The quantity that survives a change of shape

A figure skater comes out of a slow turn with her arms out, pulls them to her chest, and speeds up by a factor of four. It is the standard demonstration of conservation of angular momentum, it is in every textbook, and the standard explanation is correct as far as it goes — as is the spinning wheel that refuses to fall over, for the same reason and with the same gap in the telling. What it does not mention is that her kinetic energy has also gone up by a factor of four, and that a conservation law cannot explain an increase in anything.

The two statements are not in competition. Angular momentum is conserved because no torque acts about the axis; kinetic energy rises because the pulling does work. Both are true at once, and the reason a physics course tends to mention only the first is that the second is a question rather than an answer, and answering it needs an integral.

Why nothing exerts a torque #

Torque is a force with a lever arm, and the lever arm is the perpendicular distance from the axis to the force’s line of action. A force aimed straight at the axis has no perpendicular distance at all.

What matters for a moment is not the force but the force times the perpendicular offset of its line of action, so a force whose line passes through the pivot produces no moment whatever its size. The same 120 N at 0.15, 0.30 and 0.45 m gives three different moments; along the radius it gives none. Pulling an arm inward is exactly that force — it points along the radius, its line passes through the axis, and the offset is zero.

That is the whole of the conservation argument. The skater’s muscles pull inward, along a radius. The line of action passes through the spin axis, the moment arm is zero, and no torque about that axis exists — however hard the pull, however fast the arms move, however much work is done. Angular momentum about the axis therefore does not change, and the fact that the body’s shape is changing while it does not change is exactly what makes the statement useful.

The acceleration of a body going round at constant speed points at the centre everywhere, which means the force producing it does too. A force in that direction can change how fast something is going round only by first changing the size of the circle; it can never change the sense of the going round, because it has no component along the motion. That is the whole of why a spin cannot be started or stopped by pulling on it.

There is a second, sharper way to see it. Angular momentum about a point changes at the rate of the torque about that point, and for a central force — one that always points at a fixed centre — the torque about that centre is identically zero. Every consequence of that is a consequence of the direction of a force and nothing else: it does not matter what the force law is, whether it is attraction or repulsion, whether it is constant or wildly varying in time.

What the conservation law does not say #

$L = I\omega$ fixes the product and constrains neither factor. Change the shape and $I$ changes; $\omega$ changes to compensate; and the kinetic energy, which is

$$E = \tfrac12 I\omega^2 = \frac{L^2}{2I},$$

changes as the inverse of the moment of inertia. Halve $I$ and the rate doubles and the energy doubles too. There is no arrangement of masses that keeps both the angular momentum and the kinetic energy fixed while the moment of inertia moves.

This is the point at which the usual account quietly stops. An increase in energy in an isolated system would be a serious matter, and it is not one here, because the system is not isolated in the relevant sense: something is pulling. The question is whether the pulling accounts for the whole of the increase, exactly, with nothing left over.

Where the energy comes from, computed #

Holding a mass $m$ on a circle of radius $r$ at angular rate $\omega$ requires an inward force $m\omega^2 r$. Pull it inward through a distance $\mathrm{d}r$ and the work done is $m\omega^2 r\,\mathrm{d}r$. As the arm comes in, $\omega$ rises, so the force rises too — the pull gets harder the further it goes, which is what anyone who has done this on an office chair reports.

Integrating that force over the whole excursion, with $\omega$ following the conservation law at every step:

$$W = \int_{r}^{r_0} 2m\,\omega(r')^2 r'\,\mathrm{d}r' = \int_{r}^{r_0} \frac{2mL^2 r'}{I(r')^2}\,\mathrm{d}r',$$

and with $I = I_{\text{core}} + 2mr'^2$ the substitution $u = r'^2$ turns it into an elementary integral whose value is

$$W = \frac{L^2}{2I(r)} - \frac{L^2}{2I(r_0)} = E(r) - E(r_0).$$

Exactly the increase in kinetic energy. Not approximately, not to leading order in something small: the two expressions are the same expression. The hero figure computes both independently — the work by numerically integrating the drawn force, the energy from $L^2/2I$ — and refuses to draw at all if they differ by more than a millionth of a joule. They agree to about a ten-millionth.

