Growth and Shape of a Chain Fountain

John S BigginsCavendish Laboratory, University of Cambridge, Cambridge, United Kingdom

(Dated: May 6, 2019)

If a long chain is held in a pot elevated a distance h1 above the floor, and the end of the chainis then dragged over the rim of the pot and released, the chain flows under gravity down into apile on the floor. Not only does the chain flow out of the pot, it also leaps above the pot in a“chain-fountain”. I predict and observe that if the pot is held at an angle to the vertical the steadystate shape of the fountain is an inverted catenary, and discuss how to apply boundary conditionsto this solution. In the case of a level pot, the fountain shape is completely vertical. In this caseI predict and observe both how fast the fountain grows to its steady state hight, and how it grows∝ t2 if there is no floor. The fountain is driven by an anomalous push force from the pot that actson the link of chain about to come into motion. I confirm this by designing two new chains, oneconsisting of hollow cylinders threaded on a string and one consisting of heavy beads separated bylong flexible threads. The former is predicted to produce a pot-push and hence a fountain, while thelatter will not. I confirm these predictions experimentally. Finally I directly observe the anomalouspush in a horizontal chain-pick up experiment.

The mechanics of chains is one of the oldest fieldsin physics. Galileo observed that hanging chains ap-proximate parabolas, particularly when the curvature issmall[1], while the true shape was proved to be a cate-nary by Huygens Leibniz and John Bernoulli[2] in 1690.A chain hanging in a catenary is a structure supportingits weight with pure tension. In 1675 Hooke discoveredthat a thin arch supporting its own weight with purecompression must follow the inverted shape of a hangingchain[3], that is, an inverted catenary. Ever since ar-chitects from Wren to Gaudi have incorporated invertedcatenary arches into their buildings and even used hang-ing strings to build inverted architectural prototypes. Inthis paper I show that a chain fountain forms an invertedcatenary of pure tension stabilized by the motion of thechain. Theoretically it has been known that a chain mov-ing along its own length under gravity will trace a cate-nary since the 1850s[4–6], when the laying of the firsttransatlantic cables made understanding the motion ofchains along their length a priority, but such structuresmove downwards tracing standard tensile catenaries; thespontaneous formation of a tensile inverted catenary is,I believe, new.

We might expect such a venerable field to have few re-maining surprises; however, chain dynamics has recentlyproduced several. A chain falling onto a table acceler-ates faster than g, leading inexorably to the conclusionthat the table must pull down on the falling chain[7, 8].If a pile of chain rests on a surface, and the end is thenpulled in the plane of the surface to deploy the chain, anunexpected noisy chain arch has been observed to formperpendicular to the surface of the chain immediatelyupstream of the pile[9]. There is also recent work on therich dynamics of whips and free ends[10, 11].

The most recent surprise comes via Mould’s videos of achain fountain[12], shown in fig. 1a, in which a chain notonly flows from an elevated pot to the floor under gravitybut leaps above the pot. These videos have surprised anddelighted almost 3 million viewers. The leaping of the

h1

h2

w

θp x

y

v

a) b)

T(x)

θ(x)

FIG. 1. a) Steve Mould demonstrating a chain fountain.Photo courtesy of J. Sanderson. b) Diagram of a chain foun-tain. A chain with mass per unit length λ flows at speed valong a curved trajectory from a pot tilted to an angle θp andelevated to an height h1, to the floor. The fountain has heighth2 and width w. At each point x the chain has a height y(x)a tension T (x) and makes an angle θ(x) with the vertical.

chain above the pot requires that when a link of the chainis brought into motion, it must not only be pulled intomotion by the moving chain but also pushed into motionby the pot[13]. The origin of this anomalous force can beunderstood by considering a chain of freely jointed rigidrods. The link which is about to be brought into motionis lying horizontally on the pot and is picked up by theupwards pulling of the chain on one end of the link. Thiscauses the link to both rise and rotate, causing the farend of the link to push down into the table generatingthe anomolous push that drives the fountain.

