Falling Dominos
Martha Buckley
MIT Department of Mathematics,
Cambridge, MA 02139
[email protected]
(Dated: May 16, 2003)
Abstract
The time taken for long lines of equally spaced dominos to fall was measured for several different
chain lengths (30, 50, and 89 dominos). The time for the chain to fall was found to vary linearly with
the number of dominos, which suggests that the wave of falling dominos approaches a constant
velocity. A theoretical model of falling dominos was developed in order to predict the critical
number of dominos needed for the wave to reach a constant velocity and the value of this constant
velocity based on the parameters of the system (the domino height and the spacing between the
dominos). The critical number of dominos was found to be nc = 1 −
K
ln(1−D/H) ,
where H is the
domino height, D is the domino spacing, and K = 2.7, an empirically determined constant. The
constant velocity was found to be v =
√
6gHD/H
√
1+αm / (2)
√
ln(
)
1−αm 2
, where g is the acceleration due to gravity
and αm = sin−1 (D/H). These theoretical predictions were compared to experimental results for a
variety of different ratios of D/H. The theoretical predictions were in relatively good agreement
with the experimental results for moderate ratios of D/H, but there was large discrepancies for
very large and very small domino spacings.
1
INTRODUCTION
Two basic questions come to mind when considering the toppling of dominos. The first is
“mathematically how can we think of falling dominos?” The second is “why would one want
to model falling dominos in the first place?” Interestingly, the answer to the first question
provides insight into the second.
Mathematically a line of dominos can be thought of as a regularly spaced array of
marginally stable elements. The dominos are marginally stable because they are in an
equilibrium position, but if a domino is displaced sufficiently from the equilibrium position,
it will fall. If a destabilizing force is sufficient to topple one domino, the destabilization wave
can propagate through the entire array of dominos.
Now that we know how to understand falling dominos mathematically, it becomes apparent that the general model of a line of falling dominos can be used to understand a variety of
different (and perhaps more useful) phenomenon. The domino effect can be used to describe
the propagation of neuron firing in a synapse-coupled neural network. The model of falling
dominos is essentially the discrete analogue of propagation phenomenon in exothermic (heat
releasing) chemical reactions modeled by the reaction-diffusion equations. On a macroscopic
scale, the domino effect can be used to describe the blow down of trees in ice-laden forests
(Stronge, p.155) In short, the model developed to describe falling dominos has the potential
to provide insight into a large number of similar physical phenomenon.
PRELIMINARY EXPERIMENT
In order to investigate the dynamics of falling dominos, I set up a basic preliminary
experiment (see figure 1). H is the length of the dominos, W is the thickness of the dominos,
and D is the spacing between the centers of the dominos (the numerical values for the
dominos that I used are given in figure 1). Three lines of dominos of different lengths (30,
50, and 89 dominos) with equal spacing (D/H = 1/2) were set up. The dominos were
toppled by leaning a domino (not counted as one of the line) against the first domino in the
line with an angle α = sin−1 (D/H). A metal box was placed a distance D from the final
domino so that when the final domino had fallen an angle α the noise would signify that all
dominos had fallen. The time, T , taken for each line of dominos to fall is illustrated in figure
2
2. The time was found to linearly increase with the number of dominos, and the T-intercept
occurred at T (0) = 0.194 ± .028. These two facts suggest that the wave velocity starts at a
somewhat slower speed and increases to approach a constant velocity.
THEORETICAL PREDICTIONS FOR THE CRITICAL NUMBER TO REACH A
CONSTANT VELOCITY
Our preliminary experiment led us the the hypothesis that the velocity of the wave of
falling dominos approaches a constant. Now our goal is to create a mathematical model for
falling dominos and attempt to determine the constant propagation velocity and the critical
number of dominos needed to reach this velocity in terms of the parameters of the system.
For clarity, we will label each domino with a number 1 through N and the time period over
which the jth domino falls will be labeled τj .
