Practical Geometry

[akitrom]

I don't quite know where to begin with a solution, so I'll articulate the problem.

"A right cylinder of uniform density has a radius, r, and a height, or thickness, h. If r is sufficiently larger than h, as with a coin, and the object is tossed about, the probability of it coming to rest on a base are very high. If r is relatively small, as with

Comments

[madarab]

Does it matter how it's thrown?

[laurence_skr]

It matters considerably how it is thrown. Since the object has a different moment of inertia while rotating about the two axes of symmetry, the possibility of angular momentum upon landing would significantly influence(and complicated) the calculations.

If you assume that the object has no horizontal or angular momentum, and the surface upon which it lands is frictionless, the calculation becomes reasonably simple, comparatively. These assumptions eliminate the possibility that the object might land on one face and then flip or roll onto another. The rotation caused only by the normal force will be insufficient to cause the cylinder to flip over, which is most easily shown by conservation of energy.

While you could forego these assumptions, you would need an alternate assumption regarding the probability distribution of the various initial rotational and horizontal velocities, in order to calculate a resultant probability.

[chimeramonster]

You've rendered me speechless by the amounts of logic and physics and math you've just typed. I am amazed.

[madarab]

Assuming a frictionless surface ignores the random small forces that allow a body to tumble rather than bounce and finally come to a stop. You'll need some kind of perturbation analysis to deal with friction and how it can add symmetry-breaking features to the solutions. It's not impossible to do, given the symmetries of the cylinder; just add small forces to a series of test solutions to see if they're stable.

[thestormcellar]

Random thoughts:

there are 2 lateral surfaces.

perhaps working with diameter would be easier than working with radius as it seems like a surface area issue.

Would a curve graph of results help?

[eyja]

*blink*

Ya...I'm so of no help.

[colinblackthorn]

It seems to me that in order for this to work, assuming uniformity of rigidity between the cylinder and the surface it lands on, you need to have the area presented by the profile of the cylinder (the square mentioned above) equal the area of each of the other two faces (the circles you see looking down the cylinder as if it were a hollow tube). This is the result of choosing randomly among the three faces. However, as lawrence pointed out, if you are really throwing the cylinder, how it is thrown changes this: if you're flipping it like a coin (even though it's not coin-shaped) it seems more likely to land on a flat side, but if you roll it like a pencil, it's more likely to end up on the rounded side (like a pencil would). However, if throwing the cylinder involves tossing it into a big rubber box in free fall and letting it carom around until its motion is thoroughly randomized, then disintegrate the box and letting the cylinder fall naturally as if a stone from the sky in a random direction with random rotation on both axes, all

[chimeramonster]

Forgive me, but....

How in the living hell did this come up? Is this from work? Did you just think this up one day whilst going about your normal daily activities? (Nevermind that in regard to mathematics I am quite willing to believe your daily activities are anything but normal....)