The analysis in [13] assumed the chain fountain risesvertically from the pot to the apex then descends ver-tically to the floor. I extend this analysis showing, ex-perimentally and theoretically, that if the pot holdingthe chain is tilted the chain fountain forms an invertedcatenary. Applying boundary conditions to the invertedcatenary solution allows me to predict both the fountain’s

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height and its width. Secondly I discuss the growth andinitiation of the fountain when the pot is level, and show,experimentally and theoretically, that the fountain growsto its full height in a time scale given by

√h1/g, and that

the predicted functional form of the growth matches thatexperimentally observed. The growth model is not an-alytically integrable, but in the simpler case where thefloor is absent I show the fountain height would growquadratically in time. All these results confirm that thedriver of the chain fountain is an anomalous push forcefrom the pot. I therefore conclude by testing two com-pletely different chains, one consisting of hollow cylin-ders threaded on a string and one consisting of heavybeads separated by long flexible threads. The formermimics a chain of freely jointed rigid rods and shouldtherefore produce the anomalous push while the latter isdesigned to suppress it and, indeed, the former makes afountain while the latter does not. Finally I directly ob-serve the anomalous push in a horizontal chain-pick upexperiment.

I. CHAIN FOUNTAIN CATENARY

A. Derivation of the catenary

We consider a chain fountain as sketched in figure 1b.An element of chain with extent dx in the x direction haslength ds = dx/ sin (θ) and a mass λds. In the chain’stangential direction there is no acceleration so the gradi-ent in the tension forces must balance gravity,

T ′(x) =λg

sin θcos (θ). (1)

Since cot θ = y′(x) this can be directly integrated to give

T (x) = λgy + λ(v2 − cg), (2)

where we have written the constant of integration in theform λ(v2 − cg), where c is an unknown constant, for fu-ture algebraic convenience. Perpendicular to the chaintwo forces are relevant, gravity and an inward Kelvinforce T (x)/r(x), where r(x) is the local radius of cur-vature. These two forces must give rise to the localcentripetal acceleration v2/r(x), so Newton’s second lawgives

T (x)

r(x)− λg sin (θ) = λ

v2

r(x). (3)

Recalling that in Cartesians 1/r = y′′(x)/(1 + y′(x)2)3/2

and sin (θ) = 1/√

1 + y′(x)2, this simplifies to

(T (x)− λv2)y′′(x) = gλ(1 + y′(x)2). (4)

Substituting in our result for T (x) and solving for y(x)reveals that the chain must move in an inverted catenary:

y(x) = −a cosh

(x− ba

)+ c (5)

where a, b are new constants of integration.

1.72m

a) b) θ=5.5°p

θ=12°p

θ=23°p

θ=31°p

FIG. 2. a) Two experimental chain fountains (black lines),both obtained with a drop drop h1 of 1.72m but with differentpot tilt angles (left: θp = 5.5◦ right: θp = 31◦) leading thefountains to have different widths. Both fountains are fittedwell by a scaled catenary (blue dashed line) b) Instances ofmany chain fountains, coloured according to pot tilt angle,scaled and superimposed so that they all collapse onto a singlecatenary shown with a blue dashed line.

B. Observation of the catenary

The simplest form of an inverted catenary is the curvey(x) = − cosh (x). The above solution for the chain foun-tain is simply this curve translated in x and y and with a“zoom” by a factor of a. All steady state chain fountainsshould therefore produce shapes that, after zooming andtranslating, will collapse onto a single − cosh(x) curve.

To test this prediction, a 50m long nickel-plated brassball-chain was put in a 1L plastic beaker, elevated to1.72m above the ground and tilted by an angle θp. Theend of the chain was then pulled over the rim of the potand released initiating the chain fountain. The experi-ment was repeated with the pot tilted at various angles,resulting in different widths of fountains. Photographsof these fountains were taken using a flash to ensure asharp picture. The pictures were taken towards the endof the run so that the fountain was of steady height. Ifthe chain tangled significantly the run was disregardedand the experiment was repeated.

