Problem 737

[hacatu]

How is whether or not coins are stackable determined? I figured that, for the assembly of the first n-1 coins to be stackable on the nth coin, the center of mass of the first n-1 coins should lie within the nth coin. Knowing the nth center of mass thus gives a quick way to compute the nth angle. It seems natural to store the center of mass in polar coordinates. I found a more or less closed form solution for the radius, as well as an iterative solution, and a formula for the angle of the center of mass and rotation angle in terms of the radius.

But my concern is twofold. My formulae seem a little bit off: the total angles for the numbers of coins given are about 3% off, but the error decreases for more coins. I know there is probably a precision problem as well. Unfortunately I haven't been able to find any clean form for the sum of angles. Have others used a laurent series or some other approximation?

I won't put my exact formula for the radius, but the n=2 radius can be immediately calculated assuming the center of mass interpretation is correct.

[MuthuVeerappanR]

It's very early to post values hacatu. Please keep working on the problem and remove the values. Possibly, someone will reply and you can get the chat in a private forum.

[pjt33]

My formulae seem a little bit off: the total angles for the numbers of coins given are about 3% off, but the error decreases for more coins.

I don't think you should be concerned about that, as long as it's 3% over rather than 3% under. Bear in mind that those numbers given are for the first point at which a discrete number of coins passes the nth full rotation. There's no expectation that they should exactly overlap.

[hacatu]

@hacatu, I can confirm that "center of mass of all coins stacked on top of a given coin is not outside that given coin" is the correct interpretation, I hope you were able to solve the problem!

[thedoctar]

I posted a message on the solutions thread discussing whether the solution incorrectly ignores torque.

I haven't done mechanics in a while, so I might be wrong.

Essentially, it seems to me that the answer ignores any torques on coins in the stack/loop, which might yield the physical coin stack to be mechanically unstable.

Can the developers read my message and let me know if I'm correct?

If considering torque indeed makes the configuration mechanically unstable, I think it should be stated in the problem statement that the problem ignores torque.

[RobertStanforth]

I posted a message on the solutions thread discussing whether the solution incorrectly ignores torque.

Thanks! I've responded in the solution thread.

[thedoctar]

Thanks for your response Robert. I updated my post to provide the link and explained the relevant formula in the wiki article.