dominoes race. - irpheclanet/paperfile/domino2.pdf · dominoes race. christophe clanet, institut...

Clanet Christophe 12/05/03

Dominoes race.

Christophe Clanet*,

*Institut de Recherche sur les Phénomènes Hors Equilibre, UMR 6594, Technopôle de

Château-Gombert, 49 rue Joliot-Curie, 13384 Marseille cedex 13, FRANCE.

Align some dominoes of constant height C on a horizontal plane with a fixed

spacing X<C. The race starts as the first domino is inclined. We study the

phenomenon experimentally and show that after the critical number of dominoes

Nc ª 1- 2.7 / ln(1 - X / C) , the race reaches a constant speed, V = 3gC. f (X / C) ,

where g is the acceleration due to gravity and f (X / C) a decreasing function of

order unity. These results are slightly altered by the inclination of the plane.

Genuine dominoes are black on one side and white with ocellus on the other, by

reference to the clothes of Catholics priests from which they took their name. It seems

that the game with 28 pieces first appeared in France in the eighteen century.

Connecting parts with identical number, the winner is the one who first put all his

pieces on the table, then exclaiming Benedicamus Domino†. Another way of playing

with dominoes consists in aligning them vertically (height C) one behind each other with

a fixed spacing X<C. If the first tumbles down, a chain-reaction is engaged leading to the

fall of all the pieces. This is the so called Domino Effect or Domino Reaction,

expressions often used in politics and economy. Some mathematical and physical

studies have already been dedicated to the subject 1,2,3, all focussing on the constant

speed on a horizontal plane. We analyze here the physics of the propagation on both

horizontal and inclined planes, paying a special attention to the ignition length, needed

for the constant velocity regime to be established.

† Thank God.

Clanet Christophe. 12/05/03

An example of race is presented in Figure 1, with C = 40 mm and X = 20 mm . The

phenomenon is observed with a high speed video camera Kodak HS4540. Time increases

from (a) to (j) with a constant time step Dt = 35.6 ms , starting at t = 0 in Figure 1-(a),

where domino 1 is released with no initial velocity. Only one domino is shocked during

the first half of the sequence and 6 in the second half. The number of dominoes tilted in

a unit of time is clearly not constant in the initial phase, but tends to a constant as can

be observed in (i) and (j). In both pictures, one clearly identifies two passive domains,

the unstable vertical one extending on the right and the stable horizontal region lying on

the left. In between, stands the active region made of falling dominoes.

Figure 1: beginning of the race observed with C = 40 mm , X = 20 mm and a

time step between images Dt = 35.6 ms.

To quantify the process, we assign a number N to each domino and denote tN the time

at which N is shocked. The falling rate ˙ N , is deduced as ˙ N =1/ tN+1 - tN( ) and the

propagation velocity V = X ˙ N . The time series tN corresponding to the experiment of


Figure 1 is presented in Figure 2: For N ≥ Nc , where Nc ª 7 , tN increases linearly with

N , and the race exhibits a constant velocity of propagation V @ 1.0 m.s-1 ,

corresponding to ˙ N @ 51 dominoes per second.

Figure 2: Evolution of tN for the race presented in Figure 1: n experimental

measurements, ___ asymptotic linear evolution.

Figure 2 stresses the two questions involved in the propagation on a horizontal plane:

How do the critical number Nc and the characteristic velocity V change with the

parameters C and X. In the whole study, domino 1 is released through the same

procedure and gravity is kept constant, g = 9.81 m.s-2 .

The discussion on Nc starts with the study of the potential energy of one domino U , as

it rotates around the bottom right corner O (Figure 1-(a)), with the angle a . The relation

U(a) writes:

0

0,1

0,2

0,3

0,4

0,5

0,6

5 1 0 1 5 2 0 2 5

tN (s)

NNc


U1/ 2MgC

= cosa +AC

sina = ˜ U , (1)

where M is the mass of the domino and A its thickness. The evolution described by

Equation (1) is presented in Figure 1-(b) with A / C = 0.2 . As a changes from 0 to p / 2 ,

˜ U varies from 1to A / C . Propagation only takes place if A / C <1 . Our study is

conducted in the limit A / C <<1 (in Figure 1, A = 3 mm so that A / C = 0.075 . All the

results presented satisfy the condition A / C < 0.1).

Figure 3: Stability of one domino: (a) rotation of one piece around the bottom

right corner O (b) evolution of the potential energy with the inclination angle.

Initially, a = 0 , the domino stands in a metastable state characterized by the difference

of potential D ˜ U @ 1/ 2 A/ C( )2 ‡. Beyond this barrier, the domino falls towards the stable

state a = p / 2 , separated from the initial state by D ˜ U = 1- A / C ª 1.

Let us now consider a line of equally spaced dominoes. For the first domino (Figure 4-

(a)), the inclination angle between two shocks changes from 0 to a1 = arcsin(X / C) .

