The log that floats on its corner

The weight of the water that is not there found Archimedes’ principle: a floating body sinks until it has displaced its own weight of water. Why a ship comes back upright found what decides whether it stays the right way up — the metacentre, set by the shape of the slice the water line cuts through the body — and closed with a round log, which floats on its side, spins freely about its axis and refuses to stand on end. The block the water does not lift, the body that displaces two things and the depth past which it must sink found the places where the principle is misapplied.

A round log turned about its own axis presents the same shape to the water at every angle, and has no preferred orientation. A square log does not, and asking how it floats turns out to have an answer as intricate as it is simple to compute. It depends only on how dense the log is — and the dependence has four regimes, two bifurcations and a symmetry between light and heavy that no intuition about floating suggests. This essay computes all of it, and finds the same arithmetic in the rolling of icebergs.

What a floating body settles into #

A floating body at rest must displace its own weight of water, which fixes how much of it is submerged: a fraction equal to its density relative to water. That leaves the orientation free, and the orientation the body chooses is the one with the lowest potential energy for the whole system of body and water.

That energy has two parts. The body’s centre of gravity rises and falls as it turns, changing its potential energy; and the water it displaces changes shape, changing the water’s. When the body sinks a little on one side the displaced water moves, and lifting water out of the hole the body makes costs energy in proportion to the depth of the hole’s centre — the centre of buoyancy. The total works out, up to a constant, as the body’s weight times the height of its centre of gravity above its centre of buoyancy. A floating body turns to bring its centre of buoyancy as high as possible relative to its centre of gravity, not to bring its centre of gravity as low as possible, and for a body of uniform density those are different instructions.

For a long square log of uniform density the centre of gravity is fixed at the centre of the square. The centre of buoyancy is the centroid of the submerged part of the cross-section, which changes as the log turns. So the problem reduces to plane geometry: for each angle, find the water line that cuts off the right area, find the centroid of the part below it, and see how high it sits.

The energy as it turns #

The figure computes the energy as the log is turned from flat, through corner-down at 45 degrees, to flat again at 90. For a light log, a tenth of water’s density, the energy is lowest flat and rises steadily to a maximum at 45 degrees: the log floats flat and, if pushed onto a corner, rolls back. For a log half as dense as water the picture is reversed. Lying flat is a maximum of the energy, a hilltop, and the minimum is at 45 degrees: the log floats corner-down, with a diagonal vertical and the water line running across its middle from one side corner to the other.

For a log a quarter of water’s density neither is the minimum. Its energy is lowest at 26.6 degrees, and it floats tilted, one face at a slant, in a position that looks as if it has been caught in the act of rolling over. It is not rolling. It is at rest in a stable equilibrium that exists only because of the geometry of a square cut by a line.

Why the diamond wins at half density #

The half-density case can be checked with a ruler, and doing so shows what the energy is measuring. Take a square of unit side, half submerged. Lying flat, the water line passes through the centre and the submerged part is a rectangle half a unit deep; its centroid is a quarter of a unit below the water line, and so a quarter below the log’s centre. Standing on a corner, the water line again passes through the centre — by symmetry, half the area is below it — and the submerged part is a triangle with its apex pointing down, whose height is half the square’s diagonal, 0.707. A triangle’s centroid is a third of its height from its base, so the centre of buoyancy is 0.236 below the centre.

The diamond’s centre of buoyancy is higher, by 0.014 of the side, and that is all it takes. The triangle puts more of its submerged area near the water line, where the log is widest, and less near the bottom, where it narrows to a point; the rectangle distributes its area evenly down to its flat bottom. The body settles in the orientation whose wet part carries its area highest. That is a statement about shapes, not about weights, which is why the answer depends on nothing but the density and the square.

A log that has to choose #

The tilted states have a feature worth pausing on. At a density just above 0.211, lying flat has become an energy maximum, and the two nearest minima are tilted by equal angles to the left and to the right. The log is perfectly symmetric and so is the water. Yet it cannot stay symmetric: it must lean one way or the other, and which way is decided by whatever tiny disturbance happens to push it first.

