The spool that rolls towards the hand pulling it

A cotton reel lies on its side on a table, a length of thread coming off the bottom of its hub. Pull the thread gently along the table, and the reel rolls towards the hand. That is a surprise twice over: the obvious picture of a reel is that pulling the thread unwinds it, and this one winds the thread up as it comes — it is turning the way that takes thread in, while the hand pulls thread out. Raise the end of the thread a little and pull again; at some angle the reel stops responding altogether, and slides or sits still while the hand pulls. Raise it further and it rolls away, paying the thread out as it goes.

The puzzle is old enough to have a name, Maxwell’s spool, and a famous relative: a bicycle held upright with one pedal at the bottom of its stroke, and that pedal pulled backwards. Does the bicycle go forwards or backwards? People who cycle every day split on the answer. Both puzzles are settled by one free-body diagram and one choice of point about which to take moments, and the choice is the lesson.

Choosing the point that hides the unknown forces #

Three forces act on the reel, apart from its weight: the thread’s tension, which is known; and the table’s push, up through the contact point, and the table’s friction, along the table at the same point, which are not. The slope, and the two directions that make it easy found that the right choice of directions can make a statics problem simple, and the same holds for moments. Take moments about the contact point. Both of the table’s forces pass through that point, so they have no moment about it, and they drop out of the question of which way the reel turns. Only the thread’s tension is left.

The same push, further out found that a force’s turning effect is the force times the perpendicular distance from the point to its line. The thread leaves the hub tangentially, so its line is at distance $r$ from the axle; the axle is at distance $R$ above the contact point. The perpendicular distance from the contact point to the thread’s line works out as

$$d = R\cos\theta - r,$$

where $\theta$ is the thread’s angle above the horizontal. When $d$ is positive the tension turns the reel forwards about the contact point — the reel rolls towards the hand. When it is negative the reel rolls away. When it is zero the tension has no moment at all, and the reel cannot start to roll either way.

Geometrically the test is where the thread’s line, extended backwards, meets the table. If it meets the table behind the contact point — on the far side from the hand — the reel rolls towards the hand; in front of it, the reel rolls away; at it, nothing. For a reel whose hub is half its rim, the dividing angle is 60°. The dividing angle depends on the ratio of the radii and on nothing else: not on how hard the thread is pulled, how heavy the reel is, or how its mass is distributed.

The point about which everything turns #

There is a second way to see why the contact point is the right place to stand, and it explains the answer rather than just computing it. A body rolling without slipping is, at each instant, turning about its contact point: that point is momentarily at rest, and every other point of the reel moves at right angles to the line joining it to the contact, at a speed proportional to its distance — the axle at $\omega R$, the top of the rim at $2\omega R$. The contact point is the reel’s instantaneous centre of rotation. The point that takes the blow without a jolt found a related point for a struck bat, the point about which a blow at the sweet spot makes the bat begin to turn; here the floor fixes the point, and the reel’s whole motion, for an instant, is a rotation about it.

Seen that way, the thread is a force applied to a body pivoted at the contact point, and the only question is which way it turns the body about the pivot — exactly the question of which side of the pivot its line passes. A thread whose line passes behind the pivot turns the reel one way; in front, the other; through it, not at all, however hard it pulls. That the pivot moves along the floor as the reel rolls changes nothing at the instant of starting, and the reel’s mass and how it is distributed enter only in how quickly it responds, not in which way.

How fast, and which forces do it #

The moment fixes the direction. The rate needs Newton’s laws for translation and rotation together, and for a reel rolling without slipping they combine into one line:

$$a = \frac{T(\cos\theta - r/R)}{m(1 + k^2/R^2)},$$

where $k$ is the reel’s radius of gyration. The bracket is the moment about the contact point, divided by $R$; the denominator is the reel’s resistance to being set rolling, which includes its resistance to being spun as well as moved — the mass, and where it sits found that a rolling body’s mass counts twice, once for moving along and once, weighted by where it sits, for turning.

A reel with a thin hub comes towards the hand at almost every angle; one with a thick hub, almost as large as its rims, comes only when the thread is nearly horizontal. That is why a yo-yo sitting on the floor can be made to “walk” towards its owner by a low pull and run away from them by a high one: its axle is narrow, and the dividing angle is steep.

