The egg that keeps turning after it is stopped

The mass, and where it sits found that how a body turns is decided by its moment of inertia, the sum of its mass weighted by the square of distance from the axis, and that two bodies of the same mass can turn very differently because one has its mass further out. The quantity that survives a change of shape found that angular momentum, the moment of inertia times the rate of turning, is conserved by any rearrangement inside a body that involves no outside twist. Both essays treated a body as one thing. An egg tests what happens when it is two.

The kitchen test is well known. Lay an egg on its side, spin it, then touch it briefly with a fingertip to stop it, and take the finger away. A hard-boiled egg stays where it was stopped. A raw egg, after a moment’s pause, starts to turn again on its own, as though it had changed its mind. The usual explanation, that the liquid inside keeps moving, is right; what it does not say is how much the liquid keeps, how fast it hands it back, and what fixes either.

Two bodies held together by viscosity #

A raw egg is a shell, about six grams of calcium carbonate a third of a millimetre thick, round fifty-odd grams of white and yolk. The shell and its contents have the same axis but not the same rotation. The shell can turn its contents only through the friction between them, and a liquid’s friction is its viscosity, which transmits a twist by shearing: the layer next to the shell is dragged round, drags the next layer more weakly, and so on inward. The figures model the egg as a sphere of radius 2.2 centimetres, a thin shell coupled to a uniform liquid ball by a twisting torque proportional to the difference in their rates of turning, with one parameter, the time $\tau$ the contents take to follow a change in the shell’s rotation.

A cooked egg has no such parameter. Its contents are set into a solid that turns with the shell as one rigid body, and stopping the shell stops everything. The first figure’s cooked egg simply halts.

The raw egg’s shell is brought to rest by the finger as easily, because it carries little of the egg’s rotation. The contents are not touched. They feel only the drag of the shell, now stationary, and that drag slows them at the rate their spin-up time allows: in half a second, held against a still shell, contents with a spin-up time of a second and a half lose about a quarter of their speed. Release the shell, and the turning contents drag it round again. The two settle at a common speed set by sharing the contents’ angular momentum with the shell, and since the contents hold five-sixths of the moment of inertia, the shell recovers most of what the contents kept. The raw egg restarts at 60 per cent of its old speed.

The share the shell carries #

The shell’s share of the moment of inertia controls almost everything, and the second figure computes it. A thin spherical shell has a moment of inertia of two-thirds of its mass times the square of its radius; a solid ball, two-fifths. For six grams of shell and fifty-four of contents, the shell’s share is 16 per cent. Mass for mass the shell is more effective than the contents, since all of it sits at the outer radius, but there is nine times less of it.

The share decides what happens when the shell is twisted. A twist of the fingers is an angular impulse, and it goes first into whatever the fingers are holding. A cooked egg takes it as a whole. A raw egg’s shell takes it alone, before its contents have had time to respond, and with only a sixth of the inertia it starts six times as fast. Then the contents catch up, the shell slows, and the two end at the speed the cooked egg had from the start, because angular momentum is conserved inside the egg and both received the same impulse.

What is not conserved is the energy. Kinetic energy of rotation goes as the square of the speed, and sharing a fixed angular momentum between two bodies that start at different speeds and end at the same one loses energy, exactly as a perfectly inelastic collision does: collisions are easier than forces found that two bodies that stick together lose the fraction of energy that sticking can afford, fixed by the masses alone. Here it is fixed by the moments of inertia alone: the fraction lost is the contents’ share, 84 per cent for an egg. The lost energy has gone into heating the egg’s contents by viscous shearing, and the total does not depend on how viscous they are. A thinner liquid takes longer to catch up and shears more gently for longer; the heat is the same.

The flick that sends a raw egg off fast #

The third figure follows both eggs after the same flick, with the friction of the table included. The cooked egg starts at three revolutions a second and slows steadily as the table brakes it. The raw egg’s shell starts at nineteen and within a second has fallen to the cooked egg’s speed, its contents rising to meet it. From then on the raw egg runs down almost alongside the cooked one, but not quite: the table brakes only the shell, so the shell runs slightly slower than its contents throughout, the contents pushing it round, and the raw egg’s slightly lower spin at every moment comes from that lag.