This is the simplest possible example of a symmetry-breaking bifurcation, and it has the same mathematics as a phase transition. The density plays the part of temperature; the tilt angle plays the part of the order parameter — the magnetisation of a magnet, the difference between two phases. Below the critical density the only equilibrium is the symmetric one, zero tilt. Above it the symmetric state is unstable, and the stable tilt grows from zero as the square root of the distance past the critical density, which is the characteristic behaviour of a continuous transition in the simplest theory of such things. The sand that chooses a side met the same structure in a box of shaken grains that suddenly gathers into one compartment, and the transition with nothing to order met it in materials.

A log floating near 0.211 is therefore a toy model of a magnet near its Curie point. Its preferred tilt is small and easily disturbed, its oscillations about it are slow because the energy landscape is almost flat, and a log whose density drifts slowly across the threshold — soaking up water, say — tips gradually from lying flat into leaning, rather than jumping. At 0.28, where the tilt reaches 45 degrees and merges into the corner-down state, the same thing happens in reverse.

Every density a log could have #

Sweeping the density from nothing to water’s own gives the whole story in one curve. Below 0.211 the log floats flat. At 0.211 flat stops being stable, and the stable angle grows continuously from zero — a bifurcation, in which one equilibrium splits into two tilted ones, left and right. By about 0.28 the angle has reached 45 degrees, and from there to about 0.72 the log floats on its corner. Then the sequence reverses: tilted again, and flat again from 0.789 to 1.

The curve is exactly symmetric about one-half, and the reason is one of the prettier facts in hydrostatics. A log of density $s$ has a fraction $s$ of its cross-section under water and $1 - s$ above. A log of density $1 - s$ has exactly the reverse. The energy condition — the centre of buoyancy as high as possible relative to the centre — becomes, for the part above water, the condition that the centroid of the dry part be as low as possible relative to the centre, and the dry part of one log is the wet part of the other turned upside down. Every floating position of a log of density $s$ is, rotated through 180 degrees with air and water exchanged, a floating position of a log of density $1 - s$. Light logs and heavy logs are the same problem.

What the logs look like #

The five cross-sections make the symmetry visible. The lightest log floats flat and high, a tenth submerged. At a quarter of water’s density it has turned to 27 degrees. At half density it floats as a diamond, the water line through its two side corners. At three-quarters it has turned back to the same 27 degrees, now low in the water, and at 0.9 it floats flat, with only a tenth showing — the lightest log turned over. Seen from the water line, the dry part of each heavy log is the wet part of the corresponding light one.

Woods span most of that range. Balsa, around a sixth of water’s density, floats flat. Pine and many other softwoods, around half, float on a corner if they are square and seasoned. Oak, near three-quarters, floats tilted or flat depending on its exact density and how much water it has soaked up. A square beam of seasoned pine thrown into a river will turn onto its corner within seconds, which is visible to anyone who tries it and surprising to almost everyone who does.

The metacentric test #

The same boundary can be found without computing any energies, by the test why a ship comes back upright developed: a floating body is stable in a given position if its metacentre lies above its centre of gravity.

For a square lying flat the metacentric height has two terms. The first is the restoring effect of the water line’s width, the second moment of the water-line area divided by the displaced volume: for a square of unit side, $1/12s$. It is large when little of the log is submerged, because a small displaced volume is shifted a long way by a small tilt. The second is the height of the centre of gravity above the centre of buoyancy, $(1 - s)/2$, which is large when the log floats high. Flat is stable where the first exceeds the second, and the figure shows that it fails between the two roots of $6s^2 - 6s + 1 = 0$, which are $(3 \mp \sqrt{3})/6$ — exactly where the energy curves said.

The metacentric test says only when flat stops being stable, not what replaces it. The energy calculation is needed for that, and it gives the tilted and corner-down states. The two methods answer different questions — local stability of one position against the global lowest energy — and they agree where they overlap, which is a check on both.

