Ship Hydrostatics and Stability
Ship Hydrostatics and Stability tests displacement, buoyancy, metacentric height and righting lever, plus the criteria used to judge intact and damaged stability. It is the most numerical core domain in GATE NM, and most questions reduce to a clear free-body or volume calculation.
Core rules
- Displacement Δ = ρ∇, where ∇ is the immersed volume. For a box barge, ∇ = LBT and Δ = ρLBT.
- KB is the centroid of the immersed volume above the keel; for a box barge, KB = T/2.
- BM = I/∇, with I the waterplane second moment about the centreline. For a box, I = LB³/12, so BM = B²/(12T).
- KM = KB + BM; GM = KM − KG for a solid KG. Positive GM gives initial stability; negative GM gives loll.
- Small-angle righting moment = Δ GM sinθ; for large angles use the GZ curve, where GZ = KN − KG sinθ.
- Free surface correction reduces effective GM by i/∇, where i is the free-surface second moment.
Worked example
A box barge has L = 20 m, B = 6 m, T = 2 m, KG = 2.0 m, in seawater of density 1.025 t/m³. ∇ = 20 × 6 × 2 = 240 m³, so Δ = 1.025 × 240 = 246 t. KB = 2/2 = 1.0 m. BM = 6²/(12 × 2) = 36/24 = 1.5 m. KM = 1.0 + 1.5 = 2.5 m. GM = 2.5 − 2.0 = 0.5 m. At 5°, righting moment = 246 × 9.81 × 0.5 × sin 5° ≈ 105 kN·m.
Traps
Using 1.000 t/m³ instead of 1.025 for seawater is a common error. KG is measured from the keel, not from the waterline. Free surface always reduces GM; it does not disappear when the tank is full or empty. The small-angle formula is not valid for large heel. Adding weight high on the ship raises KG and lowers GM; adding weight at the keel lowers KG.