Jenna Bratz

1
Domino RallyHow long does it take for a domino
to fall?

• Jenna Bratz
• Rachel Bauer

2
(No Transcript)
3
Intro

• First set of dominoes date back to 1120 A.D.
• First used for games of strategy
• Lining them up and knocking them down became
  increasingly popular in the 1980s with
  introduction of the game Domino Rally

4
Intro (contd)

• Our Goals
• Model the toppling time of 1 domino
• Model the toppling time of 2 dominoes
• Extend to 3, 4 or n dominoes
• Find the optimal distance that minimizes topple
  time

5
Literature

• Charles Bert modeled the topple time of both one
  and two dominoes using a conservation of energy
  argument, however the experimental time did not
  match the predicted time
• Shaw discovered that in a long chain of linearly
  equally spaced dominoes, only the 5 preceding
  dominoes contribute to the fall of the current
  domino
• Math 512 at UD verified Berts argument with
  better data, and also verified Shaws argument
  experimentally
• Heinrich and Lutz modeled molecule cascades, in a
  similar manner to modeling a chain of falling
  dominoes.

6
1 Domino (Experiment)

• Procedure
• Domino was placed on sandpaper (to ensure no
  sliding)
• A domino toppler was used to push the domino to
  its balancing point and then to just let it go
• A high speed camera was used to capture the
  topple time (250 frames per second)

7
1 Domino (Experiment)
8
1 Domino (Theory)
9
1 Domino (Theory)

• Assumptions
• The domino will not slide (the pivot point will
  remain in the same position throughout the fall)
• The domino will not rotate
• The domino will start from an initial velocity of
  zero, and will have an initial angle of
  and a final angle of

10
1 Domino (Theory)

• Conservation of Energy to obtain a model for
  theta in terms of time
• Here, kinetic energy is the sum of the rotational
  kinetic energy, and the translational kinetic
  energy
• The angle is by the choice of our
  coordinate system

11
1 Domino (Theory)

• Because of assumptions, the initial angular
  velocity is zero.
• Also assume that the final translational kinetic
  energy is zero since all the energy is
  transferred into the rotational kinetic energy
• Equation reduces to

12
1 Domino (Theory)

• Plug in I, writing w as and taking a second
  derivative of the equation with respect to time
  gives the following ODE
• Can reformulate into a first order system with
  gives

13
1 Domino (Theory)

• Stability Analysis
• Equilibria at
• Jacobian
• After analyzing equilibria, obtain that when n is
  even, there is an unstable saddle and when n is
  odd, there is a center

14
1 Domino (Theory)

• Phase Plane

15
1 Domino (Theory)

• Numerical Solution of a particular domino
• Wanted epsilon to be as small as possible, just
  enough to knock the domino off balance
• Educated guess of epsilon being 1 degree.

16
1 Domino (Theory)
17
1 Domino (Theory)

• Fit a curve to the numerical solution
• Only interested in theta up to Pi/2

18
1 Domino (Theory)

• Used this fit to estimate the time at exactly
  Pi/2.
• Theoretical total time for one domino to fall is
  .27374 seconds

19
1 Domino (Analysis)

• Experimental mean time .2667 seconds
• Theoretical time .27374 seconds
• Difference is .00704 seconds, only 2.57 error!

20
2 Dominoes (Experiment)

• Set up 2 equally spaced, equally sized dominoes
• Used domino toppler
• Did 10 trials, spaced the dominoes at 2.17 cm,
  which was half the height of the domino

21
2 Dominoes (Experiment)
22
2 Dominoes (Experiment)
23
2 Dominoes (Theory)

• Assumptions
• The dominoes will not slide (the pivot point
  will remain in the same position throughout the
  fall)
• The dominoes will not rotate
• Dominoes are parallel and equally spaced
• The second domino will receive a fraction of the
  first dominoes horizontal velocity

24
2 Dominoes (Theory)

• The first domino will hit the second domino at a
  critical angle,
• Using this critical angle, can find the time at
  which the first domino hits the second, and from
  this time,
• can obtain the speed
• at which the first
• domino is moving

25
2 Dominoes (Theory)

• Finding the velocity of the first domino at the
  hitting point, will give the initial velocity of
  the second domino
• After initial conditions of second domino are
  obtained, the same model as the one domino case
  can be used to model the fall time of the second
  domino

26
2 Dominoes (Theory)

• Chose a distance of half the domino height of the
  domino chosen in the first theory, d.0217 m
• This gives
• We fit a curve to the numerical solution of the
  angular velocity of the first domino, and found
  the velocity at this critical angle to be

27
2 Dominoes (Theory)

• Now, all of this velocity is not going to be
  transferred to the second domino . In particular,
  we claim that not all of the horizontal velocity
  will be transferred
• We discovered that any fraction less than ½ of
  the first dominos velocity did not cause the
  second domino to fall
• So we chose to use exactly half of the horizontal
  velocity

28
2 Dominoes (Theory)

• Horizontal Velocity is given by
• We want half of this velocity to be the starting
  velocity of the second domino. So the new ODE
  for the second domino becomes

29
2 Dominoes (Theory)

• Similar to the one domino case, we found a
  numerical solution.
• Still want the time it takes the second domino to
  reach Pi/2, so we fit a curve to the numerical
  solution for theta.

30
2 Dominoes (Theory)

• Finding the time at Pi/2 gives t.40069, and
  adding this to the time it took the first domino
  to reach the critical angle
• (t.19805).
• So the total time for two dominoes to fall should
  be t.59874 seconds

31
2 Dominoes (Analysis)

• Theory did not match experiment
• Most likely due to random choice of the amount of
  horizontal velocity that is transferred
• Adjust the starting velocity of the second domino
  to match data

32
2 Dominoes (Full Velocity)

• New assumption all of the horizontal velocity is
  transferred to the second domino
• New problem for domino 2

33
2 Dominoes (Full Velocity)

• Numerical solution is shown below
• Steepness can be seen from the phase plane

34
2 Dominoes (Full Velocity)

• Found curve of best fit (again) and got new
  topple time for 2 dominoes to be .35764 seconds
• This number is still higher than experimental
  time
• Reason could be that the first domino may have
  had some small initial velocity in the
  experiment, which would decrease the topple time.

35
Conclusion

• Model for the topple time of one domino was
  confirmed by data
• Topple time for 2 dominoes is very dependent on
  the amount of velocity transferred
• Appears that having all of the horizontal
  velocity transferred gives an accurate estimate
  for topple time

36
Further Work

• Improve 2 domino model
• Model n dominoes
• For 3 dominoes, incorporate the effect of both
  the first and second dominoes
• Find the distance that minimizes topple time
• Explore different spacings, both nonlinear and
  not equal spacing

37
Apologies

• We would like to apologize to Patrick C. Rowe,
  for not spelling his name in dominoes on top of a
  slab of jello.