DECEMBER 2019

PHYSICS TODAY 71
above and below the ﬂuid surface have
been established by the density contrast,
it will ﬂoat stably in any orientation. But
a cubic body has welldeﬁned stable ori
entations relative to the ﬂuid surface that
over a wide range of densities do not in
clude the intuitive orientations—ﬂoating
with a face or edge parallel to the ﬂuid
surface or with a corner pointing upward,
perpendicular to the surface. To test that
assertion experimentally, just place a cube
of wood in water and let it stabilize.
How will an iceberg shaped like the
one shown in ﬁgure 1 establish a stable
equilibrium orientation? A geometry
useful for examining stability quantita
tively is the circular cylinder of length H
and diameter D, as shown in ﬁgure 2.
When H < D, the cylinder is a disk, and
when H > D, the cylinder is elongated,
like a wine cork or pencil. The question
raised by ﬁgure 1 is whether cylindrical
bodies of ice whose lengths exceed their
diameters ﬂoat with their long axes per
pendicular to the water surface? The an
swer is generally no, particularly for
elongated cylinders with H > 2D.
The shape of stability
In 2004 D. S. Dugdale presented a useful
discussion of the ﬂoatingcylinder prob
lem. He deﬁned four domains of stable
equilibrium in density–shape space: (I) The
cylinder ﬂoats with its rotational axis perpendicular to the
water surface and its circular faces parallel to the water surface.
(II) The cylinder ﬂoats with its rotational axis parallel to the
water surface and its circular faces partially submerged to a
depth dependent on the density of the cylinder. (III) The rota
tional axis of the cylinder is tilted at an angle neither parallel
nor perpendicular to the surface. (IV) The orientations de
scribed in I and II are both stable.
Figure 2 illustrates the complexity of those domains of sta
ble equilibrium as a function of density contrast and cylinder
shape. For ice ﬂoating in water (ρ = 0.9), stable equilibria exist
only under the conditions of domains I, II and IV; the density
conditions that yield stable equilibria within domain III (that
is, in the range 0.2 < ρ < 0.8) exclude the ice–water density con
trast. Therefore, ice cylinders ﬂoating in water will stabilize in
only two orientations—with the cylindrical axis either perpen
dicular or parallel to the water surface.
In 1991 Edgar Gilbert showed that for a stable equilibrium
with the cylindrical axis perpendicular to the water surface, the
following condition must hold: ρ(1 − ρ)(2H/D)
2
< 0.5. For ρ =
0.9, the equation requires that H/D < 1.1785. A glance at the ice
berg in ﬁgure 1 leaves li"le doubt that its H/D ratio is greater
than that. So the iceberg violates the stability condition re
quired for the cylinder to ﬂoat with its rotational axis perpen
dicular to the water surface. It would therefore spontaneously
reposition itself to an orientation with its rotational axis hori
zontal. In that equilibrium orientation, the waterline on the
cylinder is a rectangle of length H and width w.
The width is determined by the density diﬀerence between
the ﬂoating body and the ﬂuid—that is, by how much of the
cylinder is submerged. Gilbert showed that the equilibrium is
stable if w < H. For ρ > 0.5, w < D, and all cylinders where H/D >
1, the condition for stability is met because w < D < H. The actual
stability ﬁeld, as determined from the condition w < H, is H/D >
0.7266, as shown in ﬁgure 2. Therefore, along the dashed line ρ
= 0.9 and in the range 0.7266 < H/D < 1.1785, both cylinder ori
entations are stable and can coexist. For H/D > 1.1785, certainly
the case for the iceberg in ﬁgure 1, the equilibrium orientation is
intrinsically unstable and does not occur in nature.
Additional resources
‣ D. S. Dugdale, “Stability of a ﬂoating cylinder,” Int. J. Eng.
Sci. 42, 691 (2004).
‣ E. N. Gilbert, “How things ﬂoat,” Amer. Math. Monthly 98,
201 (1991).
‣ J. F. Nye, J. R. Po"er, “The use of catastrophe theory to
analyse the stability and toppling of icebergs,” Ann. Glaciol. 1,
49 (1980).
‣ J. C. Burton et al., “Laboratory investigations of iceberg cap
size dynamics, energy dissipations and tsunamigenesis,”
J. Geophys. Res. Earth Surf. 117, F01007 (2012).
PT