We model a falling domino as a rigid body which rotates about its right edge (see figure
3). The surface on which the the dominos are lined up is assumed to be flat and sufficiently
rough that the dominos only pivot (do not slide) against this surface. The potential energy
of the domino is given by
P E = mg/2(cos α + W/H sin α)
(1)
where H is the height of the domino, W is the thickness of the domino, m is the mass of
the domino, α is the angle which the domino have rotated from the vertical, and g is the
acceleration due to gravity. We define a dimensionless variable
U=
PE
= (cos α + W/H sin α)
1/2mgH
(2)
which is the ratio of the potential energy of the domino after rotating an angle α to the
potential energy of the domino when it is in the vertical position. Because dominos are much
taller than they are thick, W/H << 1 and we can neglect the term W/H sin α. Therefore,
U ≈ cos α
(3)
For the rest of the analysis, we will be using this approximation and therefore will essentially
be treating the dominos as lines, rather than rectangles. This has profound implications because it reduces the number of parameters which describe the dynamics of the falling dominos
3
from 4 (H, D, W , and g) to 3 (H, D, and g). Using dimensional analysis (Buckingham’s Pi
Theorem), we know that this will reduced the number of dimensionless groups that we can
form from 3 to 2.
Now we must consider a line of equally spaced dominos. The first domino falls, and
when it hits the second domino, it makes an angle α1 = sin−1 (D/H) with the vertical (see
figure 4). The height of the position of impact above the plane on which the dominos rest
is y1 = H cos α1 . The change in potential energy of the system after the first domino falls is
∆U1 = Ui − Uf = 1 − cos α1 .
Now the second domino falls and hits the third domino (see figure 5). Just as the first
domino, the second domino now makes an angle α1 with the horizontal and the height above
the plane is y1 . However, the first domino continues to fall as the second domino is falling.
We assume that contact is maintained between the dominos after impact and the dominos
slide against each other without friction. The angle that the first domino makes with the
vertical at the moment that the second domino impacts the third is α2 and the height above
the plane is y2 = H cos α2 . We now seek to determine the change in potential energy of the
system over the period τ2 . The important aspect to notice is that while the second domino
rotates by an angle α1 , the first domino rotates by an angle α2 − α1 . Since the dominos
are identical, this is equivalent to a single domino rotating by an angle α2 . Therefore, the
change in potential energy of the system during the period τ2 is ∆2 U = Ui − Uf = 1 − cos α2 .
In general, the angle that the first domino makes with the vertical after n dominos have
toppled is αn and the height above the plane is
yn = H cos αn
(4)
. The change in potential energy of the system in the interval τn is
∆Un = Ui − Uf = 1 − cos αn
(5)
Now that we have described a line of falling dominos mathematically, our goal is to find
the critical number, nc , of dominos after which the wave of falling dominos reaches a constant
velocity. Qualitatively, we would expect this to occur when αn is close to π/2. In this case
∆Un ≈ 1 and does not change from shock to shock. Quantitatively, we must relate αn to the
ratio D/H in order to find nc . By similar triangles we find the relationship
∆sn
H
=
yn−1 −yn
,
yn−1
where ∆sn is the diagonal distance between consecutive intersections (between dominos n
4
and n − 1, see figure 6). As one moves backward from the leading domino (the leading
domino refers to the domino on the leading edge of the wave–ie all the dominos behind the
leading domino have been knocked down and all those ahead are still standing), the dominos
are closer and closer to horizontal. Therefore, ∆sn → D. We can substitute this into the
expression that we derived using similar triangles to conclude that
yn−1 −yn
yn−1
= D/H. This
equation can be manipulated to yield yn /yn−1 = 1−D/H, which states that the ratio between
the heights of consecutive intersections of dominos is constant. This recursion relation
between yn and yn−1 can be converted to an explicit expression for yn : yn = (1−D/H)n+1 y1 .
