The Domino Effect J. M. J. van Leeuwen 4 Instituut–Lorentz, Leiden University, P. O. Box 9506, 0 0 2300 RA Leiden, The Netherlands 2 n February 2, 2008 a J 7 Abstract ] h p The physics of a row of toppling dominoes is discussed. In particular the - n forces between the falling dominoes are analyzed and with this knowledge, the e effect of friction has been incorporated. A set of limiting situations is discussed g . in detail, such as the limit of thin dominoes, which allows a full and explicit s c analytical solution. The propagation speed of the domino effect is calculated si for various spatial separations. Also a formula is given, which gives explicitly y the main dependence of the speed as function of the domino width, height and h interspacing. p [ 1 1 Introduction v 8 1 Patterns formed by toppling dominoes are not only a spectacular view, but their dy- 0 1 namics is also a nice illustration of the mechanics of solid bodies. One can study the 0 problem on different levels. Walker [1] gives a qualitative discussion. Banks [2] con- 4 0 siders the row of toppling dominoes as a sequence of independent events: one domino / s undergoes a free fall, till it hits the next one, which then falls independently of the c others, and so on. He assumes that in the collision the linear momentum along the sup- i s y porting table is transmitted. This is a naive viewpoint, but it has the advantage that h the calculation can be carried out analytically. A much more thorough treatment has p been given by D. E. Shaw [3]. His aim is to show that the problem is a nice illustration : v of computer aided instruction in mechanics. He introduces the basic feature that the i X domino, after having struck the next one, keeps pushing on it. So the collision is com- r pletely inelastic. In this way a train develops of dominoes leaning on each other and a pushing the head of the train. One may see this as an elementary demonstration of a propagating soliton, separating the fallen dominoes from the still upright ones. Indeed Shaw’s treatment is a simple example how to handle holonomous constraints in a com- puter program describing the soliton. As collision law he takes conservation of angular momentum. We will demonstrate, by analyzing the forces between the dominoes, that this is not accurate. The correction has a substantial influence on the solition speed, even more important than the inclusion of friction, which becomes possible when the forces between the dominoes are known. The setting is a long row of identical and perfect dominoes of height h, thickness d and interspacing s. In order to make the problem tractable we assume that the dominoes only rotate (and e.g. do not slip on the supporting table). Their fall is due to 1 the gravitational force, with acceleration g. The combination √gh provides a velocity scale and it comes as a multiplicative factor in the soliton speed. Typical parameters of theproblemaretheaspectratiod/h,whichisdeterminedbythetypeofdominoesused, and the ratio s/h, which can be easily varied in an experiment. Another characteristic of the dominoes is their mutual friction coefficient µ which is a small number ( 0.2). ∼ The first domino gets a gentle push, such that it topples and makes a “free rotation” till it strikes the second. After the collision the two fall together till they struck the third and so forth. So we get a succession of rotations and collisions, the two processes being governed by different dynamical laws. Without friction the rotation conserves energy, while the constraints exclude the energy to be conserved in the collision. In fact this is the main dissipative element, more than the inclusion of friction. The goal is to find the dependence of the soliton speed on the interdistance s/h. In the beginning this speed depends on the initial push, but after a while a stationary pattern develops: a propagating soliton with upright dominoes in front and toppled dominoes behind. The determination of the forces between the dominoes requires that we first briefly outline the analysis of Shaw. Then we analyze the forces between the dominoes. Knowing these we make the collision law more precise. With the proper rotation and collision laws we give the equations for the fully developed solitons. The next point is the introduction of friction and the calculation of its effect on the soliton speed. As illustration we discuss the limit of