Buoyancy and stability of floating bodies

Archimedes' principle and definitions

Archimedes' principle: a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced. $F_B = \rho g V_{disp}$.

• Centre of buoyancy (B): centroid of the displaced volume of fluid; the line of action of $F_B$ passes through it.
• Metacentre (M): the point where the line of action of buoyancy, for a small angle of heel, cuts the original vertical centre line of the body.
• Metacentric height (GM): distance between the centre of gravity G and the metacentre M.

Expression for metacentric height

When a floating body heels through a small angle $\theta$, the shape of the displaced volume changes: a wedge emerges on one side and an equal wedge is immersed on the other. B shifts to a new position $B'$.

        M
        |\
        | \
   G ---|--\---
        | B  B'
   ~~~~~~~~~~~~~~~~ water line

The horizontal shift of the centre of buoyancy is found from the moment of the wedge buoyancy about the centre line. For small $\theta$, the moment of the wedge volumes equals $I\theta$, where $I$ is the second moment of area of the waterline plane about the longitudinal axis of tilt. Then

$$V \cdot BB' = I\theta \quad \text{and} \quad BB' = BM \cdot \theta$$

so

$$BM = \frac{I}{V}$$

where $V$ is the volume of fluid displaced. Hence

$$GM = BM - BG = \frac{I}{V} - BG$$

$BG$ is positive when G is above B, and negative when G is below B. The metacentric height is taken as positive when M is above G.

Conditions of equilibrium

For a stable floating body, the weight and buoyancy form a righting couple $W \cdot GM \sin\theta$ that restores the body to its original position. For an unstable body the couple is an overturning couple.

For a submerged body, the shape of the displaced volume does not change when the body tilts, so B stays fixed relative to the body and there is no metacentre in the same sense; stability needs B above G.