The question the moment argument hides is what actually turns the reel. The thread pulls forwards, towards the hand, and if nothing else acted the thread’s pull on the underside of the hub would spin the reel backwards — unwinding it — while dragging it forwards. A reel that rolls towards the hand is spinning forwards. The spin comes from the table. The friction on the reel points away from the hand, backwards, and its moment about the axle, acting at the rim’s radius $R$, spins the reel forwards, overcoming the thread’s opposite moment at the hub’s smaller radius. The force that takes what it needs found static friction adapting to whatever is required of it, up to a limit, and here it supplies exactly the backwards force that makes the forward translation and the forward spin agree.

Friction it cannot do without #

That gives the reel a second way to fail. The friction needed is the difference between the thread’s forward pull and what the reel’s forward acceleration accounts for, and the table can supply at most $\mu$ times the normal force — which the thread itself reduces, as it lifts the reel when it pulls upwards.

A light pull, a fifth of the reel’s weight, rolls it at any angle on any ordinary surface. A pull of half its weight needs a friction coefficient above 0.3 at every angle, and on a polished table the reel slides towards the hand without rolling. A heavy pull, lifting the reel almost off the table at steep angles, needs friction no surface provides. The reel that “will not move” at the critical angle is the one case where no friction is needed at all: at $\cos\theta = r/R$ the thread’s pull passes through the contact point, the reel has no reason to turn, and the friction needed is just the thread’s horizontal component, holding the reel in place against it — slide or topple found the same kind of competition, between a push’s tendency to slide a block and its tendency to tip it about an edge, decided by where the push’s line passes relative to that edge.

Where the hand’s work goes #

There is a last check that the accounting is right, and it uses energy. The friction acts at the contact point, which in rolling without slipping is momentarily at rest, so the friction does no work — the floor that does no work made that point for a jumper’s feet, and it is exactly true here. All the reel’s kinetic energy must come from the hand.

The hand moves along the thread at the reel’s speed times $\cos\theta - r/R$: the speed of the thread where it leaves the hub, which is the reel’s own motion along the thread’s direction combined with the hub’s surface turning. That factor is the same bracket that set the acceleration, so the power the hand delivers — tension times the hand’s speed along the thread — has the sign of the motion it produces, and equals the rate at which the reel’s kinetic energy grows. When the reel rolls towards the hand, the hand still backs away, more slowly than the reel approaches; the gap between them closes, and the difference is thread wound onto the hub. When the reel rolls away, both factors have changed sign, the hand again backs away — now following the thread as the hub pays it out faster than the reel recedes — and again does positive work. At the critical angle the hand holds the thread at a fixed length and does no work, and the reel stays put: a reel that will not move is a reel to which no work is being done.

That reel has a practical use. A drum winch on a vehicle stuck in mud, a cable laid off a reel at a fixed angle, a carpet roll pulled from the floor — each is a spool, and each behaves according to where the pull’s line meets the ground. Pulling a heavy cable drum by its unwinding cable at a shallow angle rolls it towards the puller, where it lands on the puller’s feet; site manuals warn against it in exactly those terms.

The only thing that pushes a car #

The same free-body reasoning settles a question that looks quite different: what pushes a car forwards. The engine turns the wheels, the wheels turn against the road, and it is tempting to say that the engine pushes the car. But the engine is inside the car, and nothing inside a body can move its centre of mass. Taken as a whole, the car is acted on by gravity, by the road’s upward push, by air, and by the road’s friction at the bottom of each driven tyre. Only the last points forwards. The engine’s torque on the wheel is what makes the tyre push backwards on the road, and the road’s friction pushing forwards on the tyre is the force that accelerates the car — the grip that needs a little slipping found that a tyre transmits it only by creeping slightly against the road.

The spool is the same arrangement with the driving torque applied through a thread instead of a shaft. In both cases the visible effort is internal, or applied off the contact point, and the force that does the moving is the one at the contact point that the eye passes over. And in both cases the limit is the same: the speed at which grip hands over to power found a car’s acceleration at low speed set by friction rather than by the engine, and the reel’s rolling ends, in the same way, where the friction it needs exceeds what the table can give.

