The most well-known image associated with the "tip of the iceberg" ...
In February 2021 glaciologist Megan Thompson-Munson actually [tweet...
- A floating body is **stable** if, when it is displaced, it re...
It's fascinating how homogeneous bodies with a symmetry like a cyli...
Icebergs also "topple" and one of the causes for this is the consta...
[Stanislaw Ulam](https://en.wikipedia.org/wiki/Stanislaw_Ulam) once...
70 PHYSICS TODAY
|
DECEMBER 2019
T
he phrase “tip of the iceberg” suggests that what you see
is much less than what is hidden from view. The concept
of a tip above seawater and a much larger root below
more or less conforms to Archimedes’s principle of buoy-
ancy: The force exerted on a body partially or completely
immersed in a fluid of higher density is directed upward
and is equal to the weight of the fluid that the body displaces.
When a totally submerged lower-density body is released,
the buoyancy force causes it to rise until it reaches a floating
equilibrium. The tip then rests above the surface and the root
below it, with the mass of each determined by the density con-
trast between the floating solid and the surrounding fluid. Fig-
ure 1 depicts a common representation of such an equilibrium.
But the configuration is pure artistic license—it does not dis-
play a stable orientation and does not exist in nature.
Sphere, cube, and cylinder
What’s wrong with the image? A floating elongated iceberg can
satisfy the buoyancy requirements of Archimedes’s principle in
many orientations, but most, including that depicted in figure
1, turn out to be unstable. To see an example of such an insta-
bility, take a wine cork and immerse it in water in any orienta-
tion. Upon release, the cork will rise to the surface and float
only with its long axis horizontal—that is, parallel to the sur-
face of the water.
An equilibrium orientation of a floating body occurs when
the center of gravity (the center of mass of the whole object)
and the center of buoyancy (the center of mass of just the sub-
merged part) are vertically aligned. If perturbations from wind,
waves, or melting lead to a small departure of that alignment,
a torque is created that reorients the body. If the torque ampli-
fies the misalignment, the orientation is unstable; if the torque
reduces the misalignment, the orientation is stable.
What are the parameters that define a stable equilibrium
orientation? A floating object must satisfy Archimedes’s prin-
ciple by displacing a mass of fluid equal to its own. Because
the object is less dense than the underlying fluid, it projects
some volume above the surface and some volume below. Thus
the first parameter that determines the stable equilibrium is
the density contrast between the floating body and the sur-
rounding fluid, here defined as a ratio ρ of the two densities,
with 0 < ρ < 1.
The density of ocean water depends on both temperature
and salinity; the density of ice depends on ambient tempera-
ture and the concentration of bubbles and structural voids. But
an ice–water density ratio of 0.90 is accurate enough to charac-
terize the stability of ice objects afloat in the ocean. It is behind
the tip-of-the-iceberg concept because at typical densities of the
two phases, about 90% of an iceberg’s volume is submerged,
leaving only the tip above water.
The second important parameter is the shape of the floating
body. Consider a floating sphere. Once its fractional volumes
Henry Pollack is an emeritus professor
of geophysics at the University of
Michigan in Ann Arbor.
QUICK STUDY
Tip of the iceberg
Henry Pollack
The conditions required for an object to float in a stable orientation sometimes
lead to surprising results.
FIGURE 1. AN ARTISTIC RENDITION OF A FLOATING ICEBERG
in apparent equilibrium. This orientation could never stably exist
because an elongated piece of ice would float on its side, not on
its head. (Image by iStock.com/the-lightwriter.)
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 well-defined 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 floating-cylinder 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
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.5
0.6
0.7 0.8 0.9
1.0 1.1 1.2 1.3
LENGTH/DIAMETER ( / )HD
()ρ
DENSITY OF FLOATING SOLID
DENSIT
Y OF SURROUNDING FLU
ID
HD/ = 0.7266 HD/ = 1.1785
HD/ = 0.5
Domain I
HD/ = 0.9
Domain IV
HD/ = 1.3
Domain II
D
D
DD
H
H
HH
IIIIII
IV
IV
FIGURE 2. DOMAINS OF STABILITY FOR FLOATING CYLINDERS. A plot of four stability
domains is shown as a function of a cylinder’s shape H/D and the solid-to-liquid density
ratio ρ. Each domain is characterized by the equilibrium orientation in which a cylinder will
float. The dotted line at the density ratio ρ = 0.9 corresponds to ice floating in water. Illustra-
tions of stable cylinder orientations in domains I, II, and IV at loci intersected by ρ =0.9 are
shown above the graph; their submerged roots are shaded and their above-water tips un-
shaded. An ice cylinder will float with its rotational axis perpendicular to the water surface
when H/D < 0.7266, and with its rotational axis parallel to the surface when H/D > 1.1785. In
the range 0.7266 < H/D < 1.1785, both equilibrium orientations can coexist. (Adapted from
D. S. Dugdale, Int. J. Eng. Sci. 42, 691, 2004.)
In February 2021 glaciologist Megan Thompson-Munson actually [tweeted](https://twitter.com/GlacialMeg/status/1362557149147058178) a plea for scientists to draw icebergs accurately. Megan's tweet inspired the creation of a [tool](https://joshdata.me/iceberger.html) to see how an iceberg of arbitrary shape would float.
Icebergs also "topple" and one of the causes for this is the constant change in shape due to non-uniform melting.
[Stanislaw Ulam](https://en.wikipedia.org/wiki/Stanislaw_Ulam) once asked if spheres are the only homogeneous bodies that can float in every orientation. To this day, no one has been able to proved or disprove the question.
It's fascinating how homogeneous bodies with a symmetry like a cylinder can float in surprising ways: with the axis of
symmetry tilted away from vertical at strange angles.
Below are some Stable orientation of cylinder of density 1/2 as function of H/R by E. N. Gilbert.
The most well-known image associated with the "tip of the iceberg" metaphor was taken by professional photographer Ralph Clevenger, who composed the image using 4 separate photographs: the shot of the clouds and underwater water was taken in Santa Barbara. The picture of the top part of the iceberg was taken in Antarctica and the bottom part in Alaska.
The image was created with a marketing plan in mind. The "tip of the iceberg" expression is a metaphor for the idea that the visible portion of an iceberg is only a small fraction of its total size.
The image has been widely used for inspirational and advertising purposes and has generated almost $1M in sales.
- A floating body is **stable** if, when it is displaced, it returns to equilibrium.
- A floating body is **unstable** if, when it is displaced, it moves to a new equilibrium.
The given diagram illustrates a floating object that is tilted at an angle Δθ. In its untilted position, the center of gravity, G, is the point where the weight of the object, W, is exerted. On the other hand, the center of buoyancy, B, represents the centroid of the volume of fluid displaced by the object, and the buoyancy force, FB, acts on this point.
When the body tilts, its centre of buoyancy shifts to a new location, B', due to the change in the shape of the displaced volume. The point at which a vertical line from B' intersects the body's line of symmetry is called the metacentre, M. The buoyancy force, FB, now acts at B'.
The central diagram in the figure shows that the weight, W, and FB create a restoring moment that brings the body back to its untilted position. However, the right-hand diagram in the figure demonstrates that W and FB also generate an overturning moment that causes the body to tilt further in the same direction.
Thus, it can be concluded that if the metacentre, M, is situated above the centre of gravity, G, the body is considered stable. Conversely, if the metacentre, M, is located below the centre of gravity, G, the body is regarded as unstable.