Two examples of a chain fountains are shown in fig.2a, one thin and one wide. As can be seen, both foun-tains undulate locally but do appear to macroscopicallytrace out a catenary. In fig. 2b many instances of chainfountains with different widths are rescaled to fit onto asingle inverted catenary curve. We thus conclude thatthe shape of a chain fountain is indeed well described byHook’s inverted catenary.

3

C. Boundary Conditions for the Catenary

We have yet to determine the width and height of chainfountain. We do this by applying boundary conditions tothe catenary. Unlike the hanging chain problem, the arc-length and the width of the catenary are not directlygiven as boundary conditions. To reach a fully specifiedsolution we must find the catenary parameters, a, b andc, and also the velocity of the chain v and the width ofthe fountain w, so we require five boundary conditions.The first, y(0) = 0, simply sets the coordinate origin atthe pot. The second, y(w) = −h2, fixes the drop of thechain fountain. To find the remaining three boundaryconditions we must examine the chain pickup and put-down processes in more detail.

1. Chain pickup

The conventional view of chain-pickup is that the chainis picked up by the tension in the chain directly abovethe pile. In a time dt a length of chain vdt is picked up,acquiring a momentum λv2dt. If the links are being ac-celerated solely by the tension then the tension must beT = λv2, which leads us to the third boundary conditionT (0) = λv2. However, this cannot be right: inspect-ing eqn. (4) we see that the right-hand side is alwaysfinite, but setting T (0) = λv2 will cause the left-handside to vanish unless y′′(0)→∞, implying that the foun-tain shape has an unobserved kink just above the pot.This would correspond to the chain reversing directionimmediately above the pot and falling straight to theground. For a fountain to be produced, we must haveT (0) < λv2. However, momentum balance requires thetotal force accelerating the links into motion to be λv2 so,if the tension is not providing all this force, then a secondunexpected force must also be acting. The only possiblesource for this momentum is the pot. We thus introducean anomalous reaction force from the pot pushing thelinks into motion fp = αλv2, reducing the tension justabove the pot to T (0) = (1− α)λv2.

The origin of this anomalous pushing force from thepot becomes clear if we model the chain as freely jointedrigid rods being picked up vertically from a horizontalsurface, sketched in fig. 3a. The next rod to be pickedup lies horizontally and is pulled upward into motion bythe preceding link, which acts via a tension force T assketched in fig. 3b. This tension force causes the centerof mass of the rod to lift and also causes the rod to ro-tate. If there were no horizontal surface this would resultin the far end of the rod initially moving downwards. Inreality the surface prevents this from happening by push-ing up on the far side of the rod with an upwards reactionforce fp so that, initially, the far side of the rod remainsstationary and the rod pivots about it. The initial ac-celeration of the center-of-mass of the rod is thus given

v T

fp

v T

RpFp

a) b)

c) d)

v

T

RfFf

e)

f)

FIG. 3. a) Sketch of a chain of rigid rods being picked upvertically from a surface. b) Force diagram of the rod aboutto be picked up, including a tension T from the moving partof the chain and a reaction fp from the surface. c)-d) Same asa)-b) but with a non-vertical pickup velocity. The surface cannow respond with a reaction force Rp and a frictional forceFp e) A chain being deposited on a surface at an angle. f)Force diagram of a link being brought to a halt but tensionand reaction and frictional forces from the floor.

by

mv = T + fp (6)

and the initial rotational acceleration is given by

Iω = (fp − T )l

2, (7)

where I is the rods moment of inertia. The initial acceler-ation of the left hand tip of the rod must be zero, requir-ing v+ ωl/2 = 0. Writing fp = αλv2 and T = (1−α)λv2

we can rearrange this to estimate α as

α =1

2

(1

2− I

14ml

2

). (8)

This is an overestimate of α since fp will be greatest atthe start of the pickup. However, this idea demonstratesthe pot can push the links, justifies taking the functionalform fp = αλv2 and demonstrates that how large thisforce is depends on the precise geometry of the chain —in the above calculation α depends on the specific ratioI/(ml2).

