Abstract
By using directed dimensional analysis and data fitting, an explicit universal scaling law for the speed of falling dominoes is formulated. The scaling law shows that domino propagational speed is linear proportional to the 3/2 power of domino separation, and -1/2 power of domino height and thickness.
INTRODUCTION
The falling of dominoes is a successive toppling of regularly spaced elements in a periodic array plotted in Figure 1 . The domino effect is not only an interesting game but also an important physical phenomena, and often be used to describes some social catastrophe, such as the cascading consequences of research misconduct [1] . The mechanics of domino falling has been studied extensively by number of leading scholars [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] . In 1983, McLachlan et al. [4] found a scaling law for the speed v in the limiting case of dominoes with zero thickness spaced in a straight line. With these assumptions McLachlan et al. [4] the functional relation:
INTRODUCTION
here g is gravitation acceleration, h the height of the dominoes, λ the spacing between dominoes, and f (x) an undetermined function of x. Efthimiou and Johnson [6] proposed a f (x) by complete elliptic integral of the first kind. Shi et al. [13] developed a precise numerical model with consideration of multipoint impacts between dominoes. Shi et al. [14] studies the toppling dynamics of a mass-varying domino system for which the mass of the domino changes at an exponential rate of its sequence number. Szirtes and Rozsa [6] studied domino by using dimensional analysis [15] [16] [17] [18] [19] for a domino with equal thickness δ, separation λ and height h. Hence the five variables and their diemsion are listed in Table I below: 
INTRODUCTION
The dimensional basis used is length (L) and time (T).
INTRODUCTION
Szirtes and Rozsa [6] applied dimensional analysis to find domino velocity, v = v(h, λ, δ, g). The problem has five variables and two dimensions (L, T), therefore there are 5 − 2 = 3 dimensionless variables Π as follows:
INTRODUCTION
From dimensional analysis,
INTRODUCTION
This relation is similar to Eq.(1) except the separation and height ratio δ/h. Although we have Eq.(1) and Eq. (3), there is no much useful information can be get from them, because the function f ( λ h , δ h ) is still undetermined. In the following, we will try to decode the function by using directed dimensional analysis proposed by Huntley [20] and Siano [21, 22] .
INTRODUCTION
tion. The problem has five variables and three dimensions (L x , L z and T) listed in Table IV below: The dimensional basis used is length (Lx, Lz) and time (T).
INTRODUCTION
Therefore there are 5 − 3 = 2 dimensionless variables Π as follows:
INTRODUCTION
where the exponents a, b, c and a 1 , b 1 , c 1 can be determined by following dimensionless conditions: dim(
INTRODUCTION
From dimensionless condition, 1+b = 0, −1−2c = 0 and a + c = 0, leads to a = 1 2 , b = −1 and c = − 1 2 . Hence, we have the first dimensionless variable
INTRODUCTION
Similarly, we have a 1 = 0, b 1 = −1 and c 1 = 0 and the second dimensionless variable
INTRODUCTION
From Buckingham dimensional theorem [15] , the domino speed v = v(h, λ, δ, g) can be replaced by
INTRODUCTION
This relation has not been reported in literature, however, the function f ( δ λ ) is still undetermined.
DIRECTED DIMENSIONAL ANALYSIS II
In order to reduce the number of variables, let's consider the domino's area instead of its height and thickness separately. The problem has four variables and three dimensions (L x , L z and T) listed in Table IV below: Therefore there are 4 − 3 = 1 dimensionless variable Π as follows: The dimensional basis used is length (Lx, Lz) and time (T).
DIRECTED DIMENSIONAL ANALYSIS II
The dimension dim(Π) = L 1+a+b
DIRECTED DIMENSIONAL ANALYSIS II
Since we have only one dimensionless variable Π, therefore, this Π can only be a constant, namely
DIRECTED DIMENSIONAL ANALYSIS II
where the area of rectangular domino A = hδ is used. The beauty of Eq.(10) is that it is an explicit formula for all relevant variables. The only unknown is the constant C, which can be determined by experiment. This remarkable relation reveals that, for given domino's dimensions h and δ, the domino wave prorogation speed is proportional to the 3/2 power law of domino's separation λ. The gradient of speed respect to the separation λ is dv dλ = 3 2 C λ ghδ . Therefore, the domino's separation λ is a control and/or dominate parameter for the speed of domino motion.
DIRECTED DIMENSIONAL ANALYSIS II
Stronge [9] conducted comprehensive study with highspeed photography on topping of domino array, who obtained three data for domino dimensions: h = 41.78mm, δ = 7.58mm: 
DIRECTED DIMENSIONAL ANALYSIS II
This explicit scaling law for the speed of falling dominoes has never been reported in literature before.
CONCLUSION
In conclusion, an explicit universal scaling law for the speed of falling dominoes has been formulated by using directed dimensional analysis. It is surprised to see that the domino speed is not linearly proportional to √ gh as reported in literature (McLachlan [4] and Szirtes and Rozsa [6] ). The study also shown that the domino wave prorogation speed is proportional to the 3/2 power law of domino's separation λ. which reveals that the domino's separation λ is a control parameter for domino motion.
CONCLUSION
Acknowledgement: The author appreciates the financial supports from Xi'an University of Architecture and Technology and Mr Zhe Liu for the preparation of Figure 1 .

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