The fast start is a prediction that sounds wrong and is easy to see. A raw egg flicked hard does spin very rapidly for a moment and then visibly slows, where a cooked egg flicked the same way spins steadily. People usually say the raw egg spins worse, and in the sense that it loses most of the energy of the flick, it does. In the sense of its first spin, it spins better. The disk that spins faster as it stops found a coin whose visible spin accelerates while its energy drains away; the raw egg’s shell does the opposite, slowing while its energy is merely being handed inward, and in neither case does the motion the eye follows tell where the energy is.

Real raw eggs come to rest faster still than the figure shows, for a reason the coaxial model leaves out. A spinning egg is not perfectly balanced on its axis, and it wobbles; a wobbling shell sets up flows inside its contents that do not simply follow its rotation, and those flows dissipate energy continually. The axis a leak of energy chooses found the same internal dissipation tipping a spinning satellite with flexing antennas into tumbling end over end. An egg spun on its side is turning about its axis of largest moment of inertia, the stable one, so the dissipation does not tip it but merely brings it to rest.

The time the inside takes to follow #

The fourth figure turns the kitchen test into an instrument. The longer the shell is held, the more the contents slow against it, and the less the egg recovers on release. The recovered fraction falls from the contents’ share of the moment of inertia, for an instant’s touch, exponentially with the duration of the hold, and the rate of the exponential is the spin-up time. Holding eggs for different times and measuring how fast each restarts would measure $\tau$ directly.

What sets $\tau$ is subtler than viscosity alone. If momentum spread inward from the shell only by viscous diffusion — the picture momentum going sideways gave of viscosity, a diffusion of momentum between layers — the time would be the square of the radius over the kinematic viscosity, $R^2/\nu$: eight minutes for water in an egg-sized container, longer than any egg is ever spun. A spinning container brings its contents round much faster than that, by a mechanism Vagn Walfrid Ekman found in 1905 in the context of ocean currents driven by wind.

When a container’s rotation changes, the fluid touching its walls is dragged into a thin boundary layer, the Ekman layer, a few times $\sqrt{\nu/\Omega}$ thick — a fraction of a millimetre for water spinning a few times a second. In that layer the fluid’s rotation differs from the interior’s, and the imbalance between pressure and centrifugal force drives a flow along the wall, outward or inward, which must be returned through the interior. The circulation carries fluid with the wall’s angular momentum into the interior far faster than diffusion could, and the interior spins up in a time $R/\sqrt{\nu\Omega}$: five seconds for water spinning three times a second, a second or two for liquids a few times more viscous. The fifth figure draws it. The spin-up time falls with the spin rate, because a faster spin makes a thinner Ekman layer that pumps harder.

Raw egg white is several times more viscous than water and the yolk much more, and their spin-up times in an eggshell are of the order of a second, which is the scale the figures assume. That is why the egg’s kitchen test works on a human timescale. With a liquid as thin as water inside, the restart would be stronger and slower to fade; with treacle, the egg would behave almost like a cooked one, its contents following the shell within a fraction of a second.

The race a raw egg wins #

The same two-body picture says something about rolling that is easy to try. The mass, and where it sits raced bodies down a slope and found that the one with less of its mass far from the axis wins, because less of the energy released by the fall goes into rotation: a solid ball beats a hollow one. Lay a cooked egg and a raw egg side by side on a gentle slope and let them go. The cooked egg must turn as a whole, and with the model’s shares about seventy per cent of the energy released goes into its motion down the slope. The raw egg’s shell must turn, since it rolls, but its contents need not, at least at first; they slide round inside, and the egg starts down the slope as though only its shell had any rotational inertia, with about ninety-four per cent of the energy going into forward motion. The raw egg pulls ahead in the first second, until its contents have spun up and it rolls as a whole. A slope is a more forgiving test than a spin, and it gives the same answer for the same reason.

The cooked egg that stands up #

There is a second kitchen test, older as a party trick than as physics, and it separates the two eggs for the same underlying reason. Lay a hard-boiled egg on its side on a table and spin it fast about the vertical, and it rises: within a second or two it is spinning upright on its pointed or blunt end, its centre of mass lifted by a few millimetres. A raw egg spun the same way stays on its side. The rising was not explained until 2002, when Keith Moffatt and Yutaka Shimomura showed that it is driven by friction at the point of contact. A spinning egg tilted on its side slides against the table where it touches, the friction force on that sliding point exerts a torque that tilts the spin axis towards the egg’s long axis, and the energy that lifts the centre of mass comes from the spin itself. The motion is the same family as a tippe top’s, which turns over onto its stem when spun, and it happens only above a critical rate of spin, below which friction merely slows the egg down.

