c 2007 Society for Industrial and Applied Mathematics

Size: px

Start display at page:

Download "c 2007 Society for Industrial and Applied Mathematics"

Damon Gardner
5 years ago
Views:

1 SIAM REVIEW Vo. 49,No. 1,pp c 7 Society for Industria and Appied Mathematics Domino Waves C. J. Efthimiou M. D. Johnson Abstract. Motivated by a proposa of Daykin [Probem 71-19*, SIAM Rev., 13 (1971), pp ], we study the wave that propagates aong an infinite chain of dominoes and find the imiting speed of the wave in an extreme case. Key words. dominoes, waves, modeing, mechanics AMS subject cassifications. 7B99, 7F35, 97A9 DOI /S Introduction. Everyone is famiiar with dominoes and has used them for fun. A common game is to arrange the dominoes in a row (Figure 1.1) and give a push to the first. This generates a peasing wave of faing dominoes. The propagation happens at some speed v (not necessariy constant). A quaitative discussion for genera audiences is given by Waker in [3]. Given the game s simpicity it is perhaps surprising to discover that an exact computation of the speed v is quite difficut. Daykin reaized this in the foowing 1971 proposa ([1]; see aso []) to the readers of the SIAM Review: How fast do dominoes fa? The domino theory of Southeast Asia says that if Vietnam fas, then Laos fas, then Cambodia fas, and so on. Hearing a discussion of the theory ed me to wonder about the proposed physica probem. The reader is invited to set his or her own reasonabe simpifying assumptions, such as perfecty eastic dominoes, constant coefficient of friction between dominoes and the tabe, and initia configuration with the dominoes equay spaced in a straight ine, and so on. In 1983, McLachan et a. [4] found a scaing aw for the speed v in the imiting case of dominoes with zero thickness equay spaced in a straight ine. With these assumptions the authors found the functiona reation (1.1) v = ( ) d g G. Here is the height of the dominoes, d the spacing between dominoes, and G(x) an undetermined function of x. This reation foowed from dimensiona anaysis of Received by the editor February 9, 4; accepted for pubication (in revised form) September 9, 6; pubished eectronicay January 3, 7. Department of Physics, University of Centra Forida, Orando, FL 3816 (costas@physics. ucf.edu). 111

2 11 C. J. EFTHIMIOU AND M. D. JOHNSON t d d Fig. 1.1 A uniform chain of dominoes Fig. 1. A 1 A A 3 A 4 A n A chain of massess rods carrying masses m on top. The masses are indicated as finite spheres ony for the sake of visuaization. the probem. 1 McLachan et a. proceeded to test the formua experimentay using dominoes of heights h =4.445 cm and h =8.89 cm. More recenty, Banks presented a simpified description of the effect [5]. His anaysis, among other assumptions, assumes a uniform propagation speed and conservation of momentum. A different direction was taken by Shaw in a short paper [6] describing how to mode the domino effect as a computer simuation and use it as an experiment in an undergraduate physics ab. Equation (1.1) is not an entirey satisfactory soution, of course, containing as it does an unknown function. This recognition became the motivation for the present artice. In this work we deveop an expression for the speed v. The resut can be cast as a particuar form of the scaing function G arising from a particuar set of assumptions. In order to highight the basic physics behind the probem, we repace the dominoes by massess rods topped with point masses m, as seen in Figure 1.. A simiar anaysis in the case of dominoes with the shape of paraeepipeds is straightforward, athough there are some minor differences.. The Mode..1. The Assumptions. We sha assume the foowing: 1. The chain of rods is uniform. This means that a rods are identica and are equay spaced aong a straight ine. Let be the ength of a rod and m the mass on top.. The coisions are head-on. This means that, seen from above, a the rods are and remain aigned on the same ine. If this is not the case, then additiona 1 This paper shoud aow the reader to write down a compete ist of assumptions needed to reach this concusion.

