## Abstract

The motion of weights attached to a chain or string moving on a frictionless pulley is a classic problem of introductory physics used to understand the relationship between force and acceleration. Here, we consider the dynamics of the chain when one of the weights is removed and, thus, one end is pulled with constant acceleration. This simple change has dramatic consequences for the ensuing motion: at a finite time, the chain ‘lifts off’ from the pulley, and the free end subsequently accelerates *faster* than the end that is pulled. Eventually, the chain undergoes a dramatic reversal of curvature reminiscent of the crack or snap, of a whip. We combine experiments, numerical simulations and theoretical arguments to explain key aspects of this dynamical problem.

## 1. Introduction

When publishing the design of his machine in 1784, little could Atwood have known that, more than two centuries later, students would be asked to predict the outcome of his experiment. The problem, traditionally offered as an illustration of the principles of Newtonian mechanics, consists of deriving the acceleration ** a** of two masses

*M*>

*m*subject to the action of gravity

**while attached to a massless and inextensible chain passing over a frictionless pulley. The well-known result is that**

*g*Little attention has been paid to the seemingly trivial case in which both *m*=0 and the chain have a finite linear density. If the mass of the chain nevertheless remains small compared with *M*, then equation (1.1) immediately gives that the remaining mass falls with acceleration ** a**=

**. However, a simple experiment (see figure 1**

*g**b*) reveals that this apparent simplification actually has a dramatic effect on the resulting motion: the chain ‘lifts off’ from the pulley in a complex motion. This lift off has some surprising features that we explore in detail in this paper. In particular, we show that the free end accelerates

*faster*than the end that is being pulled by the mass: in this sense, the free end ‘beats’ the free fall of the mass. We also show that the chain eventually ‘snaps’ in a manner that is reminiscent of the crack of a whip [1,2].

Thin filamentary structures are as important in applications as they are ubiquitous in nature and industry; examples range from macromolecules such as DNA [3] to the kilometric transoceanic cables laid on the ocean bed from telecommunication vessels [4]. On a human scale, textiles, hair and ropes are other examples of thin elongated structures in which one dimension greatly exceeds the two others (so that the objects may be modelled as one-dimensional). The thinness of such rod-like structures also makes them flexible, so that they are frequently subject to large deformations in the three-dimensional environment in which they evolve. This flexibility in shape, in turn, leads to rich bifurcation landscapes [5], striking pattern formation [6] and intricate dynamical behaviour.

Despite the range of length scales and materials encountered, many filamentary structures are well modelled by the Kirchhoff equations for elastic rods [7]. A particularly interesting limit of these equations is the case of inextensible strings in which inextensible rods have negligible resistance to bending and twisting: their behaviour is dictated to a large extent by the geometric constraint of inextensibility. In fact, this constraint by itself is enough to give rise to complicated dynamics: while the simplest case of a straight string accelerating along its length may be understood by a simple application of Newton's second law, any closed shape is a solution of the governing equations with any constant tangential velocity [8]. It is therefore the combination of acceleration and ‘turning a corner’ that gives rise to the most interesting dynamics. In particular, accelerating inextensible strings form surprisingly complicated shapes including arches [9] and the mesmerizing ‘chain fountain’ [10,11].

In this paper, we consider the planar motion of a chain moving around a pulley subject to a constant acceleration at the one end and free at the other end. We model the chain as an inextensible string in partial contact with a disc (the pulley) and investigate key features of the motion. This set-up is perhaps the simplest geometry in which one can study how a string or chain ‘turns a corner’, because the curvature of this corner is simply that of the pulley. We first describe in some detail the different phenomena that are observed experimentally and then analyse the lift off in detail by combining theoretical arguments with numerical simulations. Finally, we describe the snapping that is associated with the reversal of curvature that ultimately occurs close to the free end.

## 2. Statement of the problem and experimental observations

In the idealized set-up of our problem, a mass *M* is attached to the end of a chain of linear density (mass per unit length) *ρ*_{ℓ} placed around a circular pulley of radius *R*. The mass *M* is released at time *t*=0 and subject to a constant acceleration *a*. The chain is initially held on the pulley with a known length of chain, *L*, hanging free from the first point of contact *C* (figure 1*a*). In a typical experiment the total length of the chain is *L*_{end}=2*L*+*πR*.

