I Why do two balls launch from the right in a Newton's cradle instead of just one?

[PeroK]

The Newton's cradle is not the Earth or a solid lump but is made out of separate parts which can be put into contact with each other. I think the experimental results speak for themselves. If you haven't already done so look at the you tube videos on the cradle. There is no noticeable rebound.

This video includes the experiment with a larger ball and a smaller ball (the smaller ball clearly rebounds).

You can also see in slow motion that clearly more that one ball moves in the normal experiment.

 

[A.T.]

(And then consider the touching case as the limiting case as the gap size goes to zero.)

I'm wondering if the gap is that relevant. If you glue the two dropped balls together, you still get a body with a specific non uniform distribution of mass and compressive stiffness, that might still result in two pulses, and thus two balls ejected to similar height.

But if you replace the two dropped balls with a single ball of twice the mass (video in post #51 at 1:16), you get just one small ball ejected very high (while the second last just moves slightly). I wonder if the result here would be clearer, if instead a bigger ball of the same material they used a denser ball of the same size and stiffness.

But this again suggests that you cannot treat this as a single collision, but have to analyze it sequentially, like in post #44.

 

[member659127]

Dimensional analysis suggests adjusting only the spring constants and masses,

Can you elaborate on that please? Dimensional analysis arguments always fascinate me...

 

[haruspex]

a tiny space between adjacent balls. (And then consider the touching case as the limiting case as the gap size goes to zero.)

That assumes there is no minimum requirement on the gap.
If the balls have mass m, and spring constant k, and the incoming ball has velocity v, the oscillation amplitude during each collision is proportional to $v\sqrt{\frac mk}$. The gap needs to exceed that amplitude if we are to avoid three balls being in contact simultaneously.
So in a typical arrangement with slow moving steel balls the gap can be extremely small, but not arbitrarily so.

 

[Dadface]

This video includes the experiment with a larger ball and a smaller ball (the smaller ball clearly rebounds).

You can also see in slow motion that clearly more that one ball moves in the normal experiment.

I think this video is better in that it considers a cradle of the type that I thought this discussion was about, a cradle where all the balls are identical.

 

[haruspex]

Can you elaborate on that please? Dimensional analysis arguments always fascinate me...

Spring constants, k, have dimension MT^-2, masses, m, have dimension M. The qualitative behaviour of a system depends on dimensionless thresholds, i.e. on how some dimensionless function of the variables compares with some magic constants.
To get rid of the M term, k and m must combine as m/k, producing dimension T². This means that if you quadruple both m and k then you will get the same qualitative behaviour but at a factor of two of the speed. E.g. if there is also a velocity involved then you will need to adjust that correspondingly to be sure of the same behaviour.
See also post #54.