‡ The difficulty in aligning dominoes thus increases as A / C( )2

.


The difference of potential energy between these two positions simply writes

D ˜ U 1 = 1- cosa1 . With two pieces (Figure 4-(b)), the leading domino inclination still

changes from 0 to a1 whereas the second varies from a1 to a2 . The difference of

potential between the initial state (thin line) and the final state (thick line) now writes

D ˜ U = 1- cosa 2 . This difference is 1 - cosa3 for three dominoes (Figure 4-(c)), and

1 - cosaN for N pieces. Qualitatively, if aN is close to p / 2 , the difference of potential

is close to 1 and does not change from shock to shock. We thus reach a state invariant

by translation which defines the constant velocity regime.

Figure 4: Evolution of the potential energy during the fall of one (a), two (b) and

three (c) dominoes.

Quantitatively, cosa N = yN / C , where yN is the ordinate of the intersection point

between domino N and domino N -1 . In the rectangular triangle (x2 , y2 , p2 ) presented

in Figure 4-(c), we observe that (y2 - y3 )/ y2 = Ds3 / C , where Dsi is the distance

between the two consecutive intersections xi -1, yi -1( ) and xi , yi( ) . As we move away

from the leading domino 1, Dsi becomes closer and closer to X . In the limit

Dsi = X, "i , we deduce yN ª 1 - X / C( )N -1 y1 , so that cosa N ª 1 - X /C( )N -1 y1 / C . The

condition a ª p / 2 , leads to:

Nc = 1 +K

ln 1- X / C( ) , (2)


where K is a constant. Equation (2) is compared with experimental measurements in

Figure 5, with K = 2.7 . As it was expected, Nc is a decreasing function of X / C ,

reaching 17 dominoes for X / C = 0.15 and 2 dominoes for X / C = 0.9 . The evolution

predicted by Equation (2) is compatible with our experimental observations.

Figure 5: Evolution of the critical number of dominoes Nc , with X / C: n

experimental measurements, ___ Equation (2).

We now move to the study of the constant velocity of propagation V , observed after

Nc dominoes. In this region, we just saw that the difference of potential of all the

dominoes as domino 1 rotates from a1 = 0 to a1 = arcsin X / C( ) simply writes

DU = 1/ 2MgC . Without friction, this energy is transferred to the kinetic energy of the

entrained domino 1, so that:

16

MC2 ˙ a 12 =

12

MgC , (3)

0

5

1 0

1 5

2 0

0 0,2 0,4 0,6 0,8 1

Nc

X / C


where ˙ a 1 is the angular velocity of domino 1 when it reaches a1 = arcsin X / C( ) .

Equation (3) induces that ˙ a 1 = 3g / C , which means that the angular velocity of the

leading domino as it reaches the next domino is independent of the ratio X / C .

Figure 6: Evolution of the angular velocity ˙ a 1 as a function of X / C:

oexperimental values with C = 160 mm , l experimental values with

C = 40 mm , the continuous lines represent 3g / C .

For two sets of dominos characterized by C = 40 mm and C = 160 mm , we

present in Figure 6, the measured values of ˙ a 1 as a function of X / C . These measures

remain almost constant around the theoretical values presented in continuous line.

0

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

0 0,2 0,4 0,6 0,8 1

X / C

˙ a 1 (rad.s-1)


Figure 7: Evolution of the constant propagation velocity V / 3gC , as a function

of X / C: n experimental measures, ___ Equation (5).

Assuming that during the fall, domino 1 is mainly accelerated by gravity, the evaluation

of the time between two shocks t , must be done through the integration of the

differential equation:

d2a1

dt2 =32

gC

sina1 , (4)

with the conditions a1 (0) = 0 , a1 (t ) = arcsin(X / C) and ˙ a 1 (t ) = 3g / C . In the range

X / C Œ 0, 0.9[ ] , the solution of Equation (4) is well approximated by

t @ C / 6g ln 1 +a M / 2( ) / 1 - aM / 2( )[ ], where aM = arcsin X / C( ) . This expression

of the time between two shocks enables the calculation of the propagation velocity as

V = X / t , which can be written :

0,2

0,4

0,6

0,8

1

1,2

0 0,2 0,4 0,6 0,8 1

V3gC

X /C


V3gC

=2 X / C

ln 1+ a M / 21- a M / 2

Ê

Ë Á ˆ

¯ ˜

(5)

This theoretical expression for the propagation velocity is compared to our experimental

measurements in Figure 7. The values obtained with Equation (5) are in close agreement

with the experimental values over the whole range of X / C .

