Mechanics of Fluids

centre of buoyancy on to the transverse plane represented in the diagram.)
The depth of each element of volume is now z + y tan θ, so
Subtraction of eqn 2.20 from eqn 2.21 gives
�
Vy = y(z + y tan θ)dA (2.21)
�
V(y − y0 ) = y 2 tan θdA = tan θ(Ak 2 )Ox
where (Ak 2 )Ox represents the second moment of area of the plane of flotation
about the axis Ox (see Fig. 2.26c).
But, for small angular displacements, y − y 0 = (BM) tan θ and therefore
or
V(BM) = (Ak 2 )Ox
(BM) = (Ak2 )Ox
V
The stability of bodies in fluids 75
(2.22)
The length BM, sometimes known as the metacentric radius, is therefore
equal to the second moment of the plane of flotation about the centroidal
axis perpendicular to the plane of rotation divided by the immersed volume.
BM must not be confused with the metacentric height GM.
For rolling (i.e. side to side) movements of a ship the centroidal axis about
which the second moment is taken is the longitudinal one. Stability in this
direction is normally by far the most important. For pitching movements (i.e.
stern up, bow down or vice versa) the appropriate axis is the transverse one.
The metacentres corresponding to different axes of rotation in general have
different positions. The position of B can be calculated since the contours of
the hull at various levels are normally known, and hence the position of the
metacentre may be determined from eqn 2.22.
The equation strictly applies only to very small angular displacements,
and this limitation is more important if the body does not have vertical sides
(although for ships the sides are usually approximately vertical at the waterline).
The result may legitimately be used to indicate the initial stability of the
body. It is nevertheless sufficiently accurate for most calculations involving
angles up to about 8 ◦ .
The value of BM for a ship is of course affected by change of loading
whereby the immersed volume alters. If the sides are not vertical at the
water-line the value of Ak 2 may also change as the vessel rises or falls in
the water. Naval architects must design vessels so that they are stable under
all conditions of loading and movement. Wide ships are stable in rolling
movements because (Ak 2 )Ox is then large and the metacentre high.