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In this DIY project, the Möbius strip is used to make a spill-proof coffee cup carrier. The author uses a Möbius strip as the handle of this carrier and says

If you attach a Möbius strip to an object, say a bowling pin, and swing it around your head, the twisted strip resists kinking and curling, proving itself a superior attachment to a nontwisted one.

Could someone provide a proof or a physical explanation of why the quoted phrase is true?

shamisen
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    The explanation follows from system dynamics: let $S$ be a set of people called "shamsters", $G$ a "gullible" set, $I$ the class of impressive sounding but groundless statements and $M$ the quantity of money held by $G$, then the rate equation $\dot{I} \propto -\dot{M}$, where $\dot{I}$ is the rate of transmission of statements from $I$ to $G$. More seriously, if a wide strip has twists in it, rather than being held taut between two axes, then it can bend easily in all planes, whereas if it is taut and flat it can't bend easily in a direction in its plane (similar to why a thin, wide beam ... – Selene Routley Nov 24 '14 at 05:44
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    ... can bend much more readily in one plane than the other). But if this is the mechanism, then a strip with a full twist in it, one half in each of the vertically hanging strips, would work better than a Möbius strip. I suspect it's a Möbius strip rather than a full twist loop for marketting purposes. – Selene Routley Nov 24 '14 at 05:46

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I suspect that the optimal arrangement is not in fact a Möbius strip but a loop with a full twist in it, and with half of the twist in each of the carrier's vertical sections, like I have plotted with Mathematica below.

Optimal Strip

Even better would probably be several twists in each vertical section.

The principle of working is as follows. If the loop is untwisted, as below

Untwisted Strip

then it can swing fairly freely if the deformed loop stays in the $X\wedge Y$ plane, but it cannot bend easily out of this plane. This is simply because, if we think of the cross section of the strip of leather, as below:

Strip cross section

then its area moment of inertia $I_{YY}$ about the $Y$ axis is greatly more than the area moment of inertia $I_{XX}$ about the $X$ axis. The stiffness of a beam in Euler-Bernoulli and Timoshenko beam theory is $E\,I$ where $E$ is the Young's modulus of the material in question and $I$ the area moment of inertia about the neutral axis of bending.

So, if there is a twist in the beam, then there is always some position along the beam where the relevant area moment of inertia is very small. Try it yourself with a strip of paper with and without twists. You'll find that the twisted one can bend freely in all directions, but the flat one will have a preferred bending plane.

So the question now arises as to why the sold device is a Möbius strip rather than a better design. I suspect this is likely for marketting purposes, not physics. I'm guessing the marketting people think that it sounds slicker to say the design is grounded on the Möbius strip rather than a twisted loop.

Selene Routley
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