tool.