thin dominoes d 0, with permits for → small interseparations a complete analytical solution. Finally we present our results for the asymptotic soliton speed for various values of the friction and compare them with some experiments. We also give an explicit formula, which displays the main dependence of the soliton speed on the parameters of the problem. The paper closes with a discussion of the results and the assumptions that we have made. 2 Constraints on the Motion The basic observation is that domino i pushes over domino i+1 and remains in contact afterwards. So after the contact of i with i +1 the motion of i is constrained by the motion of i+1. Therefore we can take the tilt angle θ of the foremost falling domino, n as the only independent mechanical variable (see Fig. 1). Simple goniometry tells that hsin(θ θ ) = (s+d)cosθ d. (1) i i+1 i+1 − − To see this relation it helps to displace domino i+1 parallel to itself, till its bottom line points at the rotation axis of domino i (see Fig. 1). By this relation one can express the tilt angle θ in terms of the next θ and so on, such that all preceding tilt angles i i+1 are expressed in terms of θ . The recursion defines θ as a function of θ of the form n i n θi = pn−i(θn), (2) i.e. the functional dependence on the angle of the head of the train depends only on the distance n i. The functions p (θ) satisfy j − (s+d)cospj−1(θ) d pj(θ) = pj−1(θ))+arcsin − , (3) h ! with the starting function p (θ) = θ. They are defined on the interval 0 < θ < θ , 0 c where θ is the angle of rotation at which the head of the train hits the next domino c θ = arcsin(s/h). (4) c 2 We will call θ the angular distance. From the picture it is clear that the functions c are bounded by the value θ∞, which is the angle for which the right hand side of (1) vanishes d cosθ∞ = . (5) s+d θ∞ is the angle at which the dominoes are stacked against each other at the end of the train. We call θ∞ the stacking angle. α = θ − θ i i+1 A θ c h µ f f α i i θ θi i+1 C d s+d B Figure 1: Successive dominoes. The tilt angle θ is taken with respect to the vertical. i In the rectangular traingle ABC the top angle is α = θ θ , the hypotenuse has i i+1 − the length h and the base BC the length (s+d)cosθ d. Expressing this base in i+1 − the hypotenuse and the top angle yields relation (1). In the picture the tilt angle of the head of chain θ has reached its final value θ = arcsin(s/h). The first domino has n c almost reached the stacking angle θ∞. The normal force fi and the friction force µfi that domino i exerts on i+1 are also indicated. The picture shows that the functions p (θ) are monotonically increasing functions. j They become flatter and flatter with the index j and converge to the value θ∞ (at least not too close to the maximum separation s = h, see Section 10). The functions are strongly interrelated, not only by the defining recursion (3). The angle θ can be i calculated from the head of the trainθn by pn−i but also froman arbitraryintermediate θk by pk−i. This implies pn−i(θ) = pk−i(pn−k(θ)), e.g. pj(θ) = pj−1(p1(θ)). (6) One easily sees that p (0) = θ . Therefore one has 1 c pj(0) = pj−1(p1(0)) = pj−1(θc), (7) a property that will be used later on several times. An inmediate consequence of (1) is the expression for the angular velocities ω = i dθ/dt in terms of ω . From the chain rule of differentiation we find n dθ dθ i n ωi = = wn−iωn, (8) dθ dt n 3 with dp (θ) j w (θ) = . (9) j dθ Computationally it is easier to calculate the w recursively. Differentiation of (3) with j respect to θ yields n (s+d)sinp (θ) j wj(θ) = wj−1(θ) 1 . (10) − hcos[pj(θ) pj−1(θ)]! − Another useful relation follows from differentiation of the second relation (6) wj(θ) = wj−1(p1(θ))w1(θ) wj(0) = wj−1(θc), (11) ⇒ since p (0) = θ and w (0) = 1. 1 c 1 3 Rotation Equations Without friction, the motion between two collisions is governed by conservation of energy, which consists out of a potential and a kinetic part. The potential part derives from the combined height of the center of mass of the falling dominoes, for which we take the dimensionless quantity n H (θ ) = [cosθ +(d/h)sinθ ]. (12) n n i i i X The kinetic part is given by the rotational energy, for which holds n 2 2 2 K (θ ,ω ) = (I/2) ω , I = (1/3)m(h +d ), (13) n n n i i X where I is the angular moment of inertia with respect to the rotation axis and m is the mass of the dominoes. We write the total energy as 1 1 I 2 E = mghe = mgh H (θ )+ I (θ )ω , (14) n 2 n 2 n n mgh n n n! where the dimensionless effective moment of inertia I(θ ) is defined as n n 2 I (θ ) = w (θ ). (15) n n j n j X We have factored out mgh/2 in (14) as it is an irrelevant energy scale. This has the advantage that the expression between brackets is dimensionless. The factor I/mgh I h(1+d2/h2) = (16) mgh 3g provides a time scale that can be incorporated in ω . From now on we put this factor n equal to unity in the formulae and remember its value when we convert dimensionless velocities to real velocities. 