The last part of the article is dedicated to dominoes races on inclined planes. The

discussion on the critical number of dominoes Nc is not reproduced and we focus on

the constant propagation velocity. An example of domino climbing is presented in

Figure 8-Up with C = 40 mm , X = 20 mm and an inclination q = 63.2° . Time increases

from (a) to (e) with a constant time step Dt =124.45 ms . The 17 initially vertical

dominoes, fall at a constant rate in 4Dt , leading to the falling rate ˙ N @ 34 dominoes per

second and rising velocity V = X ˙ N / sinq @ 0.76 m.s-1 . With similar conditions but

q = p / 2 we observed in Figure 1, ˙ N @ 51 dominoes per second and V @ 1.0 m.s-1 .

Falling rate and propagation velocity thus decrease as q is decreased. Perhaps a less

intuitive behavior is observed in the descending case, an example of which is presented

in Figure 8-Down, where C = 40 mm , X = 20 mm and q =116.8° . Time increases from

(a) to (e) with a constant time step Dt = 80 ms . The 14 dominoes fall at the constant

rate ˙ N @ 44 pieces per second, that is with a descending velocity

V = X ˙ N / sinq @ 0.98 m.s-1 . The descending falling rate is thus also smaller than the one

observed on the horizontal plane.


Figure 8: Dominoes race on inclined planes. Up: C = 40 mm , X = 20 mm and

q = 63.2° . Constant time step between images Dt =124.45 ms . Down:

C = 40 mm , X = 20 mm and q =116.8° . Constant time step between images

Dt = 80 ms.


The falling rate observed horizontally appears to be close to a maximum…

To analyze these results we just need to generalize the expression of the potential

energy in Equation (3) and pay a special attention to the maximal angle aM , in the

integration of Equation (4).

Going through the same reasoning as for the horizontal case, we get that the available

potential energy is DU = 1/ 2MgC 1- cosq( ) . Using Equation (3), the angle velocity ˙ a 1

is changed to ˙ ˜ a 1 = 1 - cosq , where ˙ ˜ a 1 = ˙ a 1 / 3g / C . This value is used in the

integration of Equation (4), leading to the new expression of the velocity of propagation:

V3gC

=1

sinq2X / C

ln1 + aM / 2 ˙ ˜ a 1( )1 - aM / 2 ˙ ˜ a 1( )

Ê

Ë

Á Á

ˆ

¯ ˜

, (6)

where aM = arcsin(X / C) in the range q Œ qMin ,q *] [ and

aM = p / 2 - arctan C / X +1/ tanq( ) , when q Œ q* ,qmax[ [ . Equation (6) obviously

reduces to Equation (5) for q = p / 2 .

Figure 9: limit angles: (a) minimum angle qMin, (b) separation angle q * , (c)

maximum angle qMax .


The limiting angles qMin , q * and qMax are respectively presented in Figure 9-(a), (b) and

(c): The angle qMin represents the stronger slope dominoes can climb for a fixed ratio

X / C . For q smaller than qMin , one domino falls but does not reach the next piece. At

the angle qMax , one domino falls and impacts perpendicularly to the next piece. This

configuration is statically stable and can stop the propagation. For angles q larger than

qMax the hit domino can even move backwards. The climbing region is thus defined as

q Œ qMin ,p / 2] [ and the descending region as q Œ p / 2,qMax] [ . The intermediate angle q *

is the limit where the falling domino hits the next piece at the top. This angle is related

to C and X through the relation 1/ tanq * = (C/ X )2 -1 - C/ X .

The evolution of the propagation velocity with the inclination is presented in Figure 9-

(a), for X / C =1/ 2 . Equation (6) is the continuous line and the black square points the

measured values obtained with C = 40 mm . The corresponding falling rate ˙ N is

presented in Figure 9-(b).

Figure 9: Evolution of the velocity (a) and falling rate (b) with the inclination

angle q , for X / C =1/ 2 : n experimental measures, ____ Equation (6).


The theory is satisfactory in the climbing region but overestimates experimental results

in the descending region. This can be related to impact losses, which become more and

more important as the falling domino 1 hits the next domino with an angle close to p / 2 .

In this case, the potential energy is not entirely converted in kinetic energy and

Equation (3) should be modified, leading to a smaller value of ˙ a 1 , and thus of the

velocity.

1. B.G.McLachlan, G.Beaupre, A.B. Cox and L.Gore. Falling Dominoes. SIAM Review.

25 (2), 403-404 (1983).

2. Charles W Bert. Falling Dominoes. SIAM Review. 28 (2), 219-224 (1986).

3. W.J.Stronge and D.Shu. The domino effect: successive destabilization by cooperative

neighbours. Proc. R. Soc. Lond. A.. 418, 155-163 (1988).

Small place but sincere thanks to David Quéré who mentioned this subject to me. I also do not forgetthat everything would have been much more difficult without the skillful technical assistance of FranckDutertre and Jacky Minelli.

Correspondence and requests for materials should be addressed to C.C. (e-mail:

[email protected]).

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