[math-fun] High school physics problem
Suppose we have a large container not 100% filled with ball bearings (all of exactly the same material & hence density) but with a wide & random variety of radii. The largest ball bearings are << the smallest dimension of the container, and the smallest ball bearings can get into every corner of the container. Initially, the bearings of all sizes are equally distributed w.r.t. location within the filled portion of the container. We're on Earth with its standard gravity g and standard atmospheric pressure & temperature (if that matters). Now shake the container vigorously for quite a while (but not so vigorously enough that the balls become dented or heated!). What is the resulting distribution of sizes after this shaking? A. The same as before. B. The larger balls become more likely in the upper layers. C. The larger balls become more likely in the lower layers. I don't have a proof, but I have an intuition about which answer is correct. Has this problem been studied before?
https://en.wikipedia.org/wiki/Granular_convection "Granular convection, or granular segregation, is a phenomenon where granular material subjected to shaking or vibration will exhibit circulation patterns similar to types of fluid convection. It is sometimes described as the Brazil nut effect when the largest particles end up on the surface of a granular material containing a mixture of variously sized objects; this derives from the example of a typical container of mixed nuts, where the largest will be Brazil nuts. The phenomenon is also known as the muesli effect since it is seen in packets of breakfast cereal containing particles of different sizes but similar density, such as muesli mix." Cool! This sounds like a great topic for a math physics PhD (theoretical) or a terrific HS science project (experimental). There must be some *critical* numbers -- e.g., in a mixture of ball bearings of two different sizes, there must be a critical ratio of sizes, or a ratio of percentages (or both), where the increased density (per unit volume) of the larger ball bearings outweighs (!) the Brazil Nut Effect. Alternatively, adjust the mass density ratio of the larger to the smaller ball bearings until the effect disappears. Also, I don't know if the *slipperiness* of the ball bearing surfaces have any effects; e.g., what if all the ball bearings are covered in a light oil? One could make a Carnot-like "heat engine" in which the larger ball bearings *do work* as a result of the shaking -- e.g., perhaps by some magnetic coupling. Extra credit: what happens in 2D (shaking "discs/rings" instead of ball bearings) or 4D ? This is exactly the sort of problem that would have interested Maxwell -- I wonder if he thought about it? At 09:35 AM 12/10/2018, Veit Elser wrote:
On Dec 10, 2018, at 12:16 PM, Henry Baker <hbaker1@pipeline.com> wrote: Has this problem been studied before?
Try searching "Brazil nut effect".
-Veit
I say B, because the largest cornflakes are at the top of the cereal box and the bottom is all crumbs and dust. On Mon, Dec 10, 2018 at 10:19 AM Henry Baker <hbaker1@pipeline.com> wrote:
Suppose we have a large container not 100% filled with ball bearings (all of exactly the same material & hence density) but with a wide & random variety of radii. The largest ball bearings are << the smallest dimension of the container, and the smallest ball bearings can get into every corner of the container.
Initially, the bearings of all sizes are equally distributed w.r.t. location within the filled portion of the container.
We're on Earth with its standard gravity g and standard atmospheric pressure & temperature (if that matters).
Now shake the container vigorously for quite a while (but not so vigorously enough that the balls become dented or heated!).
What is the resulting distribution of sizes after this shaking?
A. The same as before.
B. The larger balls become more likely in the upper layers.
C. The larger balls become more likely in the lower layers.
I don't have a proof, but I have an intuition about which answer is correct.
Has this problem been studied before?
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I doubt that a proof is possible without more assumptions, e.g. about the contact friction and the shape of the container. But the problem has been studied and given a name "the museli effect" for the larger objects tending to the top. https://en.wikipedia.org/wiki/Granular_convection Brent On 12/10/2018 9:16 AM, Henry Baker wrote:
Suppose we have a large container not 100% filled with ball bearings (all of exactly the same material & hence density) but with a wide & random variety of radii. The largest ball bearings are << the smallest dimension of the container, and the smallest ball bearings can get into every corner of the container.
Initially, the bearings of all sizes are equally distributed w.r.t. location within the filled portion of the container.
We're on Earth with its standard gravity g and standard atmospheric pressure & temperature (if that matters).
Now shake the container vigorously for quite a while (but not so vigorously enough that the balls become dented or heated!).
What is the resulting distribution of sizes after this shaking?
A. The same as before.
B. The larger balls become more likely in the upper layers.
C. The larger balls become more likely in the lower layers.
I don't have a proof, but I have an intuition about which answer is correct.
Has this problem been studied before?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The granular separation process is related to "spontaneous stratification", which is also only partly understood: https://www.simonsfoundation.org/2014/05/23/mathematical-impressions-spontan... George http://georgehart.com On 12/10/2018 4:12 PM, Brent Meeker wrote:
I doubt that a proof is possible without more assumptions, e.g. about the contact friction and the shape of the container. But the problem has been studied and given a name "the museli effect" for the larger objects tending to the top.
https://en.wikipedia.org/wiki/Granular_convection
Brent
On 12/10/2018 9:16 AM, Henry Baker wrote:
Suppose we have a large container not 100% filled with ball bearings (all of exactly the same material & hence density) but with a wide & random variety of radii. The largest ball bearings are << the smallest dimension of the container, and the smallest ball bearings can get into every corner of the container.
Initially, the bearings of all sizes are equally distributed w.r.t. location within the filled portion of the container.
We're on Earth with its standard gravity g and standard atmospheric pressure & temperature (if that matters).
Now shake the container vigorously for quite a while (but not so vigorously enough that the balls become dented or heated!).
What is the resulting distribution of sizes after this shaking?
A. The same as before.
B. The larger balls become more likely in the upper layers.
C. The larger balls become more likely in the lower layers.
I don't have a proof, but I have an intuition about which answer is correct.
Has this problem been studied before?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (5)
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Brent Meeker -
George Hart -
Henry Baker -
Kerry Mitchell -
Veit Elser