Because cos αn = yn /H, we have that
cos αn =
(1 − D/H)n+1 y1
H
(6)
In order to find the critical number of dominos to reach a constant velocity, we must solve
for the value of n (which we will denote nc ) where the αnc ≈ π/2 and cos αnc = ² << 1.
Solving for nc yields
nc = 1 −
K
ln(1 − D/H)
(7)
K is a constant that depends on H and ², but rather than attempt to compute K we
leave K to be determined experimentally. In reality, the value of K would depend on the
thickness W (or a dimensionless group W/H) in addition to the ration D/H. However,
our current model assumes that the thickness of the dominos is essentially zero (they are
lines rather than rectangles). Therefore, it is most reasonable to allow K to be determined
experimentally.
EXPERIMENTAL DETERMINATION OF nc
Our next goal was to determine experimentally the critical number of dominos needed for
the wave to reach a constant velocity. In order to determine nc it was essential to measure
the time period τi between impacts very accurately. Because this time period is on the order
of 10−2 to 10−3 seconds, a high speed camera was necessary. A digital high speed camera
was used at a frame rate of 500 frames per second with a 1/3600 second exposure time.
Because the integration time was so short (in order to minimize blurriness), it was necessary
to illuminate the (white) dominos strongly against a black background using a high-watt
halogen lamp. The first 8-10 dominos were placed in the field of view of the camera, and the
5
first domino was toppled by leaning a domino (not one of the dominos whose falling time τ i
is to be measured) at an angle α1 = sin−1 (D/H) against the first domino. The time at which
each domino impacted the next domino was determined by stepping sequentially through
each frame and determining the first frame in which the next domino was displaced. These
times were subtracted in order to find the time period τi between the impacts of dominos
i-1 and i and dominos i and i+1. Since the dominos are separated by a distance D, the
velocity of the ith domino is vi = D/τi . The evolution of vi for D/H = .5 is illustrated in
figure 7. As expected from theoretical predictions, experimentally vi was found to approach
a constant within a relatively short number of dominos. The experimentally determined
value of nc for a range of different ratios of D/H is plotted and compared to the theoretical
predictions in figure 8. The experiment matched theoretical predictions relatively closely,
although due to the finite resolution of the camera and difficulties in determining the exact
frame where impact has occurred, there is likely a fair amount of error in the determination
of nc . Unfortunately, this error is difficult to quantify so I chose not to include error bars.
One thing to notice is that the theoretical model seems to predict that nc → ∞ as
D/H → 0. However, it is important to keep in mind that D is the distance between the
centers of the dominos. Since our theoretical model currently neglects the finite thickness
of the dominos, it is possible for D/H → 0, but in reality the minimal value of D/H is
W/H, where W is the thickness of the dominos. The vertical red line in figure 7 indicates
the smallest possible value of D/H.
THEORETICAL PREDICTIONS FOR THE CONSTANT PROPAGATION VELOCITY
Now we attempt to make a theoretical prediction of the constant propagation velocity
that will be reached after the nc dominos have fallen. Our basic method will be to use an
energy balance argument in order to determine the instantaneous angular acceleration ω ∗ of
the leading domino (after nc dominos have fallen) when it impacts the next domino. Then,
we will use Newton’s law to find a second order differential equation describing the dynamics
of the system, and the geometry of the system and the value of ω∗ will provide the initial
conditions necessary to uniquely solve this differential equation.