4 We see (14) as the defining expression for ω as function of θ n n 1/2 e H (θ )) n n n ω (θ ) = − . (17) n n In(θn) ! As mentioned e is a constant during interval n. So we can solve the temporal behavior n of θ from the equation n dθ (t) n = ω (θ ). (18) n n dt The initial value for θ is 0 and the final value equals the rotational distance θ . The n c duration of the time interval where n is the head of the chain, follows by integration θc dθ n t = . (19) n Z0 ωn(θn) In this time interval the soliton has advanced a distance s + d. The ratio (s + d)/t n gives the soliton speed, when the head of the train is at n. In order to integrate the equations of motion (18) we must have a value for e which basically amounts to n finding an initial value ω (0) as one sees from (14). In the next section we outline how n to calculate successively the ω (0). n Putting all ingredients together we obtain the asymptotic soliton speed v as as 1/2 3 s+d 1 v = gh lim . (20) as q 1+d2/h2! h n→∞ tn In this formula the time t is computed from the dimensionless equations (setting n I/mgh equal to 1). 4 The Collision Law, first version We now investigate what happens when domino n hits n + 1. In a very short time domino n+1 accumulates an angular velocity ω (0). The change in ω takes place n+1 n+1 while the tilt angles of the falling dominoes hardly change. Shaw [3] postulates that the total angular momentum of the system is unchanged during the collision. This is not self-evident and we comment on it in Section 6. Before the collision we have the angular momentum n L = w (θ )ω (θ ). (21) n j c n c j X After the collision we have n+1 L = w (0)ω (0). (22) n+1 j n+1 j X Equating these two expressions yields the relation n n+1 ω (0) = ω (θ ) w (θ )/ w (0). (23) n+1 n c j c j j j X X With the aid of this value we compute the total energy e and the next integration n+1 can be started. For the first time interval holds e = 1 +ω2(0) since only the zeroth 0 0 domino is involved and it starts in upright position with angular velocity ω (0). The 0 value of ω (0) has no influence on the asymptotic behavior. After a sufficient number 0 of time intervals, a stationary soliton develops. 5 5 Forces between the Dominoes Conservation of energy requires the dominoes to slide frictionless over each other. Before we can introduce friction we have to take a closer look at the forces between the falling dominoes. Without friction the force which i exerts on i+1 is perpendicular to the surface of i+1 with a magnitude f (see Fig. 1). Consider to begin with the head i of the train n. Domino n feels the gravitational pull with a torque T n T = (sinθ (d/h)cosθ )/2, (24) n n n − and a torque from domino n 1 equal to the force fn−1 times the moment arm with − respect to the rotation point of n. The equation of motion for n becomes dω n = Tn +fn−1h[cos(θn−1 θn) (s+d)sinθi+1]. (25) dt − − Domino n 1 feels, beside the gravitational pull Tn−1, a torque from n which slows it − down and a torque from n 2 which speeds it up. Generally the equation for domino − i has the form dω i = Ti +fi−1ai−1 fibi. (26) dt − The coefficients of the torques follow from the geometry shown in Fig. 1. a = hcos(θ θ ) (s+d)sinθ , b = hcos(θ θ ). (27) i i i+1 i+1 i i i+1 − − − Note that the first equation (25) is just a special case with f = 0. Another interesting n features is that a < b . So i gains less from i 1 than i 1 looses to i. Therefore i i − − dominoes, falling concertedly, gain less angular momentum than if they would fall independently. This will have a consequence on the application of conservation of angular momentum in the collision process. We come back on this issue in the next section. We can eliminate the forces from the equation by multiplying (25) with r = 1 and 0 the general equation with rn−i and chosing the values of rj such that