3 DOMINO WAVES 113 parameters are necessary to describe the coisions from point to point. The higher the asymmetry, the more parameters are needed, and the probem becomes highy compicated. 3. There is enough static friction between the rods and the foor to keep the rods from siding reative to the foor. Thus the rods pivot about fixed axes. 4. No energy is dissipated at the contact point between the rods and the foor. This condition is independent of the previous one; it is possibe for an object to rotate about an axis and yet dissipate energy. 5. Coisions are instantaneous. This means that the time interva t during which a coision occurs is zero. The change in tota anguar momentum is then L = t τ =. That is, tota anguar momentum is conserved during the coision, even though gravity is an externa force and produces a nonzero torque. 6. The coisions are eastic. This means that energy is not dissipated during the coisions. 7. The rods are stiff. This means that there is no deformation of the rods and thus no energy is converted to eastic potentia energy of deformation. This condition is independent of the previous one; it is possibe for the rods to be stiff and sti dissipate energy during a coision... Definition of Symbos. To faciitate the cacuations in the next section, we present our notation in advance. We abe the rods sequentiay with the numbers 1,, 3,...,k,...,starting with 1 from the eft end. We aso abe A 1,A,A 3,...,A k,..., the pivot points of the rods as seen in Figure 1.. Then θ k is the anguar dispacement of a rod from the vertica, and ω k = dθ k / is the corresponding time-dependent anguar veocity; Ω 1 is the initia anguar veocity of the first rod immediatey after it is pushed; Ω k is the initia anguar veocity of the kth rod just as it begins to move; this is of course the resut of the coision with the (k 1)th rod (for k>1); Ω fk is the anguar veocity of the kth rod just before coision with the (k + 1)th rod (the subscript f means fina or faen ); Ω bk is the anguar veocity of the kth rod just after coision with the (k + 1)th rod (b because the rod has just bounced ); β 1 is the ange a rod forms with the vertica at the point of coision: β 1 = sin 1 d ; T k is the time the kth rod takes to fa from the vertica to its coision with the (k + 1)th rod. In this notation, Ω k is the anguar veocity of the kth rod at θ k = and Ω fk,ω bk its anguar veocities after it has faen to θ = β 1, just before and just after coision with the next rod, respectivey..3. Study of the Two-Rod Coisions. Now examine the coision between the kth and (k + 1)th rods. Our assumptions guarantee that during the coision kinetic energy and anguar momentum are conserved, and that whie a rod fas its tota energy (kinetic pus potentia) is conserved. These conservation aws determine the soution. For the foowing argument, refer to Figure.1. Just before the coision the kth rod has anguar veocity Ω fk and the (k + 1)th rod is at rest. After coision the kth rod has anguar veocity Ω bk and the (k + 1)th

4 114 C. J. EFTHIMIOU AND M. D. JOHNSON Ω k β 1 β Ak 1 A k A k+1 (a) Ak 1 A k A k Ak 1 A k A k+1 Ak 1 A k A k+1 Fig..1 (c) β 1 Ω f,k 1 Ω k+1 Ω bk Ak 1 A k A k+1 Ak 1 A k A k+1 (e) (f) (b) (d) β 1 Ω b,k 1 The series of coisions between the (k 1)th, kth,and (k +1)th rods. In particuar,the figure shows the state of the rods (a) just before the (k 1)th and kth rods coide; (b) just after the (k 1)th and kth rods have coided; (c) whie the kth rod rotates towards the (k +1)th rod; (d) just before the kth and (k +1)th rods coide; (e) just after the kth and (k +1)th rods have coided; (f) whie the (k +1)th rod rotates towards the (k +)th rod. rod has anguar veocity Ω k+1. Appying conservation of kinetic energy, 1 IΩ fk = 1 IΩ bk + 1 IΩ k+1, where I = m is the moment of inertia of a rod. Therefore (.1) Ω fk = Ω bk +Ω k+1. Next we appy conservation of anguar momentum with respect to point A k+1. Just before the coision the kth rod has anguar veocity Ω fk and thus transationa veocity ṽ k = Ω fk. From Figure. one can see that ony the component ṽ k cos β 1 contributes to the anguar momentum cacuated around the point A k+1, and that it does so with impact parameter cos β 1. The (k + 1)th rod has no anguar momentum initiay. Therefore L initia = m (ṽ k cos β 1 )(cos β 1 ) = m Ω fk cos β 1. Ω fk