As derived earlier (see figure 2 and equation 1) the potential energy of a single domino
6
that makes an angle α with the vertical is P E = 1/2mgH(cos α + W/H cos α), and for
W/H << 1, P E ≈ 1/2mgH cos α. The constant propagation velocity is observed after n c
dominos and αnc ≈ π/2. Therefore,
(∆P E)nc = 1/2mgH
(8)
Assuming that the collision is perfectly elastic, all this potential energy will be transferred
to the kinetic energy of the leading domino. The kinetic energy of a body rotating about a
fixed point is
KE = 1/2Iω∗2
(9)
where I is the moment of inertia of the domino about the axis of rotation and ω∗ =
dα
dt
is the
instantaneous angular velocity of the leading domino about the fixed point at the instant
that it impacts the next domino. The moment of inertia of a domino rotating about its edge
is I = 1/3m(H 2 + W 2 ) ≈ 1/3mH 2 for W << H. Equating the expressions for the potential
and kinetic energy and solving for ω∗ yields the expression
ω∗ =
s
3g
H
(10)
This result is interesting because it hypothesizes that the angular velocity of the leading
domino at the instant of impact is independent of the spacing between the dominos. However, an experimental determination of the instantaneous angular velocity upon impact is
quite difficult and verification of this result with experiment was not attempted at this time.
However, the angular velocity at impact found using energy conservation can be used in
order to solve for τ , the time between successive collisions in the constant velocity regime.
We assume that during its fall, the leading domino is accelerated primarily by gravity. The
torque Γ (the letter τ is generally used to indicate torque, but since we have defined τ to
be the time between collisions, we will use Γ for torque) created by gravity about the axis
of rotation is
Γ = mg sin α
(11)
where α is the angle that the leading domino makes with the vertical. By Newton’s law
Γ = Iω 0
where I is the moment of inertia about the axis of rotation and ω =
(12)
dα
dt
is the instantaneous
angular velocity of the leading domino. Substituting (11) into Newton’s law yields the
7
differential equation
3g
d2 α
=
sin α
2
dt
2H
(13)
This differential equation must be solved subject to the conditions that
−1
0
α(0) = 0, α(τ ) = sin (D/H), and α (τ ) =
q
3g/H
(14)
The first two conditions state that the domino starts off in a vertical position (at t = 0) and
at time τ it impacts the next domino, spaced at a distance of D. The third condition states
that the instantaneous angular velocity at the time of impact is ω(τ ) = ω∗ =
exactly the expression that we derived above.
q
3g
,
H
which is
For D/H in the range [0,.9] (which is certainly the case for a chain of dominos since for
D < H no propagation will occur), the solution to (12) subject to the initial conditions (13)
is well approximated by
q
1 + αm / (2)
√ )
τ = (3g/H)1/2 ln(
1 − αm / 2
(15)
where αm = sin−1 (D/H). Therefore, the constant propagation velocity is V = D/τ , where
√
τ is defined as above. A dimensionless propagation velocity V∗ = V / 3gH can be defined
and
V∗ =
√
2D/H
√
1+αm /√2
ln( 1−α
)
m/ 2
(16)
The theoretical prediction of the constant propagation velocity is shown by the solid line in
figure 9.
EXPERIMENTAL MEASUREMENT OF THE CONSTANT PROPAGATION VELOCITY
We now attempt to experimentally verify that a constant propagation velocity is reached
and experimentally determine this velocity for a variety of different ratios of D/H. A long
line of equally spaced dominos was set up (the number of dominos N >> nc ), and the
digital camera focused on 5-7 dominos near the end of the chain. Because N >> nc , the
propagation velocity certainly should have reached a constant. As before, the time that
each successive domino impacts the next was determined by stepping sequentially through
each frame, and the time τi between the collisions of dominos i-1 and i and i and i+1 was
determined by subtracting. The values of τi were averaged and the error in the measurement
8
was determined by finding the standard deviation, στ , of the measurements. The velocity
v = D/τ was calculated and the error was determined using the error propagation formula
√
σv = D/τ 2 στ . The experimentally determined dimensionless velocity V∗ = V / 3gH with
appropriate error bars is compared to the theoretical predictions in figure 9. The theoretical
predictions were in relatively good agreement with the experimental results for moderate
ratios of D/H, but there was large discrepancies for very large and very small domino
spacings.