If the chain is being picked up at an angle, as sketchedin fig. 3c., the chain velocity and the tension still pointalong the departing chain’s tangent. Momentum conser-vation thus requires the force on the link from the surfaceto be tangential, requiring the surface to provide someforce in the in-surface direction. Such a force cannotarise as a reaction force, but it could arise as a frictionalforce. However, since the link will start to move in thevelocity direction and friction points opposite to velocity,as shown in fig. 3d, the sum of the two surface forces willonly point in the tangential direction if pickup is perpen-dicular to the surface. We conclude the chain must bepicked-up in the direction normal to the pot, giving afourth boundary condition θ(0) = θp.

4

2. Chain put-down

Similar considerations apply at the floor end. A totalforce λv2 is required to bring the moving chain to a halt.The traditional view is that this force is provided by thereaction force from the floor is ff = λv2 and therefore thetension at the end is T (w) = 0. However, careful obser-vations of free-falling chains dropping onto hard surfacesreveal that, remarkably, they fall with an accelerationfaster than g. This demands that the tension just abovethe floor is finite so that the total force on the falling por-tion of the chain is greater than its weight. We thereforeset T (w) = βλv2, meaning that ff = (1− β)λv2.

If the chain arrives at the floor at an angle, the totalforce from the floor must point along the chain’s tangent.In this case the friction and the normal reaction are bothopposite in direction to the chain’s velocity, as sketchedin fig. 3e-f. so they can be added to point tangentially.Provided the floor can provide sufficient friction there isno constraint on θ(w).

3. Applying the boundary conditions

The preceding sections have yielded five boundary con-ditions:

y(0) = 0 (9)

y(w) = −h1 (10)

θ(0) = θp (11)

T (0) = (1− α)λv2 (12)

T (w) = βλv2. (13)

Taking the cot of eqn. 11 gives y′(0) = cot θp, which isimmediately solved by

b = a sinh−1(cot(θp)). (14)

The first boundary condition now gives c = acosec(θp)while the fourth gives T (0) = λ(−cg + v2) = (1− α)λv2.Solving these two equations for a and c gives

a =αv2 sin(θp)

gc =

αv2

g. (15)

The fifth boundary conditions reads −λgh1 + T (0) =βλv2, which can be solved for v to give

v =

√gh1

1− α− β. (16)

A unit length of chain thus releases gravitational energygh1λ and receives kinetic energy 1

2gh1λ/(1 − α − β). Inthe traditional regime (α = β = 0) half the gravita-tional energy is thus lost in the chain pickup process, asclassically expected when picking up a chain at constantvelocity[13]. The additional anomalous forces reduce this

α=0.05α=0.10α=0.15α=0.2

α=0

-π/2 -π/4 π/4 π/2θp

w/h

0.5

-0.5

1

FIG. 4. Prediction for the width of the fountain w, in unitsof the fountain drop h1, plotted against pot tilt angle θp. Thelines correspond to different anomalous pushes from the potduring parameterized by α. If α = 0 then w = 0. The plotis drawn with β = 0.2. Reducing β only has only a modesteffect on the curves: e.g. β = 0 reduces the curves to abouttwo thirds of the shown height.

energy loss. The requirement that the kinetic energy notbe larger than the released gravitational potential energyimposes the bound α+ β < 1

2 .

Finally the second boundary condition is −h1 = c −a cosh

(w−ba

), giving w = a cosh−1

(c+h1

a

)+b. Substitut-

ing in our results for a, b c and v yields,

w =αh1 sin(θp)

1− α− β× (17)(

cosh−1(

(1− β)cosec(θp)

α

)+ sinh−1(cot(θp))

).