The realization of this thought experiment was carried out by using conventional ball and link chains purchased from the local do-it-yourself shop and hung over horizontal Pyrex beakers (Fisher Scientific) that were used as the frictionless pulley (the surface of the beaker being smooth, the chain slides over the surface with very little friction). The mass *M*≫*ρ*_{ℓ}*L* is released at time *t*=0 resulting in an acceleration *a*≈*g*. Although not restricted in any other way than by a frictionless contact with the pulley, the motion of the chain was observed to be planar, i.e. the motion remains in the *xy*-plane normal to the axis of the pulley and containing the chain at *t*=0. In the subsequent modelling, we therefore assume that the motion remains planar throughout.

Typically, the time evolution of the chain passes through three, qualitatively different, phases shown in figure 1*b*:

— For 0≤

*t*<*t*_{LO}, the chain follows the dynamics that might naively be expected: it moves around the pulley at the speed that is imposed by the accelerating mass.— At a time

*t*=*t*_{LO}, the chain starts to partially lose contact with the pulley, lifting off, so that for*t*>*t*_{LO}the hanging part of the chain goes faster than the rest of the chain. Owing to the excess length of the chain, an arch forms between points*C*and*C**(*t*) defining an angle*θ**(*t*) (figure 1*b*).— After the free end goes past the last contact point

*C*(first image of the last row in figure 1*b*), the arch flares up, its curvature increasing rapidly until the free end eventually snaps at time*t*=*t** (the curvature there changes sign).

To complement our experiments, numerical simulations were carried out using the discrete elastic rod method that provides a discretization of the equations of motion for thin elastic rods using a Lagrangian formulation [12]. This method, and its counterpart for thin viscous threads, have been used and validated previously [13,14]. In the simulations, we use a string model (i.e. the bending and twisting stiffnesses are both set to zero) so that the: Kirchhoff equations [7] simplify to the conservation of linear momentum:
** x**(

*s*,

*t*) is the centre-line of the chain as a function of the arc length

*s*and time

*t*,

**(**

*t**s*,

*t*)=∂

**(**

*x**s*,

*t*)/∂

*s*and

**(**

*n**s*,

*t*) are the unit tangent and normal vectors to the chain axis, respectively (see figure 1

*a*). Further, the variables

*T*(

*s*,

*t*) and

**(**

*v**s*,

*t*)=∂

**(**

*x**s*,

*t*)/∂

*t*denote the tension and velocity of the chain and

*P*(

*s*,

*t*) represents the (frictionless) reaction from the pulley on the chain, such that

*P*=0 whenever there is no contact. This frictionless contact with the pulley is implemented numerically using a geometrical method, i.e. by alternating dynamic steps that ignore contact forces, with projection steps in which the configuration is projected onto the manifold of admissible configurations.

We close the system of differential equations (2.1)–(2.2) using the boundary conditions at the two ends of the chain. Namely, the free end of the string is stress free and we prescribe the acceleration *a* of the chain's end attached to the mass in our experiments, that is
*e*_{y} is in the vertical direction and *L*_{end}=2*L*+*πR* is the arc-length corresponding to the point of which the mass driving is attached. Figure 2 shows a direct comparison between experiments and numerical simulations without any adjustable parameter. The favourable agreement between the two validates the approximations made in modelling the chain as a string that is driven at constant acceleration (our experiments are in fact conducted with a constant force *M*** g** but the two methods are approximately equivalent, because

*ρ*

_{ℓ}

*L*≪

*M*). Of particular interest is the possibility offered by the simulations to access the successive space derivatives of the basic physical variables (such as curvature) with controllable time resolution covering long periods of time. Additionally, physical quantities that are otherwise challenging to measure experimentally, such as the tension within the chain, are readily available from simulations.