A raw egg fails the test because its contents do not take part. Spinning it on its side means spinning the shell; the liquid inside, as the figures here showed, follows only over its spin-up time, which is about as long as the rise itself would take, and while it lags, its shearing against the shell takes energy from the spin and turns it into heat. The shell alone has too little inertia, and the liquid drains what it has. Both tests — the one that stops an egg and lets go, and the one that spins it on its side — are measuring whether the inside turns with the outside, and a solid egg is the only kind that answers yes at once.

A planet with a liquid inside #

The Earth is a raw egg. Its iron core, 3,500 kilometres in radius, is liquid, inside a solid mantle, and the two are coupled by viscosity, by electromagnetic forces between the conducting core and the mantle, and by the pressure of the core against the mantle’s slightly flattened shape. Because the coupling is imperfect, the core and the mantle can turn slightly differently. When the Moon and the Sun pull on the Earth’s equatorial bulge and make its axis precess and nod, the core does not quite follow the mantle, and a free oscillation of the core relative to the mantle, the free core nutation, shows in the Earth’s rotation with a period of about 430 days as seen in space. The wobble that is slow because the Earth gives found a different free motion of the Earth’s axis, the Chandler wobble, lengthened by the planet’s elasticity and by its oceans; the free core nutation is the egg test applied to a planet, and its measured period, shorter than a core in equilibrium would give, showed the boundary between core and mantle to be slightly more flattened than a fluid planet’s should be.

The same physics turns up in engineering wherever a container of liquid is spun. A spacecraft’s tanks of propellant behave like the egg’s contents, and the attitude-control engineers who model them use exactly this picture of a rigid shell coupled to a slower-responding interior; a spinning wheel of liquid in a centrifuge takes its spin-up time to reach full speed, and the energy lost in the process heats the sample.

Where the coaxial model stops #

The figures treat the egg’s contents as a single rigid lump coupled to the shell by one time constant, and they are neither. The white and the yolk have different viscosities, and the yolk is suspended in the white by two twisted cords, the chalazae, so that its rotation lags the white’s in turn. The Ekman spin-up time is itself a property of a whole flow, not a constant: it depends on the spin rate, as the last figure shows, so a single $\tau$ is a simplification for an egg whose speed changes by a factor of six. The egg is not a sphere, and its long axis makes it wobble when spun on its side. And the coupling torque is linear in the difference of speeds only for small differences; an impulsive flick that sets the shell turning six times faster than its contents creates violent shear near the wall that the linear model does not describe.

None of this changes the two results that do not depend on the model’s details: that the energy lost in sharing a flick is the contents’ share of the moment of inertia, and that a stopped shell restarts with the angular momentum the contents kept. Both follow from conservation laws applied to two bodies.

What a speed curve leaves out #

The figures plot rates of turning, and the interesting physics happens in flows the curves do not show: the thin boundary layers against the shell, the circulation they drive through the interior, the shearing that turns the flick’s energy into heat. Nor do the figures show the most striking thing a hard-boiled egg does, which is to rise up and spin on its end when spun fast enough on its side — a rise driven by friction at the point of contact, which Keith Moffatt and Yutaka Shimomura analysed in 2002, a cousin of the gyroscopic rise of the top that nods before it settles, and which a raw egg, its energy draining into its contents, cannot manage. The test that separates the two kinds of egg has a second form, and it rests on the same difference.

Still open: how the contents of a wobbling shell dissipate #

The coaxial model captures the egg’s response to twists about its axis, but not the steady loss of a wobbling one, and the general problem — how a liquid inside a precessing, wobbling container moves and how much energy it dissipates — is still being worked out. It matters for planetary cores, whose flows driven by precession may help power their magnetic fields, and for spacecraft whose sloshing propellant has destabilised their rotation. Experiments with liquid-filled spheres precessing in laboratories show the interior flow becoming unstable and turbulent above a threshold of precession rate, and which regimes apply to the Earth’s core, or to a wobbling egg, has not been settled.

The habit worth carrying away is to ask how many bodies a body is. A raw egg is a shell carrying 16 per cent of its moment of inertia, coupled to its contents by viscosity over about a second, so it starts a flick six times faster than a cooked egg, loses 84 per cent of the flick’s energy as its contents catch up, and, stopped for half a second, restarts at 60 per cent of its old speed from the angular momentum its contents kept. Anything that holds a liquid while it turns keeps a memory of its past rotation for as long as its contents take to forget.