Substituting eqn. 16 for v into 15 gives

a =αh1 sin(θp)

−α− β + 1c =

αh1−α− β + 1

. (18)

We have now determined the chain’s shape and veloc-ity. Its trajectory (parameterized by a, b, c and w) isindependent of g although the chains speed is not. Theexpression for w, the width of the fountain, is rather com-plicated so we plot w as a function of θp in fig. 4. Wesee that for small tilt angles the width of the fountainis zero, but that the gradient dw/dθp is divergent, albeit

weakly as sinh−1(cot(θp)) ∼ log(θp). For large tilt anglesthe fountain width starts to decrease.

The fountain height above the pot is simply h2 = y(b):

h2 =αh1(1− sin(θp))

1− α− β. (19)

Both the height and the width of the fountain vanish ifα = 0 (but doesn’t if β = 0) establishing that the pushfrom the pot during the chain pickup is the driver of achain fountain.

If the pot is level (θp = 0) then w = 0 so the fountain isvertical. However, the height of the fountain is maximal,h2 = αh1/(1− α− β).

5

FIG. 5. Experimental snapshots of chain fountains with pottilt angles of (left to right) 5.5◦, 12◦, 23◦ and 31◦ overlaid withthe predicted catenary shape using α = 0.12 and β = 0.11 ina blue dashed line.

4. Experimental test of the boundary conditions

We test our conclusions about the width and height ofthe chain fountain in fig. 5 using the snapshots of chainfountains with different pot tilt angles shown in fig. 2b.Fixing h1 to be the actual drop of 1.72m and θp to be themeasured tilt angle of the pot, a good match is achievedwith the two large-angle fountains, and for the heightof all the fountains, by taking α = 0.12 and β = 0.11(consistent with [7] and [13]). However, the small an-gle fountains are predicted to be substantially thinnerthan observed. This result is not a consequence of thechoices for α and β: although the shapes produced bysmall angle fountains are convincingly catenary, physi-cally reasonable values of α and β never produce suchwide fountains. In fact the chain must exit at a slightlylarger angle than the θp. This may be permitted eitherby the pots finite width, which for small angles is anappreciable fraction of the width of the catenary at potlevel, or by the fact that the links do not all lie flat inthe pot. Divergence of dw/dθp when θp is small makesthe system very sensitive to any deviation in the chain’sexit tangent from the pot normal when the pot is closeto flat.

II. DYNAMICS OF THE CHAIN FOUNTAIN

We now turn our attention to the growth and initiationof the fountain. For simplicity, we focus on the case ofa level pot, θp = 0, where the steady state is a verticalfountain, with vertical rising and falling legs connectedby a very small region of high curvature at the apex. Wesketch such a fountain in fig. 6a and label the tensionin the chain just above the pot TP , that at the apex TTand that just above the floor TF . We also distinguish theupward velocity of the rising leg v1 and the downwardvelocity of the falling leg v2 since during fountain growththese velocities will not be equal. Since we are studyingthe dynamics of the fountain, all of these quantities are

h1

h2

TF

TP

TT

v1

v2

v1 2vl

v1 dt

v2dt

v1

2vl+dl

a) b)

TT TT

TT

TT

FIG. 6. a) Diagram of a growing vertical chain fountain.While the fountain grows the velocities of the two legs arenot equal. b) Diagram of a small length (2l) of chain flowingaround the apex of the fountain. The same material is shownon the left, and a short time dt later on the right.

now functions of time. Applying Newton’s second law tothe rising and falling legs gives:

−λgh2 + TT − TP = h2λv1 (20)

−λg(h1 + h2) + TT − TF = −(h1 + h2)λv2. (21)

Our previous results about chain pickup and putdowntell us that the tensions just above the pot and the floorare

TP = (1− α)λv21 TF = βλv22 . (22)

Finally we must consider the apex of the fountain. In fig.6b we show a section of chain of length 2l that traversesthe apex in two diagrams separated by a short time dt.Since the chain cannot change length, 2dl+v2dt−v1dt =0. However, dl/dt = h2, so this constraint can be writtenas

h2 = 12 (v1 − v2) . (23)

In a time dt an amount of material of length v1dt− dl =12 (v1 + v2)dt is converted from moving upward withspeed v1 to downward with speed v2, so the total mo-mentum of the material around the apex changes bydp = 1

