11 April 2010

The mathematics of a twisted rope

This is VERY interesting.  I didn't know that ropes are supposed to be made with the strands under tension.  The math explains why:
Mathematicians prove that a three-stranded rope is always 68% the length of its component strands, regardless of the material from which it is made.

Ropes are generally useful things to have around, not least because the best of them tend not to stretch under tension.

The ancient Egyptians can attest to that and thoughtfully left pictures on the inside of various tombs showing how they made their ropes. It turns out they do it in a way that it essentially identical to way we do it today, with the strands under tension as they are wound.

Despite rope's obvious geometric properties, the art of rope making has been strangely neglected by mathematicians over the centuries. Today, Jakob Bohr and Kasper Olsen at the Technical University of Denmark put that right by proving the remarkable property that ropes cannot have more than a certain number of turns per unit length, a number which depends on the diameter of the component strands.

And that's just the start. They go on to show that a rope with a smaller number turns than this maximum will always twist in one direction or another under tension. So they call this maximum number of turns the zero-twist configuration...

The work also explains why ropes are best made with the strands under tension. The force causes the pitch angle to be less than ideal so that when the force is relaxed the rope 'relaxes' into the zero twist configuration, which cannot be further stretched under tension...
You learn something every day.
Credit. Explanatory text and image from MIT's Technology Review.

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