At any given time after lift-off but before the chain snaps (*t*_{LO}<*t*<*t**), the chain may be divided into four regions: (I) and (IV) denote the two straight, vertical parts, respectively, (II) denotes the part of the chain not in contact with the pulley and (III) the contact region. For example, in figure 2*b*–*d*, only a small portion of the chain (III) remains in contact with the pulley. In this region, the chain's speed is prescribed by the free fall of the mass, as in region (IV). In region (I), the chain moves significantly faster than the imposed acceleration, whereas remaining perfectly straight and tangential to the pulley. In addition, the tension in the chain is larger than if the lift off were artificially suppressed (shown as a dashed line in figure 2*d*). The area (II) bridges the areas (I) and (III)–(IV) and their corresponding physical quantities, resulting in a ballooning shape (figure 2).

The length of the chain that is initially hanging freely, *L*, turns out to play an important role in the problem and, therefore, we use *L* as the length scale of the problem. It is then natural to use the quantity (*L*/*a*)^{1/2} as the time scale of the problem. If the acceleration of the freely hanging part of the chain were to remain at *a* throughout the motion, the length of the chain hanging beneath the point C then shrinks to zero at a time

From the six dimensional quantities of interest, we identify three dimensionless parameters:
*M* is chosen such that *π*_{1}≪1; *ρ*_{ℓ} therefore enters in the problem as a multiplying factor for forces only. The value of *π*_{2} may easily be varied experimentally by varying the radius of the pulley or, more simply, by varying the length of chain that initially hangs freely. By contrast, varying *π*_{3} is more difficult. Our experiments are always performed with *a*=*g* and hence *π*_{3}=1. However, the inclusion of a body force acting on the chain complicates some of the analysis without changing the qualitative behaviour. Therefore, in much of the analysis, we shall neglect the acceleration owing to gravity, i.e. we assume *π*_{3}=0. Experimentally, this case could be obtained by placing the chain, pulley and the mass on a smooth horizontal surface: at *t*=0, the mass is then pushed off the table into the air [15] and allowed to fall under gravity, so that the acceleration of the chain is *a*=*g*, but the effective gravity acting directly on the chain vanishes. In the simulations, the value of *π*_{3} is set to 1 or 0 depending on whether the results are to be compared with experiments or with our analysis.

## 3. Prior to lift off

We consider a pulley of radius *R* around which a chain with linear density *ρ*_{ℓ} is hung. We begin by including gravity as a body force, though later we will neglect it to simplify the ensuing calculations. Initially, the length of chain between the free end (at arc-length position *s*=0) and the chain's first point of contact with the pulley is *L* (i.e. the point C in figure 1 has arc-length coordinate *s*=*L* at *t*=0^{−}).

At *t*=0^{+}, the end at arc length *s*=*L*_{end} is pulled with constant acceleration *a*.

Assuming that the chain remains in contact with the pulley, ** v**=

*v*

**and projecting (2.1) onto the tangential and normal directions, and then using the Frenet formula ∂**

*t***/∂**

*t**s*=−

*κ*

**with**

*n**κ*=∂

*ϕ*/∂

*s*the curvature, we find that

*ϕ*is the angle between the local tangent to the chain and the upward pointing vertical. In particular, while the chain remains in contact with the pulley,

*t*<

*t*

_{LO}, we know that

*C*in figure 1). In the hanging part

*ϕ*(

*s*,

*t*)=0, and so, integrating (3.1) subject to

*T*(0,

*t*)=0, we find that the tension

*T*may be written

*C*

_{1}=

*ρ*

_{ℓ}

*g*

*s*

_{C}is found by requiring the tension to be continuous at

*s*=

*s*

_{C}.

In the freely hanging portion of the chain, (3.2) is identically satisfied by *P*=0. Elsewhere, (3.2) can be viewed as an equation for the reaction of the pulley on the chain required for this motion to occur. From a simple rearrangement, we find that
*P* is a function of arc length *s* and time *t* through the angle *ϕ* and arc length of the contact point, *s*_{C}(*t*), in addition to the explicit dependence on *s*. The minimum value of *P* at any particular time *t*<*t*_{LO} is attained at *s*=*s*_{C}

