# The Naked Scientists Forum

### Author Topic: What is the largest domino that a smaller domino can topple in a chain reaction?  (Read 2293 times)

#### cheryl j

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##### What is the largest domino that a smaller domino can topple in a chain reaction?
« on: 18/04/2014 01:49:13 »
Imagine a length of dominoes standing on end, and although proportional,  they are not the same size. The next domino is slightly bigger than the one behind it - the one that is closer to you. If you push the first domino over, how increasingly bigger in mass can the next ones be and still fall over?  And does it matter how close they are?
« Last Edit: 20/04/2014 04:03:37 by CliffordK »

#### RD

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##### Re: I have no idea what to title this, sorry.
« Reply #1 on: 18/04/2014 03:23:50 »
Domino-toppling  chain-reaction ...

#### alancalverd

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##### Re: I have no idea what to title this, sorry.
« Reply #2 on: 18/04/2014 08:34:32 »
Consider an infinitely thin domino (a mathematician's domino!). An infinitesimal lateral force at any point above the base will make it topple.

Now think about a cube of side length 2a (a civil engineer's domino?). In order to topple it, you need to move the center of gravity outside the base, so any lateral force less than the weight of the cube must be applied above the highest point that the CG will reach (√2 × a). That gives you the minimum criterion for toppling.

The static lateral force applied by one domino to the next depends on the ratio of the length of the falling domino to the separation between them.

However the above only applies to the static case where the dominoes fall infinitely slowly. The real dynamic case requires the transfer of momentum from the moving domino (and all its predecessors that are still moving) to the stationary one.

The general equation is "left as an exercise to the reader" (apparently I have a lawn to cut!) but it's a hell of a good question, and the answer will depend on the aspect ratio of the dominoes, their separation, and the ratio of weight to mass (g).

#### Bored chemist

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##### Re: I have no idea what to title this, sorry.
« Reply #3 on: 18/04/2014 11:02:08 »
Well, someone checked, and videoed it.

#### CliffordK

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##### Re: What is the largest domino that a smaller domino can topple in a chain reaction?
« Reply #4 on: 20/04/2014 04:52:32 »
Great Video BC & RD.  I might have to find a set of dominoes to play with, or perhaps makes some pseudo-dominoes.

Somebody has probably worked out all the domino equations.

The tipping point of the domino is where the center of gravity is exactly over the corner, or where the two opposite corners line up vertically.  That also delineates the energy required to lift the center of mass to the tipping point.

I believe the minimum spacing for equal dominoes to sustain a chain reaction would be where the center of mass of the falling domino is at the same elevation of the center of mass of the standing domino (where it was).  Is this twice the first angle to the tipping point?  This would be slightly less than twice the thickness of the domino.  Or, somewhere thereabouts.

Is spacing them a single domino thickness apart too close?

Of course, all the points are moving, so as the next domino topples, the previous one is still pushing slightly, so you may be able to space them closer than I had originally guessed, but still, a single domino width is probably close to the minimum.

I'm not quite sure what the optimal spacing is, and it may well be different for equal sized dominoes, and different sized dominoes.

However, there may be some acceleration of toppling of equal sized dominoes.

So, the film suggested about 1.5X for successive larger dominoes.

ABCDE

Now, what if one put multiples in?

AAA BBB CCC DDD E

Would one experience slight acceleration of the toppling of identical dominoes, and thus being able to topple slightly larger dominoes, perhaps 1.6 or 1.7X?

#### cheryl j

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##### Re: What is the largest domino that a smaller domino can topple in a chain reaction?
« Reply #5 on: 20/04/2014 05:25:22 »

However, there may be some acceleration of toppling of equal sized dominoes.

I wondered about that too. In demonstrations with equally sized dominoes, it seems to speed up towards the end, but I didn't know if that was an illusion.

#### The Naked Scientists Forum

##### Re: What is the largest domino that a smaller domino can topple in a chain reaction?
« Reply #5 on: 20/04/2014 05:25:22 »

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