As mentioned before, our current theoretical model neglects the finite thickness of the
dominos, essentially treating the dominos as line segments. However, in reality the dominos
have finite thickness W and since D is the distance between their centers, the minimal value
of D/H is W/H (as indicated by the vertical red line in figure 8). It is clear that neglecting
the thickness of the dominos is a worse and worse approximation for smaller ratios of D/H
(and the approximation is completely invalid for D/H < W/H. This provides some insight
into why the for small domino spacing there was a considerable discrepancy between the
theoretically predicted velocity and the experimentally determined velocity.
The theoretically predicted velocity and the experimentally determined velocity also disagree significantly for very large spacings (D/H ≈ 1). This is due to the fact that the
approximation that the dominos fall by pivoting only and do not slide on the surface upon
which they are resting becomes worse as the angle that the dominos fall through before
impacting on the next domino get larger. The fact that the dominos were not merely falling
by pivoting for large spacing was apparent when they were viewed with the digital camera.
Dominos were sliding all over the place, and the final resting positions were quite scattered
rather than being in a neat line.
CONCLUSIONS AND FURTHER WORK
The theoretically determined critical number of dominos needed to reach a constant
velocity was found to be in close agreement with the experimentally determined values for a
wide range of ratios of D/H. However, there are still problems due to the finite resolution of
the camera because for very small spacings the time between the frames is of the same order
of magnitude as the time that it takes a domino to fall. There is also considerable difficulty
in visually determining the frame in which the impact occurs. It might be possible to write
9
a program to determine the time of impact, in order to both achieve greater accuracy and
save the considerable headache of stepping through the frames one by one.
The theoretically determined constant propagation velocity matched the experimental
results well for moderate spacing, but the theory broke down for both small and large spacing
(D/H ≈ W/H and D/H ≈ 1). The breakdown for small spacings was due to the fact that
the current theory neglects the finite thickness of the dominos. It would certainly be possible
to develop similar theory, albeit more complicated, to take into account the thickness of the
dominos. This theory might also allow the value of K to be calculated theoretically (because
K depends on the final angle at which the dominos rest, which depends on the ratio W/D,
in addition to depending on the ratio D/H ). The breakdown for large spacing is most
likely due to sliding of dominos on the plane that they rest upon. This could be rectified by
placing sand paper on the table in order to reducing sliding. Developing a theory to account
for sliding would be quite difficult.
Another shortcoming of our model is that it does not take into account the coefficient of
restitution. When a domino impacts on another, it bounces back slightly. By assuming sliding contact is maintained between the dominos we have completely neglected this bouncing.
Finally, our model does not consider any dissipative effects (All collisions are assumed to be
elastic, and the dominos slide against each other without friction.) However, in reality, the
collisions between the dominos will not be completely elastic and energy will be dissipated
due to friction as the dominos slide against each other.
Despite these shortcomings, our model did a decent job of predicting the critical number
and the constant propagation velocity. The theoretical predictions of nc matched experiment
well for a range of values of D/H. The theoretical predictions for the constant propagation velocities were within the error bars of the experimentally measured values for both
experimental data points with moderate spacing between the dominos. Finally, it must be
noted that there were significant errors in the experimental technique which could be reduced significantly using more sophisticated techniques and equipment. It is possible that
further experiment might yield better agreement between the theoretical predictions and
the experimental results (ie the disagreement may be due to inaccurate experiments as well
as shortcomings of the model).
Acknowledgments: Thanks to Thomas Peacock for suppling the dominos and the lab in
which the experiments were done, and helping me set up the experiment. A special thanks
10
W
H
D
FIG. 1: A regularly spaced array of dominos. The height of the dominos is H, the thickness W ,
and the distance between the centers of the dominos is D. In our experiment H = 5cm, W = .8,
and D was varied.
to the Edgerton Center for the use of their high-speed digital camera.
[1] Clanet, Christophe, “Dominos Race” Beyond Science, p.1-13. [2001]
[2] Stronge, W.J. and D. Shu, “The domino effect: successive destablization by cooperative neighbors”. Proceedings of the Royal Society of London, vol A 418, p.155-163, 1988.
