Heavy pendant on weightless chain of length L is hung on frictionless cone of angle α+α=2α?

Question

The axis of the cone is vertical, but it is not very important.

What is minimal distance between the chain and the tip of the cone?

Answer 1

Without giving full proof, the path of the chain will be a geodesic on the cone. Because a cone can unroll flat into a sector of a circle, the geodesic is the straight line from one corner of it to the other. The angle of the sector is just 2πSin(α), where α is the half angle of the cone, and thus

Tan{πSin(α)) = ((1/2)L) / M, or

M = ((1/2)L) / Tan{πSin(α))

where L is the length of the chain, and M is the minimum distance from the chain to the apex. The curious result is that while for small α, this minimum approaches infinity, but for α = π/4, or a right cone, the minimum is 0. Thereafter it's 0 for all cones of half angle α = π/4 or greater.

As for why the path should be a geodesic, consider a path from one corner to the other in a rolled out sector of a circle. For a given path length L, it is a straight path that maximizes the distance of the pendant from the apex.

I give this one a star for having to think on this discontinuous result.

Answer 2

conic equation is always a problem to me. and to next answerers, would you please tell me what "cone of angle α+α=2α" means. i know Alex might explain after selecting BA but i'd like to know as earliest as possible.