August 16, 2010
Los Angeles, CA
IYPT 1993 # 8: Domino Wave: Balance numerous domino tiles on a table or
floor with their long axis vertical with horizontal spaces that can be varied.
When the first tile is knocked over a domino wave occurs. Predict and
measure the speed of this wave and its dependence on various parameters.
I. Conceptual picture
I-1. The problem involves motion and collision of solid bodies – dominoes. Basic
knowledge about solids (moment of inertia, angular velocity, torque, angular
momentum, and kinetic energy of rotating bodies) is in multiple algebra or
calculus based books (see [1] or similar texts.) The problem does not require
knowledge about waves. The phenomenon looks like a wave, because dominos
do not move along the table, but the perturbation propagates from domino to
domino by impact. Te perturbation itself does not look like an oscillation. It is an
irreversible transition form one state of equilibrium to a new one. So, do not get
confused by the title of the problem.
I-2. Jearle Walker’s famous book [2] contains a problem 1.68. The problem is
similar to our problem, but at the end Walker invites the readers to think about
dominos of different sizes. Walker created an electronic bibliography for his book
and you can find it on the Internet [3]. Some of the articles on Walker’s
bibliography are available for download (see [4-7].) To download [5] you need to
have an access to the American Journal of Physics publications.
Articles [4] and [5] describe some basic ideas of the theory and they present
experimental results. Articles [6] and [7] derive equations for the domino waves.
II. References
[1] Serway, Vuille, and Faughn, “College Physics”, 8th ed. (BROOKS/COLE,
2010)
[2] J. Walker, “The Flying Circus of Physics”, (Wiley, 2006)
[3] www.flyingcircusofphysics.com/News/NewsDetail.aspx?NewsID=51
[4] Larham, R., “Validation of a model of the domino effect," (2008) available at
http://arxiv.org/abs/0803.2898v1
[5] Shaw, D. E., "Mechanics of a chain of dominoes," American Journal of
Physics, 46, 640-642 (1978)
[6] Efthimiou, C. J., and M. D. Johnson, “Domino waves,” (2008) available at
http://arxiv.org/abs/0707.2618v1
[7] van Leeuwen, J. M. J., “The domino effect,” American Journal of Physics, 78,
721-727 (2010)
(also available at http://arxiv.org/abs/physics/0401018v1)
III. Experiment
There are several ways to perform an experiment with domino waves. The most
difficult part is to measure time between collisions. In ref. [4] you will find how
Larham used sound recorder and microphone. You can try also camera Casio
XF1. This is the first affordable (less than $1,000) high-speed camera, and it can
take up to 1200 fps video.
IV. Theory
Theory that describes the domino wave should take into account the following
phenomena
Each domino is a uniform rectangular
solid
Dominos are arranged in a row at equal
distances
Each domino has three states of
equilibrium states on a flat horizontal
table (see Figure 1)
o Metastable (state 1 on Figure 1).
At this state domino can stay long
and it is stable with respect to small shakings. However it will start
falling if the deviation from equilibrium is significant.
o Unstable (state 2 on Figure 1). At this state domino cannot stay
long. It will move after any small perturbation is applied.
o Stable (state 3 on Figure 1)
Different kind of collisions could occur in the wave
o Elastic collision. After elastic collision moving domino slows down
and domino at rest starts motion and moves faster. So, the slow
domino will join the group of falling dominos (see Figure 2). This
kind of wave develops when dominos are put on a big distance
from each other. The mechanism is described in the Reference [6].
They considered the wave using thin domino approximation. At this
approximation there are only two states of equilibrium, 1 and 3 on
Figure 1.
o Inelastic collision. During inelastic collision, a new domino joins the
group and they move all together (see Figure 3). This happens
when dominos are put on the table on a short distance from each
other. Inelastic mechanism is described in Reference [7].
Good luck!
T.Bibilashvili