Listen! A Smartphone Inquiry on the Domino Effect

In this work, we present the phenomenon of a chain of falling dominoes (“domino effect”), investigated by acoustical means with a smartphone. In general, the domino effect consists of systematically placing a series of dominoes in a row (see Fig. 1), sometimes forming artificial figures and mechanisms (“Rube Goldberg”-like devices),^1–3 and by giving the initial domino a slight push, causing a chain effect with a propagating wave of toppling dominoes. The domino effect has been met with some interest as an example of “everyday physics” (Ref. 4, example 1.68) and as a model system for collective dynamics.^5–8 

As an alternative to the use of more complex apparatus (discussed below), acoustical measurements are taken with a smartphone to find the time of collisions between the leading domino piece and the next one, and to deduce the propagation speed from it. Results are compared with models published in the literature, and a simplified one derived below (“Approximate model of the toppling process” section).

We first give a brief account of previous treatments and then present an approximate model allowing for a simplified estimation of the relevant parameters.

There are two main difficulties of an accurate determination of t_t: First, anharmonicity: the toppling of a domino (sometimes referred to as a “rod”) usually takes place over a large angle range of 90° (from upright to horizontal) and is described by an anharmonic pendulum, requiring special functions (elliptic integrals) for the solution (Ref. 9, Chap. 21). Second, collective dynamics: the “toppling” is not that of an isolated domino, but that of the group of foremost dominoes, pushing the leading one.⁸ An additional complication occurs if one does not assume “infinitely” thin dominos (i.e., disregarding all powers of d/h), leading, e.g., to changes for the moment of inertia, a “potential barrier” in the toppling motion, etc.⁸ 

Depending on various assumptions about geometry, collision dynamics, and additional effects like friction and sliding, a number of theoretical and numerical studies^5–13 have been published, almost exclusively at very high levels of sophistication. Banks⁹ has provided a model fully taking into account anharmonicity (but not collective effects). We note incidentally that Banks erroneously uses v_as = (l – d)/t_t for the propagation speed (Ref. 9, p. 267), involving the interspacing s = l – d instead of the spacing l. Indeed, the leading domino only covers the distance s in the time t_t, but the next domino it hits then immediately starts to topple around the pivot point at its leading edge [right edge in Fig. 2(a)], i.e., with an additional displacement of the size of its thickness d. The small delay for the elastic wave propagation through the dominoes is disregarded here [it is of the order of d/v_{el. wave} ∼ (some millimeters/some hundred meters per second) ∼ 10 µs, while the t_t ∼ 10 ms; see “Results and comparison to theory” section]. This leads to correction factors 1 + d/s for v_as and better consistency with experiment of the approach of Banks⁹ than thought previously.⁸ 

van Leeuwen⁸ has published a very complete theory taking into account collective effects and friction, and making no assumptions about the relative dimensions of the system (in particular, covering finite thickness of the dominoes and also anharmonicity effects; however, a constant spacing and parallel alignment of the dominoes is assumed).

In that theory, an important assumption made is that the leading domino, after hitting the following one, keeps pushing on it and does not bounce back. This is consistent with observation, and it means that the collisions are inelastic. Moreover, this also takes into account the fact of the domino wave approaching a finite asymptotic velocity, while elastic collisions would lead to an ever-increasing velocity as more and more energy is in the system by more and more dominoes having fallen and released their potential energy. Moreover, a couple of further simplifying assumptions are made on the shape, arrangement, and motion of the dominoes, which

• all have the same shape, dimensions, and distance;

• are in a straight line and the end position is exactly stacked (no sidewise deviation);

• do not slip on the surface, nor “take off” during the collision.

The meanings of the functions P, Q, and K introduced by van Leeuwen⁸ are as follows: P is a dimensionless measure of the energy released by a domino falling from its initial to final position (mghP/2; m: mass of a domino, h: height difference of the center of mass; the factor 2 is for convenience). The function Q is derived from a detailed model of the inelastic collisions between dominos; its explicit form is one main result of the work by van Leeuwen, and its derivation can be found in his paper.⁸ The factor K is a correction factor close to 1 for usual working conditions of the domino effect (up to a 10% increase for separations as large as s/h ≥ 0.9, which will be not considered here). Note that the unit for v_as in Eq. (2) is given by the factor (gh)^1/2, while Q and K are unitless. Below, we will compare results of our smartphone experiment and of a simplified theory (described in the next section) to van Leeuwen’s predictions.

