Problem of the Week 1152

A Better Cable Connection

A uniform cable attached to a ceiling at one end and holding up a weight at the other end will always break at the top, because the top has to support the weight plus the cable below it. This is wasteful because the bottom part of the cable need not be as thick as the top. An efficient cable would be no thicker than needed to support the total weight below it.

Imagine a tapered cable of circular cross-section and length L that hangs freely, where at every point the cable is exactly thick enough to support the cable below it plus the hanging weight, which we take to be 1 unit.

Let r(x) be the radius of the cable at distance x from the bottom. Assume that a unit area of cable is strong enough to hold a weight of k units below it (i.e., the tensile strength of the cable is k) and that a unit volume of cable weighs d units (density is d).

What is the function r(x)?

Source: Julien Beasley, Seattle, who writes: "The inspiration for this problem came to me at a rock concert. I was looking at the lamps suspended from the ceiling on cables, and thinking 'The lower part of the cable is holding less weight than the top. The cable will always break at the top. What form should the cable take so that it would break anywhere with equal probability?'"

© Copyright 2012 Stan Wagon. Reproduced with permission.