5 DOMINO WAVES 115 BEFORE AFTER Fig.. A k π β 1 π β 1 A k+1 A k A k+1 ṽ k cos β 1 π β 1 Ω k+1 π β 1 ṽ k = Ω fk ṽ k cos β 1 ṽ k = Ω bk The coision between the kth and (k +1)th rods. cos β 1 After the coision the (k + 1)th rod rotates around the point A k+1 with anguar veocity Ω k+1 and thus has anguar momentum IΩ k+1. The kth rod has the new anguar veocity Ω bk, again around the point A k. Therefore it wi contribute an anguar momentum m Ω bk cos β 1 with respect to A k+1. Therefore L fina = m Ω bk cos β 1 + IΩ k+1. Conservation of anguar momentum (L initia = L fina ) yieds (.) Ω fk cos β 1 = Ω bk cos β 1 +Ω k+1. The system of equations (.1) and (.) can be soved easiy for Ω k+1 and Ω bk : (.3) Ω k+1 = f + Ω fk, Ω bk = Ω k+1 f, where f ± cos β 1 ± 1/ cos. β 1 Now consider the kth rod as it fas from the vertica to ange β 1, its position just before the coision. Conservation of tota energy yieds 1 IΩ k + mg = 1 IΩ fk + mg cos β 1,

6 116 C. J. EFTHIMIOU AND M. D. JOHNSON or (.4) Ω fk = Ω k + g (1 cos β 1). Combining equations (.3) and (.4) we find (.5) where Ω k+1 = f + Ω k + b, b = g f + (1 cos β 1 ). Equation (.5) is a mixed progression (i.e., a combination of an arithmetic and a geometric progression) and can be soved by we-known techniques (see Appendix A). The resut is Ω k = f (k 1) + Ω 1 + b 1 f (k 1) + 1 f+. Reca that Ω 1 is the initia anguar veocity of the first rod caused by the initia externa push. We show now that f + < 1. Since β 1,π/, x = cos β 1 1,. Then (x 1/x) > x +1/x > x +1/x >. From the ast inequaity it foows that f + =/(x +1/x ) < 1. Since f + < 1 it foows that im n + f+ n = and therefore im k + Ω k = g (1 cos β 1) 1 f+ Ω. Thus deep into the chain we find transationa invariance: the initia anguar veocity imparted to a rod by its neighbor becomes independent of position. Notice that in this imit the initia push given to the first rod becomes irreevant..4. Wave Speed. We can obtain the imiting speed of the wave by computing the time between coisions, working we into the chain where this becomes independent of position. Appy conservation of energy for the nth rod as it begins moving and after it fas through an arbitrary ange θ. This yieds 1 IΩ n + mg = 1 Iω n + mg cos θ. Setting ω n = dθ/, we can separate t from θ and sove for the time required for the rod to move from θ =toθ = β 1 : Tn = β1 f + dθ Ω n + g g cos θ. This integra can be expressed in terms of the compete eiptic integra of the first kind K(k) (see Appendix B): T n = [ ( )] π β1 K (k n ) F,k n, an + c where a n =Ω n + g, c = g, and k n = c a n+c.

7 DOMINO WAVES 117 Fig..3 The scaing function G(d/). In the imit of arge n, the time T n approaches a imiting vaue ( )] π β1 T = [K (k) F,k, a + c where a =Ω + g, c = g, and k = c a+c. The wave therefore approaches a imiting speed v = d/t given by v = d a + c K(k) F ( π β1,k). A itte agebra ets us write the wave speed in the scaing form of equation (1.1): with and G ( ) d k = v = g G, = d 1 k[k(k) F ( π β1,k)] (1 f +) (1 cos β 1 )f + + (1 f +). Since f + and thus k depend ony on β 1 = sin 1 (d/), this G is indeed a function ony of d/, as required by scaing. The scaing function G is potted in Figure.3. No simpe cosed expression exists for compete eiptic integras, but some insight into our soution comes from ooking at the imit of very cosey spaced rods (d ).