How wide a plank has to be #

A plank is a square stretched sideways, and stretching it widens the water line, which strengthens the first term.

For a rectangle of width $b$ and thickness $a$, the same condition becomes $(b/a)^2 > 6s(1 - s)$. The required ratio is largest at half density, where it is $\sqrt{1.5}$, 1.22. Any timber more than 1.22 times as wide as it is thick floats broad face up whatever its density, which is why planks, boards and rafts behave as expected and square beams do not. A square is simply not wide enough to be sure of lying flat, and the range of densities over which it fails is exactly the range over which the curve rises above one.

Icebergs that roll over #

The same arithmetic decides the fate of icebergs. Ice has about 0.89 of the density of seawater, and a large tabular iceberg — a slab broken from the front of an ice shelf, often hundreds of metres thick and kilometres wide — floats flat, with about a ninth of its thickness above water. As it drifts it melts and fractures, and the pieces it breaks into are narrower. The plank condition says that a slab of ice floats flat only if its width exceeds $\sqrt{6 \times 0.89 \times 0.11}$ times its thickness: about 0.77. A piece narrower than that is unstable lying flat, and capsizes.

Iceberg capsize is violent. A block of ice hundreds of metres tall rolling through ninety degrees releases a large fraction of its potential energy in under a minute, generating waves, tsunamis in narrow fjords, and seismic signals. The glacial earthquakes recorded at the calving fronts of Greenland’s glaciers — seismic events detectable across the world — are produced when newly calved icebergs, thicker than they are wide, capsize against the glacier front. The same instability that turns a square pine beam onto its corner in a stream moves the ground under Greenland.

Where the model stops #

The calculation is for a long log, a two-dimensional cross-section, of uniform density, in still water, with surface tension ignored. A short log or a cube has ends, and the water line is then a rectangle or square whose second moments about different axes differ; the three-dimensional problem has more stable orientations — face up, edge down, corner down and tilted states between — and is correspondingly richer. A real log is not uniform: its heartwood and sapwood differ in density, and a log that has soaked up water becomes heavier over days, sliding along the curve through its regimes and rolling as it passes each boundary. Waves rock a floating body and can tip it from one stable state to another when the energy barrier between them is small, as it is near the bifurcations. And for very small bodies surface tension at the water line changes the forces entirely, which is why a small cube of balsa or a needle can float in positions the buoyancy calculation forbids.

What the pictures cannot show #

The figures show equilibria — where the log comes to rest — and not how it gets there. A log released in the wrong orientation rolls towards the nearest minimum, oscillates about it, and is damped by the water; the time it takes, and whether it overshoots into a neighbouring equilibrium, depend on the water’s drag and the log’s rotational inertia, which the energy calculation does not use. They also cannot show that near each bifurcation the energy landscape is almost flat, so a log whose density is close to 0.211 or 0.28 has only a weak preference and wanders slowly in response to the smallest disturbance. The equilibria are exact; the approach to them is a separate problem in dynamics.

Still open: how often icebergs capsize, and what it costs the ice sheets #

Whether an iceberg capsizes as it calves, or later as it melts, matters for how fast ice shelves and glacier fronts retreat. A capsizing iceberg pushes against the glacier front with a force large enough to move it measurably, can trigger further calving, and mixes the water around it, bringing warmer deep water to the surface. How much of the ice lost from Greenland’s fast glaciers passes through capsize, how much energy it delivers to the ocean and the solid Earth, and whether the process speeds up the retreat of calving fronts in a warming climate are questions being worked on with seismometers, satellite imaging and laboratory models of floating blocks. The stability criterion for a floating block is the simple part of that problem.

The habit worth carrying away is to ask what a floating body actually minimises. A body at rest in water turns to raise its centre of buoyancy relative to its centre of gravity, and for a square that gives four regimes — flat, tilted, corner-down, tilted, flat — with a symmetry between densities s and 1 − s, because the dry part of one log is the wet part of the other. A plank lies flat because it is wider than 1.22 of its thickness, not because it is flat; an iceberg narrower than 0.77 of its height rolls over.