So the energy has an unremarkable source. Whoever pulls the arms in does work against the force that would otherwise fling them out, and every joule of that work appears as rotational kinetic energy. The four-times-faster spin is paid for at full price, and the price is felt: a skater pulling in from a fast spin is doing real physical work, and it is why the manoeuvre is tiring in a way that looks as though it should not be.

Two features of that integral are worth pausing on, because both are easy to state backwards. The first is that the force rises during the pull, and rises steeply: with the arms coming in from three-quarters of a metre to a tenth, the rate of turn goes up by a factor of four and the required force, which carries $\omega^2$, goes up by sixteen for the same radius and by rather less once the shrinking radius is allowed for. The last few centimetres cost far more than the first few, which is the opposite of how a fixed force behaves and is the reason the manoeuvre has a natural stopping point that has nothing to do with anatomy.

The second is that letting the arms back out returns the energy. The same integral run the other way is negative: the arms do work on whoever is holding them, the rate falls back to where it started, and the kinetic energy comes back down to its original value. Nothing is lost in an idealised cycle, which is what makes this a conservative exchange rather than a dissipative one — and it is why the demonstration can be repeated on the same office chair a dozen times without anybody having to wind anything up. What is being traded is stored work, not fuel.

Two conservation laws that cannot both be used #

The trap this essay exists for is the habit of reaching for whichever conservation law is nearest. In a collision, momentum is conserved and kinetic energy generally is not; in a shape change of a spinning body, angular momentum is conserved and kinetic energy is not. The structures are identical.

Run the same two masses elastically and the kinetic energy does come out equal to what went in. Both collisions conserve momentum exactly, and they differ entirely in what happens to the energy — so momentum conservation cannot be what decides the outcome of either, and the extra information has to come from somewhere else. In the shape-change problem that extra information is the work done pulling, which is why an ice skater’s spin speeds up and a collapsing star’s does not violate anything.

The reason the two laws behave differently is that momentum and angular momentum are conserved by the symmetries of space — nothing about the situation picks out a preferred position or a preferred direction — while energy conservation only constrains the total including every internal form. Kinetic energy alone is not a conserved quantity and never was; it is a conserved quantity in the special case where nothing does work on it and nothing turns it into anything else. Shape changes and collisions are precisely the cases where that is false.

The same argument at astronomical scale #

The most consequential use of this is not a skater but a cloud.

Orbits under four different central force laws are all different curves and none of them but the inverse square closes — and every one of them conserves angular momentum exactly, because every one of them is central. The conservation is a statement about the direction of the force and not about its strength, so it survives changes to the force law that destroy everything else: the shape of the orbit, whether it closes, whether it precesses, whether it is bounded at all.

Kepler’s second law — a planet sweeps equal areas in equal times — is angular momentum conservation, written before angular momentum had a name. The areal rate is $L/2m$, so a statement that it does not change is a statement that $L$ does not change, and the reason it does not change is that gravity points at the Sun. Nothing else about gravity is used: the same law holds for a planet on a spring.

At a collapse of seventy thousand to one in radius the arithmetic becomes violent. The moment of inertia falls as the square of the radius, so the rate of turn rises as its inverse square: a slowly turning object the size of a star, collapsed to the size of a city, ends up turning thousands of times a second. Its rotational kinetic energy has risen by the same enormous factor, taken out of the gravitational energy released in the fall — which is why a pulsar spins as it does, and why it is the fastest-rotating object anybody has measured.