an−j n−1 aj rj = rj−1 , (r0 = 1), or rn−i = . (28) bn−j j=i bj Y Then adding all the equations gives dω i rn−i Ti = [fi−1rn−iai−1 firn−i−1ai] = 0. (29) " dt − # − i i X X Now observe that the recursion for the r is identical to that of the w as given in (10). j j With r = 1 we may identify r = w . It means that if we multiply (29) with ω and 0 j j n replace rn−iωn by ωi, we recover the conservation of energy in the form d 1 2 ω = ω T . (30) dt2 i i i i i X X It is not difficult to write the sum of the torques as the derivative with respect to time of the potential energy, thereby casting the conservation of energy in the standard form. So if conservation of energy holds, the elimination of the forces is superfluous. However, equation (29) is more general and we use it in the treatment of friction. 6 6 The Collision, second version We have assumed that inthe collision of the head of chain n with the next domino n+1 conserves angular momentum. Having a more detailed picture of forces between the sliding dominoes we reconsider this assumption. In this section without friction and in Section 8 with friction. The idea is that in the collision domino n, exerts a impulse on n+ 1 and vice versa with opposite sign. In other words: one has to integrate the equations of motion of the previous section over such a short time that the positions do not change, but that the velocities accumulate a finite difference. However, not only the jump in velocity propagates downwards, also the impulses have to propagate downwards in order to realize these jumps. Denoting the impulses by capital F’s, domino i receives Fi from i + 1 and Fi−1 from i 1. So we get for the jumps in the − rotational velocity ω (0) = F a , n+1 n n  w1(0)ωn+1(0) w0(θc)ωn(θc) = Fn−1an−1 Fnbn,  − ··· = ··· − (31)  wn+1−i(0)ωn+1(0)−wn−i(θc)ωn(θc) = Fi−1ai−1 −Fi bi. The functions a and b are the same as those defined in (27). If we would have a = b i i i i we could add all equations and indeed find that the angular total angular momentum is conserved in the collision. But only a = b since θ = 0. The impulse F can be n n n+1 i eliminated in the same way as before by multiplying the ith equation with rn+1−i and adding them up. For the coefficient of ω (0) we get n+1 n+1 n+1 rn+1−iwn+1−i(0) = rjwj(0) = Jn+1, (32) i j=0 X X and for the coefficient of ω (θ ) one finds with (10) n c n n n rn+1−iwn−i(θ) = rn+1−iwn+1−i(0) = rjwj(0) = Jn+1 1. (33) − i i j=1 X X X As general relation we get J ω (0) = (J 1)ω (θ ). (34) n+1 n+1 n+1 n c − In our frictionless case r = w and therefore J = I (0). So the desired relation j j n+1 n+1 reads I (0)ω (0) = (I (0) 1)ω (θ ) = I (θ )ω (θ ). (35) n+1 n+1 n+1 n c n c n c − We have added the last equality since it smells as a conservation of angular momentum using the effective angular moment of inertia I(θ). This inertia moment is however linked to the energy and not to the angular momentum. The true angular momentum conservation is given in Section 4. It is also not conservation of kinetic energy. Then the squares of the angular velocities would have to enter. The difference with the earlier relation (23) is that the sum involves the squares of the w’s. This has a notable influence on the asymptotic velocity. 7 7 Fully Developed Solitons After a sufficient number of rotations and collisions a stationary state sets in. Then we may identify in the collision law the entry ω (0) with ω (0). This allows to solve n+1 n for the stationary ω (0). We use (11) to relate the effective moments of inertia n n n−1 2 2 2 2 2 I (θ ) = w (θ ) = w (0)+w (θ ) = I (0) w (0)+w (θ ). (36) n c j c j+1 n c n − 0 n c j=0 j=0 X X For large n the last term vanishes and we may drop the n dependence in I . So n I(θ ) = I(0) 1. (37) c − The collision laws thus may be asymptotically written as, I(0)ω (0) = [I(0) 1]ω (θ ). (38) n n c − The rotation is governed by the conservation of energy, which we write as 2 2 I(θ)ω (θ)+H (θ) = I(0)ω (0)+H (0). (39) n n n n We can use (9) to relate the height function H (θ ) to its value at θ = 0. n c n d d H (θ ) = [cosp (θ )+ sinp (θ )] = H (0) 1+cosp (θ )+ sinp (θ ). (40) n c j c j c n n c n c h − h j X The limiting value of pn is the stacking angle θ∞ Therefore the difference between the initial and the final potential energy reads d H(0) H(θc) = 1 cosθ∞ sinθ∞ P(h,d,s). (41) − − − h ≡ Wehave introduced the functionP asthe loss in potentialenergy in thesoliton motion. It