Buoyancy, Centres of Buoyancy and Metacentric Height

A yacht floating at rest is in vertical equilibrium when the upward buoyant force balances its weight. According to Archimedes' principle, the buoyant force corresponds to the weight of the water displaced by the immersed hull. If weight is added, the yacht must normally sink farther until it displaces enough additional water to restore equilibrium. If weight is removed, the opposite occurs. This apparently simple principle connects displacement, draft, hull volume and loading condition and forms the foundation from which more detailed stability and trim analysis develops.

The distributed hydrostatic pressure acting on the immersed hull can be represented by a single resultant buoyant force acting vertically through the centre of buoyancy. Geometrically, the centre of buoyancy is associated with the centroid of the underwater volume for the stated condition. Its position changes when the immersed geometry changes through loading, trim or heel. The centre of buoyancy is therefore not a fixed point permanently attached to the yacht. It is a hydrostatic property of the particular underwater shape.

The yacht's weight can be represented as acting vertically downward through its centre of gravity. Unlike the centre of buoyancy, the centre of gravity is determined by the distribution of actual masses: hull structure, machinery, tanks, interiors, equipment, tenders, stores and people. Adding, removing or moving weight changes the centre of gravity according to the magnitude and location of that weight. Naval architects therefore monitor longitudinal, transverse and vertical centres of gravity throughout design, construction and operation.

For a yacht floating freely without external forcing, weight and buoyancy must balance not only in magnitude but also in their moment effects. In the upright equilibrium condition, the vertical lines through the relevant centres align so that no unbalanced heeling or trimming moment remains. If weight shifts transversely, longitudinally or vertically, the yacht responds by changing heel, trim or stability until a new equilibrium is reached, subject to the hull geometry and available righting ability.

When an upright hull heels through a small angle, the shape of the immersed volume changes. One side becomes more deeply immersed while the opposite side emerges, causing the centre of buoyancy to move sideways. The buoyant force now acts through the shifted centre of buoyancy. The relationship between that new line of buoyant action and the centre of gravity determines whether a restoring moment is produced. This shifting of buoyancy with heel is the geometric basis of initial transverse stability.

For sufficiently small changes in heel, successive lines of action of buoyancy can be treated as intersecting near a point called the transverse metacentre. The location of the metacentre is related to the geometry of the waterplane and the position of the centre of buoyancy. It is therefore a hydrostatic property rather than a weight-distribution property. The metacentric concept is most useful for initial, small-angle stability and should not be extended carelessly to describe behaviour at large angles.

Metacentric height, commonly written GM, is the vertical distance between the centre of gravity G and the metacentre M under the relevant convention. A positive initial GM indicates that a small heel produces an initial restoring tendency in the conventional intact condition. Raising the centre of gravity reduces GM; lowering it increases GM if the hydrostatic geometry is unchanged. The numerical value therefore combines a hull-form property with the actual vertical location of weight.

A very small positive GM can indicate weak initial stiffness and large equilibrium angles under modest heeling moments, but an excessively large GM can create a stiff vessel with rapid roll motions and high accelerations. Comfort, structural loads, equipment and operational considerations can therefore matter alongside basic positive initial stability. More importantly, GM describes only the initial slope of stability behaviour. The full righting-arm curve and applicable criteria are required to evaluate stability over larger heel angles.

Waterplane breadth and distribution have a major influence on the transverse metacentric radius and therefore on the initial stability available before weight position is considered. A broad waterplane can generate a different metacentric characteristic from a narrow one at otherwise similar displacement. Yet beam alone does not determine stability because vertical centre of gravity, free surfaces, loading and the full shape of the hull all matter. Initial stability is consequently a coupled geometry-and-weight problem.

A naval architect uses centres of buoyancy, gravity and metacentric height to understand equilibrium and small-angle response, but the analysis does not end there. Righting arms, range of positive stability, downflooding, weather effects, free surfaces and yacht-specific loading conditions may all be relevant under the applicable stability standard. GM is therefore a foundation concept rather than a certificate that a vessel is safe in every circumstance. The approved stability information remains the controlling operational reference.