2 (v1 + v2)2

dt. This momentum must be providedby the tension at the apex TT which acts on both endsof the material, so

2TT = 12λ (v1 + v2)

2. (24)

Inserting our results for TT , TP and TF into equations(20)—(21) yields

−gh2 + 14 (v1 + v2)

2 − (1− α)v21 = h2v1 (25)

−g(h1 + h2) + 14 (v1 + v2)

2 − βv22 = −(h1 + h2)v2 (26)

12 (v1 − v2) = h2. (27)

In the steady state we expect h2 = v1 = v2 = 0 and hencev1 = v2 = v, reducing the above system of equations to

−gh1 + v2 − (1− α)v2 = 0 (28)

−g(h1 + h2) + v2 − βv2 = 0, (29)

6

0 5 10 15 20 250.0

0.2

0.4

0.6

0.8

1.0

0

0.05

0.1

0.15

0.2

0.25

h2 h1

h1tg

gh1v

h2

v2

v1

FIG. 7. Numerical solution to eqns. (25-27) calculated us-ing α = 0.2 and β = 0. The velocities of both legs of thefountain (v1 and v2) and the fountain height rise from zero totheir (equal) steady state values in a timescale comparable to√h1/g.

which can be solved for h2 and v to recover the steadystate solutions:

h2 =αh2

1− α− βv =

√g√h2√

1− α− β. (30)

The dynamic equations do not appear to be analyticallyintegrable. They contain one characteristic time-scale√h1/g and one characteristic length-scale, h1, so we ex-

pect the fountain to rise to a height proportional to h1(as confirmed by the steady state analysis) in a time pro-

portional to√h1/g which, for our fountains, is about

half a second. The full equations are however straight-forward to solve numerically using, for example, the stan-dard NDSolve function in Mathematica. The appropriateboundary conditions are h2 = v1 = v2 = 0 at t = 0. Ex-ample solutions are shown in fig. 7. As expected, thethree dynamic variables, v1, v2 and h2, rise from zero totheir steady states values in a characteristic time

√h1/g.

A. Experimental observation fountain growth

The fountain height h2(t) can be straightforwardlymeasured from a video of a chain fountain. A chain foun-tain was prepared with the same 50m brass ball-chainused in previous experiments, using a 1.8m drop and a1L, 0.17m tall glass pot. The pot was tilted to a smallangle θp ∼ 1.5◦ to ensure the chain exited reliably onone side of the pot. The end of the chain was then low-ered from the pot to the floor and held stationary beforebeing released to initiate the fountain. This was doneto match the analysis in the previous section which as-sumes that the falling leg of the fountain touches the floorthroughout. The fountain was recorded at 30fps, and thefountain height was calculated for every fifth frame asthe distance between the apex of the fountain and thefill level of the beads in the pot. The latter falls duringthe course of the fountain as the pot empties. We com-pare the data from this experiment with the theoreticalprediction in fig. 8a, using the same chain parameters(α = 0.12, β = 0.11) as in the catenary analysis. We see

0.0 0.5 1.0 1.5 2.00.00

0.05

0.10

0.15

0.20

5 10 15 20 25

0.05

0.10

0.15

h2 h1

h1tg

a)

00

2h /m

t /s

b)

FIG. 8. Comparisons of experimentally observed (joinedblue dots) and theoretically predicted (green line) fountainheight (h2) as a function of time, t. a) Experiment using adrop of 1.8m showing a steady state height and a theoreticalline calculated using α = 0.12 and β = 0.11. b) Experimentwith no floor showing quadratic growth of the fountain height.Theoretical line calculated using α = 0.12

that the experimental line is quite noisy, but the theoret-ical line accurately predicts the average properties of theexperimental data.

