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Heavy pendant on weightless chain of length L is hung on frictionless cone of angle α+α=2α?
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?
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.
HEAVY CHAIN – Ali (XPDC)
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