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A jar is filled with two types of balls, red and green. Red balls have radius $r_1$ and mass $m_1$, green balls have radius $r_2$ and mass $m_2$.

If initially the balls are randomly placed throughout the jar, and we are to shake this jar, generally one type of balls will tend to gather at the top. Why does this happen?

What are necessary conditions on $r_1$, $r_2$, $m_1$, $m_2$ such that green balls are more likely to occupy the upper part of the jar after shaking?

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would this question not be much more nutritive, when done on cookies of different size and density? :=) –  Georg Feb 2 '11 at 13:54

2 Answers 2

up vote 5 down vote accepted

I know about several attemps to formulate a theory for this problem (it would seem that this is commonly called the "Brazil Nut Problem" in the English literature, I'm familiar with the "Paranuss-Problem" from the German literature), see

  • Daniel C. Hong, Paul V. Quinn, Stefan Luding: "The Reverse Brazil Nut Problem: Competition between Percolation and Condensation" (arXiv)

and

  • Troy Shinbrot und Fernando J. Muzzio: "Reverse Buoyancy in Shaken Granular Beds", Phys. Rev. Lett. 81, 4365–4368 (1998)
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It turns out that larger balls tend to move up even if their density is larger than the density of smaller balls. Intuitively it happens because for the small balls on the bottom of a large ball it much more difficult to move than for the ones on the top.

But there is no rigorous answer to your question. It is a whole area of Granular Matter physics for one to research. Models, different simulations, experiments, e.t.c.

I will also note that the separation between different balls strongly depends on many different properties -- not only sizes and masses. Friction, air flows, boundaries, way of shaking can strongly influence the phenomenon, while there is no understanding why this is happening.

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""even if their density is larger than the density of smaller balls."" I'd prefer to say: "even if their density is slightly/somewhat larger..." Putting gold balls in a "sea" of styrofoam balls will lead to a very conventional result irrespectively of all other parameters :=) –  Georg Feb 2 '11 at 11:59
    
What happens to the net potential energy? Even with denser balls on top, if the packing density is sufficiently increased it is possible the final gravitational potential energy has been lowered. But because energy has been added by shaking (and dissipated by friction), this is not required. –  Omega Centauri Feb 2 '11 at 14:35
    
One problem is, that "shaking" is not a good aproximation to get a liquid-like behaviour. In case You place the jar on a vibrating table which You can control in frequency and amplitude, things maybe will look better. Doing so was a rather popular tecnology to convey goods some 50 years ago, but today it is used no longer, because it is rather noisy. –  Georg Feb 2 '11 at 14:51
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I know of two phenomena that depend on shaking actually sifting the lighter/finer ensembles at the bottom and bringing the larger on the surface.1)cement when as semi liquid is vibrated with some special machines, a porcelain level is created at the lowest surface which assists in making roofs impermeable to water.2) In Greece where quakes, particularly small ones, are very common, no matter how often a field is cleared of stones, more stones in time come to the surface. –  anna v Feb 2 '11 at 20:31
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here is a link for quakes in Greece : gein.noa.gr/services/info-en.html . That the earth has not washed away can be seen from the trees themselves which are very old ( some maybe 2000 years old) and their roots are not exposed. –  anna v Feb 3 '11 at 14:36

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