In the following, we will present an approximate treatment valid for a considerable range of realistic parameters of the domino effect. We start from the same simplifying assumptions as Banks⁹: (i) the propagation of the domino wave is considered as a sequence of single-domino events, where the domino before the leading one topples until it hits the leading one, which then topples independently of the other dominoes; and (ii) linear momentum during the collisions is conserved. Based on this, and following from momentum conservation, the initial angular velocity is given by θ̇₀ = cos(θ_c)θ̇_c, where the subscripts (0 and c) refer to the moments of the beginning of the toppling and of the collision with the next domino, respectively [Ref. 9, Eq. (21.24)].

(g = gravitational acceleration, and M, R_S, and I are the mass, center-of-mass position, and moment of inertia of the rod, respectively). For a thin rod of height h, one has R_S = h/2, I = (1/3)Mh², and ω = [(2/3)g/h]^1/2; we will use the latter expression for the following. Note that the right side of the second equation in Eq. (4) is positive, in contrast to the equation for an ordinary pendulum: for toppling, the initial position is the (unstable) upright position, where gravity leads to an ever-increasing inclination, whereas for the pendulum, gravity acts as a restoring force to the equilibrium point with zero inclination.

The anharmonicity in the toppling of the domino as mentioned above occurs due to the wide range of θ from 0 (upright) to π/2 (horizontal) covered in a complete toppling process, necessarily involving higher orders in the sinusoidal term in the accelerating force; this, in turn, involves elliptic integrals for the solution function.¹⁴ The dynamics of the toppling process has several interesting applications and extensions that have led to a whole series of publications, like the faster-than-gravity acceleration of the tip of the toppling object,^14–16 the breaking of falling chimneys at a specific position,^17,18 or the inverted pendulum,^19,20 in turn the basis of many further applications like the stability of the upright position of humans²¹ or of the unicycle (Ref. 4, example 1.116).

In the above approximate treatment with its very simple result, the main specific effects known to occur in the domino effect, viz. anharmonicity and collective behavior, are not present, and thus the assumptions made appear rather strong. How accurate is this approximation, then, as compared to experimental data, and to more advanced theories?

A chain of dominoes is prepared on the floor (Fig. 1), with dimensions as given above [Fig. 2(c)]. Measurements are carried out for two values of spacing (l = 2 cm and 3 cm); the number of dominoes is typically 20. A smartphone is placed close to the domino arrangement with the microphone pointed in the direction of the falling dominoes. An audio signal of the sounds emitted at the subsequent collisions is recorded in the smartphone (see Fig. 3). Data are then read out for further analysis with the application Audio Recorder,²³ transferred to a computer as a CSV file, and analyzed in Excel to extract the time of each collision.

From the recorded data, two variables are obtained:

• time of collision between domino n and n + 1: t_n

• total propagation time (since the start): T_n = Σt_n

For this case, we can compare our results to an older model by Shaw.⁶ Inferring t_t from the slope of his curve, one obtains t_t = 0.033 s, consistent with Eq. (12). Going further, we also compare the data to the prediction of the simplified model in the “Approximate model of the toppling process” section. Equation (8) yields t_t ≈ 0.031 s, still in good agreement with the measurement, even though the small-angle assumption is not justified now [s/h = 0.58, θ_c = arcsin(s/h) = 0.62, θ_c² ≈ 0.39, θ_c³ ≈ 0.24]. The propagation velocity is then v_as = 91 cm/s from experiment (97 cm/s from theory).