8 118 C. J. EFTHIMIOU AND M. D. JOHNSON In this imit β 1 d/ and f + 1 β 4 1. Using these in the above expression yieds k 4β 1 1. Then ( ) π β1 K(k) F,k From this we find = G π π β 1 ( ) d 1 k sin t 1 d/. π π β 1 = β 1. Thus the wave in very cosey spaced rods moves very fast. In a simiar way one can examine the other extreme geometrica imit, d/ sighty smaer than unity. Put β 1 = π/ ɛ. Then d = sin β 1 = cos ɛ 1 ɛ, whie f + 4ɛ and k 1 16ɛ.Fork very near unity the compete eiptic integra is approximatey K(k) n(4/k ), where k =1 k [7]. Here this gives K(k) n(1/ɛ), which diverges as ɛ approaches zero. However, in this imit F is finite: ( π β1 F ) π 4 + ɛ,k = π 1 k sin t 4 In the imit d using the above approximations yieds g v [ ], n or v g G(d/), where ( ) d G = (1+ )( d) 1 n(1 + ) + n(1 d ), cos t = n( +1). as d/ increases to 1. Thus as d gets very cose to the wave speed drops to zero. Physicay this occurs because one rod gives a very sma push to the next in ine, which as a resut takes a very ong time to fa. The reader might wish to conduct a quick experiment to verify this concusion or a more carefu one to compare our theoretica resuts with experiment. Of course, the reader has noticed the consistency of our resuts with that of McLachan et a. (1.1). 3. Concusions. In this paper we have presented a set of assumptions for the propagation of the domino wave and we have computed the corresponding imiting speed. For simpicity we have presented the soution for a simpified geometry. However, the reader can easiy transfer the soution to the case of dominoes with the shape of paraeepipeds with appropriate adjustments of course. Appendix A. Mixed Progression. Consider a sequence a k, k =1,,..., with the recurrence reation a k = ra k 1 + b.

9 DOMINO WAVES 119 This is known as a mixed progression. We want to express a k in terms of a 1, r, and b. Mutipy both sides by r n k and sum from k =ton (with n ): n r n k a k = k= n ( r n k+1 a k 1 + r n k b ). k= In the first term on the right-hand side repace k by k = k 1 and in the second repace k by k = n k. This yieds n r n k a k = k= n 1 k =1 r n k a k + b The first two sums have neary a terms in common (a but the nth on the eft and the first on the right). Canceing the terms in common and evauating the third sum yieds the desired soution: a n = r n 1 a 1 + b 1 rn 1 1 r Appendix B. The Eiptic Integra of First Kind. The eiptic integra of the first kind [7, 8] is defined by F (φ,k) = φ n k =. r k. dφ 1 k sin φ, k<1. When φ = π/ this is caed the compete eiptic integra of the first kind, denoted by K(k): The integra I(θ ) = K(k) = θ π/ dφ 1 k sin φ. dθ a c cos θ, <c<a, can be expressed in terms of the eiptic integra of the first kind as foows. First make the change of variabe θ = π t: I(θ ) = θ/ π/ θ / Using the identity cos t =1 sin t this becomes: I(θ )= = π/ π/ θ / π/ a + c a + c cos(t). (a + c) c sin t π/ θ /. 1 c a+c sin t

10 1 C. J. EFTHIMIOU AND M. D. JOHNSON Finay we set k c a + c, and we rewrite the above resut in the form I(θ )= = ( π/ a + c a + c [K(k) F π/ θ/ 1 k sin t ( π θ )],k. ) 1 k sin t Acknowedgments. We thank the referees for bringing to our attention the artices of Waker [3] and Shaw [6] and the book of Banks [5]. Note Added in Proof. After this work was competed a paper [9] with somewhat simiar anaysis appeared on the Corne archives. REFERENCES [1] D. E. Daykin, Faing dominoes, Probem 71-19*, SIAM Rev., 13 (1971), pp [] M. S. Kamkin, ed., Probems in Appied Mathematics: Seections from SIAM Review, SIAM, Phiadephia, 199. [3] J. Waker, The Amateur Scientist: Deep think on dominoes faing in a row and eaning out from the edge of a tabe, Scientific American, August 1984, pp [4] B. G. McLachan, G. Beaupre, A. B. Cox, and L. Gore, Soution for Faing dominoes (Probem 71-19*), SIAM Rev., 5(1983), pp [5] R. B. Banks, Towing Icebergs,Faing Dominoes,and Other Adventures in Appied Mathematics, Princeton University Press, Princeton, NJ, [6] D. E. Shaw, Mechanics of a chain of dominoes, Amer. J. Phys., 46 (1978), pp [7] I. S. Gradshteyn and I. M. Ryzhik, Tabes of Integras,Series,and Products, Academic Press, San Diego,. [8] M. L. Boas, Mathematica Methods in the Physica Sciences, 3rd ed., John Wiey & Sons, New York, 5. [9] J. M. J. van Leeuwen, The Domino Effect,

SEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l

SEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l Probem 7. Simpe Penduum SEMINAR. PENDULUMS A simpe penduum means a mass m suspended by a string weightess rigid rod of ength so that it can swing in a pane. The y-axis is directed down, x-axis is directed