B. A chain fountain with no floor

The above analysis can easily be extended to a fountainwith no floor, where the falling leg of the fountain fallsto ever lower depths. The first alteration is that, sincethere is no floor, it can provide no anomalous force so thetension at the bottom of the falling leg of the fountainTF = 0. The second is that h1 is now also a dynamicvariable, satisfying the trivial equation h1 = v2. Thisresults in the system of equations

− gh2 + 14 (v1 + v2)

2 − (1− α)v21 = h2v1 (31)

− g(h1 + h2) + 14 (v1 + v2)

2= −(h1 + h2)v2 (32)

12 (v1 − v2) = h2 v2 = h1. (33)

This system of equations has no steady state — thechain velocity increases continuously as the end of thechain gets lower. These equations admit a simple ana-lytic solution with the appropriate boundary condition(v1 = v2 = h1 = h2 = 0 at t = 0), namely

v1(t) =

(1− 4

√(4− 3α)

)gt

16α− 21(34)

v2(t) =

(8α+ 2

√4− 3α− 11

)gt

16α− 21(35)

h1(t) =

(8α+ 2

√4− 3α− 11

)gt2

32α− 42(36)

h2(t) =

(4α+ 3

√4− 3α− 6

)gt2

42− 32α. (37)

We see that the velocities increase linearly in time whilethe fountain height rises as quadratically. We test thisexperimentally using an 8m drop and an 8m chain infig. 8b, and achieve a good fit again using the parametervalue α = 0.12.

7

10mm

a) b) c)

FIG. 9. Experimental chain fountains with two very differ-ent sorts of chains. a) Image of the two chains. The topchain consists of hollow cylinders threaded on a nylon thread.The lower chain is heavy tungsten beads separated by longstretches of fine thread. b) The bead-chain does not producea fountain with a 1.8m drop. c) The cylinder chain does pro-duce a fountain with a 1.8m drop.

III. EXPERIMENTAL EVIDENCE FOR THEANOMALOUS PUSH

A. Different sorts of chain

The driver of the chain fountain is the anomalous ex-tra push from the pot that helps launch links of the chaininto motion. If this force is absent (α = 0) the chain ispredicted to flow from pot to floor without forming afountain. Eqn. (8) suggests that the value of α is de-pendent on the microscopic details of the chain. We testthe suggested origin of the extra push by considering twovery different chains. The first consists of small (diam-eter 4mm) spherical tungsten beads separated by 2.5cmlengths of thread. This chain should have α = 0 andhence not form a fountain. The beads will be picked upone at a time and picking up a single bead, even with aforce at its edge, will not produce a reaction from the ta-ble. This is firstly because the radius of the bead is small,so the rotation induced is small, and secondly because,even if the bead is induced to rotate, its spherical shapemeans this will not result in it pushing down on the tableand producing a reaction. The second chain consists ofshort hollow cylinders (in-reality pieces of uncooked mac-aroni pasta, each about 2cm long and 4mm in diameter)strung end to end on fine thread. This chain resemblesthe freely-jointed rod idealized chain used to derive eqn.(8), and hence should produce a reaction from the ta-ble and a chain-fountain. Both chains had similar massdensities and were 10m long. The two chains are shownin fig. 9a. Both were dropped from a pot 1.8m abovethe ground. The fountains produced are shown in figs.9b-c. The chains were too short to produce steady statefountains so we cannot determine α from the fountainheight, however, the result that the bead chain does notproduce a fountain while the freely-joined-rod chain doesis unambiguous.

B. Direct observation of the anomalous push

8m of ball chain was arranged in close packed serpen-tine rows on a table, and the end was then released over

the end of the table, causing the chain to leave the packin a horizontal direction perpendicular to the rows. Ascan be seen in supplementary video 1 (a video of a similarexperiment is reported online [14]) during the experimentthe pack of chain moves backwards, directly showing thatthe rows of chain push forward on the deploying chainwith an anomalous force.