The “frictionless” theory of van Leeuwen [Eqs. (2) and (3)] yields t_t,vLe ≈ 0.029 and 103 cm/s, with a less good, but still satisfactory, agreement with the experimental result [and with the approximate result from Eq. (12) within the error of the latter]. The deviation can be understood as follows: the larger the spacing, the flatter is the final position of the domino stack [see Fig. 2(b)], and the longer is the part of the toppling process when dominoes slide on each other; it is thus plausible that van Leeuwen’s model version neglecting friction allows for less good agreement and has to be replaced by a version including friction (which is not considered here).²⁴ 

The experimental and theoretical values for the domino experiment presented here are summarized in Table I, and we now turn to a brief discussion of its educational potential for physics education.

The results in this experiment, especially the linear increase of T_n with n (Fig. 4) and the numerical values for the propagation speed of the domino wave, show good agreement with other measurements and theory. The example can be seen as a demonstration of the use of mobile devices as experimental tools (MDETs²⁵) for a student project about a phenomenon of “everyday physics,” and at the same time of interest for current research: the domino effect as a paradigm of collective dynamics,^5–8 with application, e.g., in carbon nanotubes²⁶ or neuronal signal transduction (Ref. 27, 48.3).

On the experimental level, the idea of using the audio signal with a smartphone for the measurement has to be highlighted, an original idea of the student coauthors in the course Physics of Everyday Phenomena²² (it later turned out that audio recording was also used in a paper on arXiv, though with a different method using more advanced equipment²⁸). Indeed, smartphones come with a built-in sensor (the microphone) and recording functionality with a sufficiently good sampling quality in order to have a sufficient time resolution analysis, and the data can be easily analyzed in the smartphone itself or transferred to any computer (e.g., as a.csv file). In that respect, one can easily see the value of MDETs compared with other methods used for the investigation of the domino effect, viz., photocell timing gates⁶ or high speed videography,^5,29 which are (much) less available and (much) more expensive. On the other hand, these methods allow for higher accuracy in the measurements, not attainable with the method presented here. But in view of the satisfactory agreement found with MDETs, one may conclude that they offer a reasonable trade-off of data quality and expenditure in the context of a student research project, especially when realized autonomously and outside a lab (note that this project was carried out during the COVID university closure in 2020/21).

With these experiences, we see an interesting perspective for the use of MDETs in settings such as student projects, undergraduate research, etc. The case for such student research projects has been made repeatedly, emphasizing in particular its potential to foster competences one would like to develop along with content mastery, such as autonomy, curiosity, creativity, critical thinking, etc. [“higher-order thinking skills” (HOTs)^30–32]. The present contribution is about a shorter student research project integrated into a lecture in order to enrich the traditional format, but this entails quite limited resources and time (∼25 h). The idea is to show that even in this setting, an experimental test of a theoretical treatment of a nontrivial phenomenon is possible, with students collecting their own data with sufficient accuracy. We think that such shorter projects (embedded in a lecture) offer students the opportunity to develop their research skills and HOTs incrementally, laying a foundation for more extensive and complex projects in the future. Of course, in a more comprehensive treatment, extensions, e.g., for broader ranges of domino arrangements and improved accuracy, should be included, and we admit this as a limitation of the given setting. We mention a series of such extensions further student research projects could explore:

• testing the limits of the theoretical model, e.g., regarding the case of large domino spacings;

• improving the accuracy for the propagation velocity by using more dominoes;

• testing the “no bouncing back” condition (i.e., inelasticity of the collision), for which an acoustical method appears particularly suitable;

• investigating the influence of different surface materials;

• investigating inclined support surfaces or curved domino alignments.

On the theoretical level, a simplified model was presented here, accessible at the undergraduate level, and highlighting the central physical ideas (toppling time as determining the periodicity of the process). It yields satisfactory agreement with the measurements, and can serve as a basis for more complete models beyond the assumptions made above (in the “Theory” section), thus completing the MDET experiment in a useful way for the purposes of physics education.

As a last example of the intrinsic curiosity and interest of the domino effect, we mention the case of unequal, increasing sizes of the dominoes, suggested by Whitehead³³ as a nice demonstration of exponential (or geometric) growth (see Refs. 34 and 35).