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Puzzle for Today

Puzzle No. 311 – Wednesday 12 September

Martha is looking at necklaces in a jewellers shop window hoping that John might buy her one for her birthday. There is a thin chain on display with a heavy pendant. It's on a smooth conical stand and she wonders why the chain doesn't slip off with such a heavy pendant. What is the biggest cone angle that will prevent the weight of the pendant from pulling the necklace off the stand?

Today’s #PuzzleForToday has been set by Dr Hugh Hunt, Reader in Engineering Dynamics and Vibration at Trinity College, Cambridge

60°

When the chain is pulled hard at one point then the chain AB follows the line of shortest length, a geodesic (assuming no friction and a light flexible chain)

If you unwrap the cone so that it is flat then the shortest distance AB is a straight line. For the line to pass over the vertex V then the unwrapped cone must be a semicircle so that the line AVB lies along the diameter. If the radius of this semicircle is R then the base circumference of the cone is πR so that the base diameter is R. The side view of the cone is therefore an equilateral triangle which makes the cone angle 60°.

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