The arithmetic is worth doing rather than gesturing at. A uniform sphere has $I = \tfrac{2}{5}MR^2$, so at fixed mass the moment of inertia falls as $R^2$ and the rate rises as $R^{-2}$; the rotational kinetic energy $L^2/2I$ therefore rises as $R^{-2}$ as well, while the gravitational energy released in the same collapse rises only as $R^{-1}$. Those two exponents are the whole story of why a collapsing cloud does not simply keep falling. A cloud that turns once every ten million years, collapsing by a factor of a thousand in radius, ends up turning a million times faster. This is why nearly everything that forms by gravitational collapse spins: not because spin is common in the raw material, but because whatever small amount of it was there gets multiplied by the square of the collapse factor. It is also why the collapse usually cannot finish. The rotational energy grows faster than the gravitational energy released, and past a certain point the material cannot fall any further inward without shedding angular momentum somewhere — which is why the collapse of a spinning cloud produces a disc rather than a ball.

The cat, and the rotation that is a shape #

The falling cat is mentioned below as a caution and it deserves better, because it is the cleanest case of something the conservation law permits and nobody expects.

The animal is dropped upside down with no rotation at all. Its total angular momentum is zero at the moment of release, no torque acts on it while it falls, and its total angular momentum is therefore zero when it lands. And it lands the right way up.

There is no contradiction, and the resolution is that “zero angular momentum” does not mean “no change of orientation”. It means the rate of turning, weighted by the moment of inertia, sums to nothing at every instant. A body that changes shape can turn one part of itself one way while turning another part the other, with the weighted sum zero throughout, and if the shape change is arranged as a cycle — returning the body to the shape it started in — the parts do not cancel over the cycle and a net rotation is left.

The classical description is a front half and a back half counter-rotating about a bent spine, with the moments of inertia arranged so that one turns further than the other; several sequences work and the animal uses more than one. What is common to all of them is that the shape returns to where it started and the orientation does not.

That is a geometric phase, in exactly the sense of the optical one: a closed loop in a space of configurations, leaving behind a rotation whose size depends on the loop’s area rather than on how fast it was traversed. Do the manoeuvre slowly or quickly and the same turn results, because nothing dynamical is happening — the total angular momentum is zero throughout, so there is no dynamics to speed up.

The same statement is what lets an astronaut turn round in free fall with nothing to push against, and it is what a satellite with reaction wheels does when it reorients: shape changes, in a system with a fixed total angular momentum, produce reorientations. The conservation law does not forbid it, and reading it as though it did is the standard mistake.

Why a collapsing cloud makes a disc #

The astronomical version has a consequence the essay’s arithmetic already contains, and it is worth extracting because it explains a shape.

A parcel of gas falling toward a centre conserves its angular momentum about that centre, since gravity is central. Its specific angular momentum — angular momentum per unit mass — is therefore fixed, and the speed at which it must circle to carry that fixed amount grows as the radius shrinks.

Set the two effects against each other. Gravity’s inward pull per unit mass falls as $1/r^2$; the outward requirement of the circular motion, expressed at fixed angular momentum, goes as $j^2/r^3$ — one power steeper. So however far out the parcel starts, there is a radius at which the second overtakes the first, and inside that radius the parcel cannot go while keeping its angular momentum.

The catch is that this applies only to the component of the motion perpendicular to the rotation axis. Along the axis there is no angular momentum to conserve, and material falls straight in without any obstacle at all.

So a rotating cloud collapses freely along one direction and stalls in the other two, and what it makes is a disc: thin along the rotation axis, extended in the plane, with its radius set by the specific angular momentum the material started with. Every star, every planetary system and every accreting object is surrounded by one for that reason.

And nothing can proceed further until angular momentum is removed, which the conservation law forbids from happening internally. Something has to carry it outward — an outflow, a magnetic field threading the disc, a spiral arm, or friction between neighbouring annuli moving at different rates. The whole question of how a disc feeds what is at its centre is the question of what does that carrying, and the conservation law’s role in it is to make it a question at all.

Which point, and whether it moves #

One technicality is worth stating properly because it produces errors that look like physics.