The key to understanding why lift off occurs is held by the expression for the smallest reaction force *P*_{min}(*t*) (3.8). At sufficiently early times *P*_{min}>0, and the contact pressure is everywhere positive (it prevents the chain from penetrating the pulley). However, *P*_{min} vanishes at a time *t*=*t*_{LO} given by
*t*>*t*_{LO}, the assumption (3.3) that the string is in contact with the pulley over half a circle breaks down, as it predicts a negative pressure (an adhesive force would be required to maintain contact). The negative values of *P* are first attained in the neighbourhood of *C*, which suggests that lift off takes place there. At time *t*=*t*_{LO}, the length of the hanging chain is

In (3.9), *t*_{LO} is less than the time

Apart from factors obviously required by dimensional analysis, both (3.9) and (3.10) appear to depend only on the ratio *π*_{3}=*g*/*a* of the acceleration owing to gravity to the imposed acceleration. They do not depend on the dimensionless group *π*_{2}=*R*/*L* involving the size of the pulley. Furthermore, we note that in the limit of zero gravity, *π*_{3}=0, one-third of the original hanging chain has passed around the pulley with two-thirds still hanging; when pulling the chain using a heavy mass in free fall, i.e. imposing *π*_{3}=1, then one-half of the original hanging chain has passed around the pulley with one-half still hanging. Here, incorporating gravity makes a quantitative, but not qualitative, difference; either way, an appreciable fraction of the initially hanging chain remains below the first contact point C at lift-off.

From the above argument, we also gain some intuition into *why* the chain lifts off: the tension at the contact point *s*_{C}(*t*) is decreasing with time (because the length of hanging chain is decreasing), whereas the tension required to turn the chain around the pulley increases (because the chain is moving faster and faster). Lift off occurs when the excess tension in the chain owing to the hanging length is ‘used up’.

## 4. Lift off

The prediction for the lift-off time, (3.9), presents a natural test of the model assumptions, in particular with regard to our neglect of the bending stiffness of the chain and the assumption of a frictionless contact between chain and pulley. Experiments were performed with *a*=*g* (i.e. *π*_{3}=1) and a number of different pulley sizes and initial hanging lengths (i.e. varying *π*_{2}). As shown in figure 3, we find reasonable agreement between the theoretical prediction for *t*_{LO} and that measured experimentally. In particular, experiments confirm the result that the size ratio *π*_{2}=*R*/*L* does not affect the onset of lift off. We also found that the value of *t*_{LO} is insensitive to the type of chain used (ball chains and link chains give the same results).

However, we note that the onset of lift off is difficult to assess accurately in experiments owing to its slow initial dynamics (to be discussed in detail in §5); it is therefore natural that experimental values of *t*_{LO} are slightly larger that theoretically predicted (though they remain within 10% of the theoretical value). Furthermore, the favourable agreement between experiments and numerical solution of the full model, which neglects any source of dissipation, shown in figure 2*a* suggests friction cannot be the cause of the discrepancy.

## 5. Behaviour just after lift off

We now turn our attention to the early phase of the motion, just after lift off. In this phase, our experiments and numerical simulations show that the chain ‘balloons’ off the pulley (figure 1*b*). For the sake of simplicity, we shall henceforth neglect the role of gravity, i.e. we set *π*_{3}=0. This analysis is conducted hand in hand with numerical simulations for *π*_{3}=0, but these results should not be compared with experiments (for which *π*_{3}=1). In appendix A, we consider how the results of this section change when *π*_{3}≠0; this analysis shows that the presence of gravity makes quantitative, rather than qualitative, changes to the motion around lift off.

### (a) Velocity after lift off

The numerical simulations of the complete theoretical model (2.1–2.4) and experiments both suggest that the portion of the chain that is freely hanging at the instant of first lift off (i.e. 0≤*s*<*s*_{C}(*t*_{LO})) remains straight (within numerical accuracy) in the immediate aftermath of lift off. We therefore attempt to repeat the calculation of §3, but this time we treat the acceleration in this straight portion, *t*_{LO}, namely *s*_{C}(*t*_{LO}), which we denote by

Setting *g*=0 in (2.1), and focusing on the straight, hanging part (zero curvature), and integrating we find that
*v*_{hang}(*t*_{LO})=*at*_{LO}, we find that
*t*→2*t*_{LO}. However, the argument above only holds in the immediate aftermath of lift off, *t*−*t*_{LO}≪*t*_{LO}. In the simulations, the reversal of curvature actually occurs at a time *b*,*c*), which is earlier than that predicted by (5.3).