IV. CONCLUSION

The observations that the chain fountain achieves acatenary shape and that the growth of the fountain satu-rates demonstrates that, although the trajectories of thechain fountain are rather noisy and variable, they areapproximating the underlying steady state outlined inthis paper. It is therefore unlikely that substantially big-ger fountains will be achieved in the future without us-ing substantially bigger drops. However, a few questionsremain. The angle the chain fountain leaves a slightlytilted pot remains unquantified. The noisy nature of thefountain trajectories is also not well understood. Theenergetic origin of the noise is clear — it comes fromthe fraction, close to a half, of the gravitational poten-tial energy released by the fountain that is “dissipated”during the pick-up process — however, its amplitude andwavelength, amongst other things, are unquantified. Itis likely that both these issues ultimately relate to thefinite width of the pot and the chaotic nature of the pileof chain in the pot. It may be interesting to investigatechain pick up from a very wide rather than a very narrowpot, and from ordered rather than chaotic chain configu-rations. The relationship between α (the push from thepot) and β (the pull from the floor) also merits furtherwork. The ball chain investigated in this paper appearsto have similar values for both coefficients, leading oneto wonder whether the two should be equal.

The work in this paper raises questions aboutthe mathematical idealization of the perfectly flexiblestring/chain. One can approach such a model as the limitof a chain of freely jointed rods as the rod length vanishes.We see from eqn. (8) that a chain of freely jointed rodshas a value of α dependent on the dimensionless num-ber I/(ma2). This will remain finite as the link lengthtends to zero, and will remain different for chains madeof different types of rod with different values of I/(ma2).Thus different limits to a perfectly flexible string will pro-duce finite fountains of different height. The behavior ofa rope fountain, with a rope without links but with finitebending resistance, is also an open problem.

The work in this paper confirms that the physical ori-gin of the chain fountain is an additional pushing forcefrom the pot acting on the links in the chain as they comeinto motion. This anomalous push is expected to applywhenever a chain is picked up from a surface, so its im-plications may go far beyond the chain fountain. In par-ticular, in any industrial or technological setting where achain is being deployed, accurately predicting how much

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force is required will require this force to be taken intoaccount. This suggests further work on how to maximizethis force may find application. Finally, picking up achain at constant speed has traditionally been thought tobelong to that class of problems in which momentum con-servations dictates that half the mechanical work done isdissipated, and only half appears in the kinetic energy of

the chain. Charging a capacitor at constant voltage is abetter known electrical example of this type of dissipa-tive phenomenon. The anomalous upwards push requiredto explain chain pickup alters this argument, increasingthe fraction of energy that is retained during pickup to1/(2(1 − α)). Perhaps it is worth revisiting other tradi-tional problems in this class to see whether similar effectscan be harnessed to reduce energy dissipation.

[1] G. Galilei, S. Drake. Madison, WI: University of Wiscon-sin Press (original work published 1638) (1974).

[2] E. H. Lockwood, A book of curves (Cambridge UniversityPress, 1971).

[3] R. Hooke, A description of helioscopes, and some otherinstruments John Martin, London (1676).

[4] Examiners and Moderators, Solutions of the problemsand riders proposed in the Senate-House examination(Mathematics Tripos) (MacMillan & Co. London, 1854).

[5] G. B. Airy, Phil. Mag. S. 4 16, 1 (1858).[6] E. Routh, Dynamics of a system of rigid bodies, with

numerous examples (MacMillan & Co. London, 1860).[7] A. Grewal, P. Johnson, and A. Ruina, American Journal

of Physics 79, 723 (2011).

[8] E. Hamm and J.-C. Geminard, American Journal ofPhysics 78, 828 (2010).

[9] J. A. Hanna and C. D. Santangelo, Phys. Rev. Lett. 109,134301 (2012).

[10] A. Goriely and T. McMillen, Phys. Rev. Lett. 88, 244301(2002).

[11] J. A. Hanna and C. D. Santangelo, ArXiv e-prints(2012), arXiv:1209.1332 [physics.flu-dyn].

[12] S. Mould, http://stevemould.com/siphoning-beads/(2013).

[13] J. S. Biggins and M. Warner, Proc. Roy. Soc. A (2014).[14] J.-C. Geminard, http://perso.ens-lyon.fr/jean-

christophe.geminard/a chain falling from a table.htm(2012).