is the difference between an upright domino and a stacked domino at angle θ∞. The functional form reads explicitly sh d(s2 +2sd)1/2 P(h,d,s) = − . (42) h(s+d) It is clear that the domino effect does not exist if P is negative, because a domino tilted at the stacking angle has a higher potential energy than an upright domino. We use (41) in the conservation law for the energy, taken at θ = θ c 2 2 I(θ )ω (θ ) I(0)ω (0) = P(h,d,s). (43) c n c − n Solving ω (0) and ω (θ ) from (38) and (43) yields n n c I(0) 1 I(0) 2 2 ω (0) = P(h,d,s) − , ω (θ ) = P(h,d,s) . (44) n I(0) n c I(0) 1 − By and large √P sets the scale for the rotation velocity. The dependence on I(0) is rather weak. For large I(0) it drops out. The minimum value of I(0) is 2 which is reached for large separations. 8 8 Friction After all this groundwork it is relatively simple to introduce friction. Let us start with the equation of motion (26). Friction adds a force parallel to the surface of i+1. For the strength of the friction force we assume the law of Amonton-Coulomb [4] f = µf, (45) friction where f is the corresponding perpendicular force. Inclusion of friction means that the coefficients a and b pick up a frictional component. The associated torques follow i i from the geometry of Fig. 1. So the values of the a and b change to i i a = hcos(θ θ ) (s+d)sinθ µd, i i i+1 i+1 − − − (46)  b = hcos(θ θ )+µhsin(θ θ ).  i i − i+1 i − i+1 Then we may eliminate the forces as before, which again leads to (29). But we cannot  identify any longer r with w . In order to use (29) we must express the accelerations i i dω /dt in the head of chain dω /dt. This follows from differentiating (8) i n dω dω i n 2 dt = wn−i(θn) dt +vn−i(θ)ωn, (47) with v given by i dw (θ) j v (θ) = . (48) j dθ n The v can be calculated from the recursion relation, that follows from differentiating j (10). Clearly the recursion starts with v = 0 (see (47)). 0 Next we insert (47) into (26) and obtain n dω n n n 2  rjwj dt =  rj Tn−j− rj vjωn. (49) j j j X X X       The equation can be transformed into a differential equation for dω /dθ by dividing n n (50) by ω = dθ /dt n n n dω n 1 n n rjwj = rjTn−j rjvj ωn. (50)   dθ   ω −  j n j n j X X X       We use this equation to find ω as function of θ and then (18) again to calculate the n n duration of the time between two collisions. The inclusion of friction in the collision law is even simpler, since relation (34) remains valid, but now with the definitions (46) for a and b . i i 9 Thin Dominoes Sometimes limits help to understand the general behaviour. One of the parameters, which has played sofar a modest role, is the aspect ratio d/h. In our formulae it is perfectly possible to take this ratio 0. In practice infinitely thin dominoes are a bit weird, because with paperthin dominoes one has e.g. to worry about friction with the air. In this limit we can vary s/h over the full range from 0 to 1. In Fig. 2 we have plotted the asymptotic velocity as function of the separation s/h. The curve is rather flat with a gradual drop–off towards the large separtions. We discuss here the two limits where the separation goes to 0 and where it approaches its maximum s = h. Both offer some insight in the overall behavior. 9 2 v as Thin Domino Limit 1/2 (gh) 1.5 1 Banks 0.5 0 0 0.5 1 s/h Figure 2: The asymptotic soliton velocity as function of the separation s/h in the thin domino limit. Also is plotted the result of Banks in the same limit. 9.1 Infinitesimal Separation If the dominoes are narrowly separated, the head of chain rotates only over a small angle θ = arcsin(s/h) s/h and the collisions will rapidly succeed each other. The c ≃ number of dominoes with a tilt angle θ between 0 and π/2 becomes very large and i slowly varying with the index i. So a continuum description is appropriate. We first focus on the dependence of θ (θ ) on the index i and later comment on the dependence i n on the weak variation with θ (which is confined to the small interval 0 < θ < θ ). n n c We take as coordinate x the distance of domino i. x = is/h (51) and use ν = ns/h for the position of the head of the train. Then θ = θ(x), θ = θ(x+dx), (52) i i+1 with dx = s/h. So for d = 0 and s/h 0 the constraint (1) becomes → sin[θ(x) θ(x+dx)] = dxcosθ(x+dx), (53) − leading to the differential equation dθ(x) = cosθ(x), (54) dx − which has the solution sinθ(x) = tanh(ν x) or θ(x) = arcsin(tanh(ν x)). (55) − − 10