Posted: Dec 20, 2012 3:53 pm
Pulsar wrote:CarlPierce wrote:i've been trying to derive the equations for the probability of a cuboid dice landing on its six faces if the X,Y,Z lengths are not equal.
I don't think it is so simple as being proportional to the area of the face.
Interesting question. The probabilities will depend on the position of the center of mass. The dice is more likely to land on the side for which the center of mass is closest to the ground. I don't know how to calculate the odds, though.
EDIT: I found a discussion about this: http://physics.stackexchange.com/questions/41297/how-to-determine-the-probabilities-for-a-cuboid-die
Very interesting.
The answer isn't the solid angle I'm sure of that. Even for the simpler problem with a perfectly rigid square cuboid onto a perfectly flat surface.
I'm sure the real world problem includes the elasticity of the material which will impart harmonic 'wobbles' as the objects rotation momentum tends to zero meaning that beyond a certain threshold ratio a long thin shaped cuboid will never land on its end if tossed.
I.e if you toss a domino giving it a fair wack of rotation momentum I think it will never land on its end. To get it to stand vertically you must carefully place it on its end. Easy experiment to try.