Angular momentum is defined about a point, and the choice matters. About a point fixed in an inertial frame the law is the familiar one: the rate of change equals the torque about that same point. About a moving point the derivation acquires an extra term, and the law as usually written is simply wrong.

There is one exception and it is why the exception is so heavily used. About the centre of mass, the extra term vanishes identically — even when the centre of mass is accelerating — so angular momentum about the centre of mass obeys the ordinary law whatever the body is doing. That is what licenses treating a tumbling, falling object’s rotation separately from its trajectory, and it is not a general property of moving points but a special property of that one.

The practical consequence is the one the rolling example gives. A ball rolling down a slope has friction acting at its contact point, which exerts a torque about the ball’s centre — so angular momentum about the centre is not conserved, and it had better not be, since the ball is spinning up. About a point on the slope, gravity has a moment and friction does not, and a different quantity changes at a different rate. Neither statement is the conservation law, and someone reaching for “angular momentum is conserved” here will get an answer and it will be wrong.

The rule to carry is short: before invoking the law, name the point, and check that the torques about it vanish. Most of the time the useful point is the centre of mass, and most of the time the reason it is useful is the exception above.

Where the model stops #

Three limits, in increasing order of how much they matter.

$L = I\omega$ is a scalar shorthand. In general the moment of inertia is a tensor: it maps the angular velocity to the angular momentum, and the two vectors need not be parallel. They are parallel only when the rotation is about a principal axis.

The axis has to be specified, and it has to be a sensible one. Angular momentum is defined about a point, and it is conserved about a point only if the torque about that point vanishes. A body rolling downhill conserves nothing about its own centre, because friction acts at the contact and has a moment about the centre; it does conserve angular momentum about a point on the slope. Choosing the wrong origin produces a conservation law that is simply false, and the error is invisible because the formula looks the same either way.

A rigid body is an idealisation, and a skater is not one. Real bodies have internal degrees of freedom, and a body that can deform can move its own angular velocity without any external torque at all. A cat dropped upside down turns over with zero total angular momentum from beginning to end, by changing shape in a cyclic sequence that returns it to its original shape with a different orientation. Nothing is conserved there but the thing that was always conserved: the total angular momentum, which was zero and remained zero, while the animal’s orientation changed by 180°.

What the pictures cannot show #

The figures draw one arm radius at a time, and the manoeuvre is a process. The work integral above assumes the arms come in slowly enough that the body is turning steadily at every instant — quasi-statically — and a real pull is not slow. Doing it fast excites vibrations in the arms and the body, which carry away energy and change the accounting without touching the conservation law at all: the angular momentum is the same whatever the timing, and the split between rotational energy and internal wobbling is not.

Nor do the figures show the direction of anything. Every quantity here is drawn as a magnitude against a radius, and the whole content of the tensor paragraph above is about directions. A figure that could show it would need three dimensions and a moving picture; what the flip mode gives instead is the consequence, which is a body doing something a scalar law says nothing about.

Where the ladder goes next #

This ladder began with torque as a force with a lever arm and continued with the moment of inertia deciding a race and the axis that will not hold. The rung immediately after this one is the gyroscope, where a torque is applied and the response is at right angles to it — the case where every intuition built on a scalar $I\omega$ fails at once, and the one place in elementary mechanics where the vector nature of angular momentum is not a technicality but the entire phenomenon.

Beyond it: the parallel-axis theorem used in earnest; angular momentum in the presence of a magnetic field, where the conserved quantity acquires a term that is not $I\omega$ at all; and the quantum version, where the same conservation law survives with the same argument and the values it takes are restricted to a discrete set.

The habit worth carrying away is the one the hero figure was drawn for. A conservation law names what does not change, and a question about a quantity that did change is not answered by it. The skater’s rate is settled by conservation; the skater’s energy is settled by an integral, and the two answers have to be produced separately or the second one goes missing without anybody noticing that it is missing.