Hydrostatics and Buoyancy

Floating body stability

A freely floating body is in equilibrium when its weight equals the buoyant force and its centre of gravity G lies on the same vertical line as the centre of buoyancy B. Equilibrium alone does not show whether the body returns upright after a small tilt, so stability must be checked separately. For a floating body the reference point is the metacentre M, the point where the line of action of the buoyant force after the tilt meets the original vertical through B. The body is stable when M lies above G, that is, when the metacentric height \overline{MG} is positive. Using the second moment of the waterplane area I and the submerged volume V_{sub}, the metacentric height is GM = I/V_{sub} - BG, where BG is taken positive when G lies above B. Setting GM = 0 gives the limiting proportion at which stability is lost. For a vertical cylinder, increasing the diameter raises I/V_{sub}, so there is a minimum ratio of diameter to length for vertical stability.

A fully submerged body has no waterplane, so the metacentre concept does not apply, and its stability depends only on the relative position of G and B. If y is measured downward from the free surface, a larger y means a lower point, so stability requires y_{CG} > y_{CB}. When the two centres coincide, a small tilt produces no restoring or overturning couple, which gives neutral equilibrium. If G lies above B, the tilt produces an overturning couple and the equilibrium is unstable. A neutrally buoyant submerged body is stable only when G lies directly below B.

For stable floatation under a small tilt, the quantity that matters is \overline{MG}, and G must lie below M.

• Restoring couple — a moment that rotates a tilted body back toward its original position. Example: a boat heeled slightly develops a restoring couple when M is above G.
• Neutral equilibrium — the body stays at any new tilt angle because no couple acts on it.
• Metacentric height — the distance \overline{MG}; it must be positive for a floating body to be stable.
• Centre of buoyancy B — the centroid of the displaced fluid volume; it is not the reference for the metacentric height.

Hydrostatic force on curved surface

Pressure on a submerged curved surface acts normal to the surface at every point, so the resultant is found by resolving it into horizontal and vertical components. The horizontal component equals the force on the vertical projection of the surface, F_H = \rho g \bar{h} A_v, and the vertical component equals the weight of the fluid column lying directly above the surface up to the free surface level, F_V = \rho g V. The magnitude of the resultant is \sqrt{F_H^2 + F_V^2}. For a circular arc, every pressure force passes through the centre, so the resultant also passes through the centre, and the components give the magnitude. When liquid rests on one side of a horizontal cylinder with its level at the top, the net vertical force per unit length is \rho g times the area of the wetted half of the circular cross-section.

For a quarter-circle arc in a half-filled pipe, the horizontal component is the force on the vertical face of water of depth R, which is \rho g R^2/2 per unit length. The vertical component is the weight of the quarter-disc of water, which is \rho g \pi R^2/4 per unit length. The force the arc supplies is the vector sum of these components, and the water exerts an equal and opposite force on the pipe section.

• Vertical projection — the flat area seen in the horizontal direction, used to find the horizontal force.
• Imaginary fluid — the volume of fluid that would lie above a surface on the opposite side; it still gives the vertical force.
• Resultant through centre — for a circular arc, the line of action of the total pressure force always passes through the centre of curvature.

Absolute, gauge and vacuum pressure

Pressure can be measured relative to a perfect vacuum, which is absolute pressure, or relative to local atmospheric pressure, which is gauge pressure. The relation is p_{abs} = p_{gauge} + p_{atm}. A vacuum gauge reads the amount by which the pressure falls below the local atmosphere, so p_{vac} = p_{atm} - p_{abs}. Because the atmospheric pressure is local and changes with weather and altitude, a gauge reading is meaningful only with the local atmospheric value. Mercury columns are often used for these readings, where 1\ \text{mm Hg} \approx 133.3 Pa.