The Taylor expansion of (5.3) for *t*−*t*_{LO}≪*t*_{LO} reads
*at*, which is the imposed pulling velocity: the correction to the imposed constant acceleration is given by the following term and occurs at order (*t*−*t*_{LO})^{2}. In figure 4*a*, we observe that our numerical results for the velocity are in good agreement with the prediction of equation (5.3) for times close to *t*_{LO} but then deviate with an error *δ* that is cubic in *t*−*t*_{LO}. This cubic error confirms that the first correction to the constant acceleration motion, namely the term of order (*t*−*t*_{LO})^{2} in (5.4), is correct.

By differentiating (5.3) with respect to time, we obtain an expression for the excess acceleration of the free part of the chain
*t*>*t*_{LO}. Thus, even though the free end is not subject to external forces, it accelerates *faster* than the end that is actually subject to the imposed (constant) acceleration.

A quantity of interest is the arc-length of the region over which the chain has lost contact with the pulley for *t*>*t*_{LO}, which we denote by Δ*s* (figure 4*b*). The above calculation does not give us enough information to determine this quantity. However, the result for *v*_{hang}, (5.3), can be used to determine the excess length of chain that has lost contact with the pulley, i.e. the difference between the length of chain that has lifted off the pulley, Δ*s*(*t*), and the arc length of the pulley from which it has lifted off, *Rθ**(*t*), with *θ**(*t*) the angle subtended between the two contact points C and C* (figure 4*b*). Denoting this quantity by Δ*L*(*t*)=Δ*s*(*t*)−*Rθ**(*t*), we find that
*s* and *Rθ** separately, but only their difference. The Taylor series of Δ*L*(*t*) for *t*−*t*_{LO}≪*t*_{LO} reveals that
*t*−*t*_{LO})^{3} here is consistent with our earlier finding that the correction to the speed of motion is at order (*t*−*t*_{LO})^{2} in (5.4), because we have integrated this quantity in time.

Comparison of the predictions of this analysis, (5.3) and (5.7), with the numerical solution of the fully nonlinear problem (figure 4) shows that these expressions are asymptotically correct immediately after lift off, *t*−*t*_{LO}≪*t*_{LO}, and validates the underlying assumption (that the hanging portion of the chain simply accelerates vertically upwards).

### (b) Characteristics

We now proceed to explain the size of the region of lift off as a function of time. We show that the point of contact *C**(*t*), as defined in figure 4*b*, follows a characteristic of the wave equation describing the evolution of the small perturbations to the motion of the string. In other words, the motion of *C**(*t*) can be viewed as a travelling front produced by the initial lift off at *t*_{LO}. To do so, we now consider the length of chain Δ*s* that makes up the ‘ballooning’ arch forming after lift off. This length is defined to be *s*_{c} and *C* and *C**, respectively (figure 4*b*).

For times close to *t*_{LO}, lift off takes place but it is mild; as a result, this lift off can be described by (i) extending beyond *t*_{LO} the solution maintaining contact with the upper half of the pulley (base solution) and (ii) perturbing it to remove any negative contact pressure. The equations governing the perturbation are obtained by a standard linearization, and are linear wave equations. In these wave equations, the wave speed *T*=*T*(*s*,*t*)). Assuming that the tension remains close to its value prior to lift off, i.e. *T*≈*ρ*_{ℓ}*as* as derived in equation (3.5), we have

When lift off occurs, the chain is locally freed from the pulley at the point *t*,*s*) plot. Within this region, the tension in the chain is altered, i.e. the tension of the true solution does not match exactly that predicted by the base solution that ignores lift off; beyond this region, information can only propagate at the wave speed at the edges, which is, by construction, given by the tension *just* before lift off. The limited propagation speed of information about lift-off results in the formation of an arch from the point *C* to a second contact point *C** (figure 4*b*). We hypothesize that the length of the arch *C* at *t*=*t*_{LO} (figure 5). Deriving Δ*s* from equation (5.8) yields
*a,* the agreement between the two is asymptotically correct immediately after lift off (with the correct prefactor). This indicates that the propagation of lift-off front is indeed limited by the rate at which the perturbations can travel through the string. This argument correctly accounts for the short time dynamics, Δ*s*∼(*t*−*t*_{LO}).

