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Packing a jar of marbles
Name: roman zabicki
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995
Question:
I saw one of those jars where you guess how many coins/jelly beans/marbles
are in it to win a prize this summer. This got me thinking, how could it be
calculated? I could easily enough measure the height and multiply it by the
radius squared multiplied by pi. But doing this would only be accurate for
finding the volume of water. With marbles/coins/jelly beans, there is empty
space in between the marbles. Is there a constant or something like that to
figure it out? If the marble (my question works better with marbles than
coins) is as tall as the height of the occupied space there is a lot of
wasted space. As the marbles get smaller, the wasted space decreases
closing in on a "steady state" ( is that the right word) and eventually it
would match the result one would get if one used water. Does anyone know of
any research done on this? I know there are not any commercial applications
so there probably is not much funding for it, but i was just wondering about
this. Any explanations would be greatly appreciated.
Replies:
You can certainly put upper and lower limits on the number of marbles. For
an upper limit you could assume that the marbles distributed their volume
like a fluid, so just divide the volume of the jar by the volume of individ-
ual marbles. For a lower limit, assume that the marbles sit a the center of
cubes whose sides are the same length as the diameter of the marbles (i.e.,
the smallest cube that can completely contain the marble. Then divide the
volume of the jar by the volume of the cube to get the number of marbles
(lower limit).
In reality the marbles will "pack" a little tighter than the perfect cube
picture. Get a bunch of marbles of uniform size and try it. Make a layer
of marbles as tightly packed as you can (just one layer). Now place another
marble on top of the layer. It does not balance directly over one of the
marbles in the first layer but nestles nicely into a pocket between three
marbles in the first layer -- thus filling the space more completely than
the cube picture above. For perfect, uniform spheres you will get a
"closest packed" arrangement (either cubic closest packed or hexagonal
closest packed) and the math for this has been worked out -- general
chemistry texts or crystallography texts can give more details. For perfect
spheres the result is that about 74% of the space is actually taken up by
the marbles. Real packing will be slightly less. I do not know about other
geometries (jelly beans).
gregory r bradburn
So the maximum packing fraction is 74%. For random packing (what you get by
just jumbling the marbles in) of spheres the packing fraction is 63-65%.
Again, I do not know about jelly beans though, but in fact this probably has
been investigated - I believe there is a part of materials science devoted
to the ways in which agglomerations of materials pack together, and this
kind of question is actually useful to companies who make large numbers of
small objects - you really do not want to count all the nails in a bin, for
example (although usually weighing the things is a better measure of the
number).
asmith
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Update: June 2012
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