The bicycle and its pedal #

Hold a bicycle upright, turn its cranks so that one pedal is at the bottom of its stroke, and pull that pedal horizontally backwards. Does the bicycle move forwards — the cranks being turned in the pedalling direction, the chain driving the rear wheel — or backwards, towards the pull?

It is a spool in disguise. The rear wheel is the rim, of radius $R$, in contact with the ground. The pedal is a point on the end of a crank of length $p$, but the crank is connected to the wheel through the chain, and its leverage on the wheel is reduced by the gear ratio: a crank of length $p$ turning a chainring of $N_f$ teeth, driving a sprocket of $N_r$ teeth, has the same effect on the wheel as a hub of radius

$$r_{\text{eff}} = p\,\frac{N_r}{N_f}$$

fixed to the wheel. Pulling the bottom pedal backwards is pulling the thread from the underside of that hub, horizontally. The bicycle rolls forwards only if $r_{\text{eff}}$ exceeds $R$.

With a 170-millimetre crank and a 340-millimetre wheel, the bicycle would roll forwards only with a sprocket twice the size of its chainring. No bicycle is geared so: the lowest gear of a mountain bike puts the effective hub at 0.82 of the wheel, and top gear at 0.16. So the bicycle always rolls backwards, towards the hand. Its cranks, which the pull was turning forwards, turn backwards instead — the rear wheel, rolling backwards, drags the sprocket backwards through the engaged freewheel, as it does whenever a bicycle is wheeled backwards — and the pulled pedal moves forwards relative to the frame while it, and the bicycle, move backwards over the ground. The pedal moves backwards over the ground more slowly than the bicycle does, which is the spool’s “hand moves back more slowly than the reel approaches” in another form.

The bicycle puzzle trips people for the same reason the reel does. The forces inside the system — the chain’s pull, the crank’s turning — are vivid, and they all point towards the bicycle going forwards. The forces that decide are the external ones: the hand’s pull and the ground’s friction, and of those only the hand’s has a moment about the contact point. Internal forces cannot move a system’s centre of mass; they can only change how the external ones are shared out. The answer comes from the free-body diagram of the whole bicycle, with the chain left inside it.

What the figures leave out #

The figures treat the reel and the bicycle as rigid bodies rolling without slipping on a flat, rigid floor, with friction acting at one point. A real reel on a tablecloth sinks slightly and rolls against a small resistance — the push a millimetre ahead of the axle found that a rolling body’s normal force acts slightly ahead of the contact point, which shifts the critical angle by an amount set by the softness of the surface. The thread is taken as weightless and the pull as steady. On the bicycle, the front wheel, the frame’s own weight and the rider-free steering are ignored, and so is the small elasticity of the chain and tyres. The domain is slow motion of a spool or a bicycle on a firm floor, with pulls small enough that friction can hold rolling.

Still open: what a spool on a soft or moving surface does #

Change the floor, and the clean geometry starts to blur. On a carpet or in sand the contact is not a point but a patch, the floor’s push acts at a position that depends on the speed and the load, and the critical angle becomes a range in which the reel may creep either way or rock. On a conveyor or a moving deck, the floor’s own motion enters the condition for rolling. How a wheel’s contact patch shares its forces with a deformable surface is the subject of terramechanics, which predicts how vehicles behave in soft soil, and it rests on measured soil properties rather than on any closed formula; where a spool’s dividing angle lands on sand is a question for an experiment rather than for the drawing.

The rigid case is settled. A spool turns about its contact point according to $T(R\cos\theta - r)$, so it rolls towards the hand when the thread’s line meets the floor behind the contact point, stays still when the line passes through it at $\cos\theta = r/R$ — 60° for a hub half its rim — and rolls away when it passes in front, with the floor’s friction supplying the spin; a bicycle pulled back by its lower pedal is a spool with a hub of the crank times the gear ratio, 0.16 to 0.82 of its wheel, and rolls backwards in every gear. Which way a pulled thing rolls is written in one line, and the line is the one through the point that does not move.