[3] Bert, Charles W. “Falling Dominos” SIAM Review, Vol 28. No 2, June 1986.
11
Time for String of Dominos to Fall
2.2
time to fall (seconds)
2
1.8
1.6
1.4
1.2
1
0.8
30
40
50
60
70
number of dominos
80
FIG. 2: The total time, T , taken for each of three lines of dominos of different lengths (N=30, 50,
and 89 dominos) to fall. A linear fit to the equation T = τ N + T0 was applied to the data, and τ
and T0 were determined to be τ = 0.0227 ± .0102 and T0 = 0.194 ± .028. The variation of T with
N was extremely linear (the fit had an error of χ2 = .01), which suggests that the wave velocity
starts at a somewhat slower speed and rapidly increases to approach a constant velocity.
12
W
H
α
α
H/2 cosα
W/2sin α
FIG. 3: A model of a domino rotating about its right edge after being toppled by a disturbance
traveling in the +x direction (from left to right). The position of the center of mass is indicated
by the large black dot.
13
y1
H
α1
D
FIG. 4: One Falling Domino: The first domino in the chain falls and hits the second domino. At
the instant of impact the first domino makes an angle α1 = sin−1 (D/H) with the vertical. The
height of the position of impact above the plane on which the dominos rest is y1 = H cos α1 . The
change in potential energy of the system after the first domino falls is ∆U = Ui − Uf = 1 − cos α1 .
14
y1
α1
H
α2
y2
α1
D
D
FIG. 5: Two Falling Dominos: After falling,the second domino now makes an angle α 1 with the
horizontal and the height above the plane is y1 . The first domino continues to fall as the second
domino is falling and makes an angle α2 with the vertical at the moment that the second domino
impacts the third and the height above the plane is y2 = H cos α2 . We now seek to determine the
change in potential energy of the system over the period τ2 . The change in potential energy of the
system during the period τ2 is ∆U2 = Ui − Uf = 1 − cos α2 .
15
H
α1
α2
α3
y1
y2
y3
D
∆ s2
H
y1
y2
∆s 3
y1 - y2
∆ s2
= y
H
1
y2
y3
∆ s 3 = y2 - y3
y2
H
FIG. 6: Similar triangles are used to relate the height of the position of impact (and hence the
angle that the domino makes with the vertical at the instant of impact) to the diagonal distance
between impacts. The similar triangles are shown for the first three impacts.
16
Evolution of Propagation Velocity for D/H=.5
110
100
90
velocity
80
70
60
50
40
30
20
10
1
2
3
4
domino number
5
6
FIG. 7: This figure shows the experimentally determined propagation velocity for the first 7 dominos for D/H = .5. The velocity starts off essentially increasing linearly with domino number and
then it levels off at a constant velocity after nc ≈ 5 dominos.
17
7
Critical Number to Reach Constant Velocity
16
D/H=W=H
14
12
Nc
10
8
6
4
2
0
0.1
0.2
0.3
0.4
0.5
0.6
D/H
0.7
0.8
0.9
FIG. 8: This figure compares the theoretical prediction (solid line) for n c (as a function of D/H) to
the experimentally determined values of nc for a range of values of D/H (indicated by *’s). Close
agreement between experiment and theory was found.
18
1
Constant Propagation Velocities
1.4
1.2
D/H=W/H
V/sqrt(3gH)
1
0.8
0.6
0.4
0.2
0
0.1
0.2
0.3
0.4
0.5
D/L
0.6
0.7
0.8
0.9
FIG. 9: This figure compares the theoretical prediction for the constant propagation velocity (solid
line) to the experimentally measured velocities (*’s with error bars). Relatively good agreement
between theory and experiment was achieved for small and moderate spacings (small to moderate
ratios of D/H), but for large spacing the theory broke down.
19
1