Mechanics of Fluids

couple acting on the body in its displaced position is a restoring couple, that
is, it tends to restore the body to its original position. If M were below G the
couple would be an overturning couple and the original equilibrium would
have been unstable.
The distance of the metacentre above G is known as the metacentric height,
and for stability of the body it must be positive (i.e. M above G). Neutral
equilibrium is of course obtained when the metacentric height is zero and G
and M coincide. For a floating body, then, stability is not determined simply
by the relative positions of B and G.
The magnitude of the restoring couple is W(GM) sin θ and the magnitude
of GM therefore serves as a measure of the stability of a floating body.
A simple experiment may be conducted to determine GM. Suppose that
for the boat illustrated in Fig. 2.25 the metacentric height corresponding to
roll about the longitudinal axis is required. If a body of weight P is moved
transversely across the deck (which is initially horizontal) the boat moves
through a small angle θ – which may be measured by the movement of a
plumb line over a scale – and comes to rest in a new position of equilibrium.
The centres of gravity and buoyancy are therefore again vertically in line.
Now the movement of the weight P through a distance x causes a parallel
shift of the total centre of gravity (i.e. the centre of gravity of the whole boat
including P) from G to G ′ such that Px = W(GG ′ ), W being the total weight
including P. But (GG ′ ) = (GM) tan θ, so
(GM) = Px
cot θ (2.19)
W
Since the point M corresponds to the metacentre for small angles of heel
only, the true metacentric height is the limiting value of GM as θ → 0. This
may be determined from a graph of nominal values of GM calculated from
eqn 2.19 for various values of θ (positive and negative).
It is desirable, however, to be able to determine the position of the metacentre
and the metacentric height before a boat is constructed. Fortunately
this may be done simply by considering the shape of the hull. Figure 2.26
shows that cross-section, perpendicular to the axis of rotation, in which the
centre of buoyancy B lies. At (a) is shown the equilibrium position: after displacement
through a small angle θ (here exaggerated for the sake of clarity)
the body has the position shown at (b). The section on the left, indicated
The stability of bodies in fluids 73
Fig. 2.25