We also note that *θ**, the angle from *C* to *C** defined in figure 5*b*, follows the same scaling law and agrees well with equation (5.9). Indeed, recall that Δ*L*=Δ*s*−*Rθ**; for *t*−*t*_{LO}≪1, we have shown Δ*L*∼(*t*−*t*_{LO})^{3} in equation (5.7) and Δ*s*∼(*t*−*t*_{LO}) above, it follows that Δ*s* and *Rθ** must be equal to leading order in (*t*−*t*_{LO}), i.e. *θ**∼Δ*s*/*R* as checked numerically in figure 6*b*.

## 6. Snapping

We now describe the kinematics of the free end during snapping. Let *ϕ*(*s*,*t*) be the angle between the tangent ** t** and the upward pointing vertical axis. We define the curvature

*κ*(

*s*,

*t*)=∂

*ϕ*(

*s*,

*t*)/∂

*s*(recall that the free end corresponds to

*s*=0). We define snapping as a sudden change in maximal curvature of the chain as depicted in figure 7

*a*,

*b*, which shows the time evolution of the largest curvature of the chain,

*κ*

_{M}. We observe that the arc-length position

*s*

_{M}at which this maximum curvature is observed travels towards the free end of the chain as snapping approaches, as shown in figure 7

*b*. During snapping,

*κ*

_{M}also has a singular behaviour: the curvature strongly increases and changes sign, as shown in the last three configurations in figure 7

*a*and in figure 7

*b*. Note that for real ropes this singularity will be regularized by their finite bending stiffness.

^{1}Therefore, the singular behaviour predicted by our string model may be seen as the ‘outer solution’ of a regularized model. Note that snapping is associated with a rapid increase of the free end velocity as illustrated in figure 7

*c*. These features are similar to what occurs in a cracking whip [1] and a falling chain [16–19].

Let us denote by *t** the time of snapping, defined as the time where the curvature diverges near the free end. For optimal accuracy, we analyse the snapping dynamics based on simulations having a relatively large value of *π*_{2}=*R*/*L*=1/2*π*. This allows us to use a fine and uniform spatial discretization of the string. To ease the comparison between this specific simulation and the rest of our results, we introduce a non-dimensionalization with respect to the space and time scales of the problem, namely *L* and *π*_{2} and *π*_{3}; in the forthcoming analysis, we focus on the particular case *π*_{2}=1/(2*π*) and *π*_{3}=0.

To measure the divergence of curvature from the simulations, we use a curvature norm that is intended to capture the largest value of curvature,
*p*≥2 is an integer. The value of the integer *p* that is used results from a trade-off: one should use values of *p* as large as possible, and *p* is too large, then *p*=6; we have verified that the results are unchanged for *p*=8.

We begin by assuming that as the singularity at *α*, to be determined, on a region of size *α*(*p*−1)>0, this contribution makes the integral diverge at the time of snapping *α*(*p*−1)>0 is indeed satisfied as *p*=6 and *α*>0, see below. One can check that the quantity *α*, independently of the particular value of *p* chosen in the definition of

The snapping time *α* entering in the power law *a*).

The points collapse onto a straight line, aside from a thin band of width *a* yields both the snapping time *t*-axis) and the exponent *α* (the reciprocal of the slope). This fit yields a value of the exponent *α*=0.659 which is numerically close to 2/3—we do not know whether this is a coincidence—and *α* have been obtained in the particular case *π*_{2}=1/(2*π*) and *π*_{3}=0. We suspect, however, that the exponent *α* can be explained by a boundary-layer theory and that it is actually independent on *π*_{2} and *π*_{3}. Unfortunately, we can offer no proof for this statement.

