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# Which weighs more, a cubic meter of large sized coal or a cubic meter of small sized coal (assume that all the pieces of coal in each cubic meter are loosely packed identical-sized spheres, each sphere touching six others)?

Question #25020. Asked by Nude Dude. (Dec 09 02 6:42 PM)

sequoianoir

I think they are both the same. (if we are talikg about a 1 metere cube - not exactly the same as 1 cubic metre)
If you put a sphere (diameter 2) inside a cube (size 2 x 2 x 2) the sphere is a perfect fit (touches all sides) but there is obviously some 'wasted space'.
This 'waste' is the same percentage as a sphere (dia 1) in a cube 1 x 1 x 1.
The cube 1 x 1 x 1 would fit 8 times in the cube 2 x 2 x 2.
So 8 spheres (dia 1) fit in the cube 2 x 2 x 2.
Both cases have the same amount of 'waste' space.
So providing the coal is spherical it should make any difference !

 Dec 09 02, 7:52 PM
Gnomon

Given the very artificial conditions outlined in the question, the two cubic metres of coal will way the same.

 Dec 09 02, 8:35 PM
Friar Tuck

You can put one 1 metre diameter sphere inside a cubic metre and the volume of the sphere would be (pi x 1 x 1)/4 = 0.25pi. You can put eight 0.5 metre diameter spheres in the same space and their volume would be ((pi x 0.5 x 0.5/4) x 8 = 0.5pi so therefore the small pieces have the greater volume and therefore the greater weight.

 Dec 09 02, 8:38 PM
sequoianoir

Sorry Friar Tuck but you got it totally wrong !

Formula is Four thirds Pi radius cubed.

Volume = ( 4 x PI x R x R x R ) divided by 3

Since the 4, Pi and the 3 is constant then the relative volumes of 2 different size spheres is the variable called diameter
If diameter = 1 then radius is 0.5 and relative volume 0.5 x 0.5 x 0.5 = 0.125

If diameter is 0.5 then radius is 0.25 and the relative volume is 0.25 x 0.25 x 0.25 = 0.015625

8 x 0.015625 = 0.125
THE SAME !!!

 Dec 09 02, 9:07 PM

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