As observed in the experiments (see §2), the string remains straight in the simulations along an interval of length ℓ(*t*) comprising the free end, at all times until *t**. This straight portion of the string is apparent in figures 7*a* and 8*c*. It defines the region denoted by (I), which is adjacent to the region (II) of large curvature (figure 2*c*). The dimensionless length *β*=1.08±0.2 (fit not shown). The numerical accuracy on this exponent *β* is not as good as that of *α* as the determination of the straight region is sensitive to numerical noise. It is clear from our numerical data, however, that the exponent *β* is larger than *α*, meaning that the length of the straight region near the free end vanishes more quickly than the minimum radius of curvature as

Next, we proceed to use our analysis to uncover a self-similar behaviour in the curvature profile. Anticipating that the point of maximal curvature plays a central role we introduce a weighting to capture its position:

Finally, we rescale the arc-length parameter to
*b*,*c*, with darker plots corresponding to times closer to the snapping time

In figure 8*d*, we plot the rescaled curvature

In summary, our analysis shows that the snapping singularity may characterized by a self-similar solution, whose scaling behaviours have been identified numerically.

## 7. Conclusion

In this paper, we have considered a degenerate version of Atwood's machine, in which a single mass pulls a chain around a pulley. In stark contrast with the apparent simplicity of the set-up, we have found that the dynamics is extremely rich, successively displaying a ballooning instability of the chain and a snapping motion of its free end, reminiscent of what is seen in a cracking whip. We have shown that the chain dynamics is well captured by a frictionless string model and that some of its features may be captured by simple arguments. In particular, we have shown that the geometry of the problem, through the imposed rotation of the chain around the pulley, is key to understanding how the end of the chain is able to ‘beat’ the free-fall that drives its motion.

Our observations can be used to speculate on the peculiar hunting techniques of a variety of amphibians. Indeed, instead of throwing their tongue in a straight motion (as observed in chameleons [20,21]), certain species of toads [22] and salamanders [23] adopt an unfurling tongue strategy. Of course, the reasons for such a mechanism are many and varied but we believe that the increase of tip velocity observed in the case of a chain is likely to reappear in problems involving a finite bending stiffness. It is then natural to wonder whether this geometrical amplification of acceleration may be used by these amphibians to allow them to maximize their chances of capturing a prey?

Finally, we note that while we have been able to rationalize some of the observations from experiment and simulation, others remain elusive. For example, predicting the shape of the ballooning region either with simple arguments, or preferably analytically, remains beyond our reach. Difficulties in doing so arise from the fact that the ‘base solution’ of the problem is unsteady (because *a*>0). An interesting avenue of research would be to explore the case where the chain is pulled at constant speed thus without any acceleration, potentially allowing for analytical developments. Similarly, there is hope that the self-similar exponents *α* and *β* identified in §6 can be explained by some boundary layer theory in future work.

## Data accessibility

All the data accompanying this publication are directly available within the publication.

## Authors' contributions

All authors designed the research, performed the research, co-wrote the paper and gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

This publication is based in part upon work partly supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 637334, GADGET to D.V.).

## Appendix A. The effect of gravity immediately after lift off

For the sake of simplicity, the main portion of the paper focuses on the problem in the absence of gravity. In this appendix, we show how the results derived in §5a for the behaviour immediately following lift off are altered once gravity is included. These results confirm that, in this case at least, gravity affects the results quantitatively, rather than qualitatively.

**(a) The unknown chain acceleration**

Immediately after lift off, the free end of the chain accelerates at an unknown rate *C* when lift off started (*ρ*_{ℓ}*v*^{2}_{hang}*κ*) for some undetermined curvature *κ*. We then have that (5.2) becomes
*v*_{hang}(*t*_{LO})=*at*_{LO} to give
*v*_{hang}(*t*) is an increasing function of *t*>*t*_{LO} and has a singularity at
*O*(*t*−*t*_{LO})^{2}, as in the case without gravity (figure 4*a*).

**(b) The excess length Δ L**

From the expression (A 2) for the acceleration of the hanging portion of the chain, we find that the excess length absorbed by the lifted off portion is

## Footnotes

↵1 For chains made up of rigid links, this singularity is regularized too, by the contact between successive chain links.

- Received March 15, 2016.
- Accepted May 17, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.