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I am learning about a model of the human spine I saw and was doing a thought experiment with a ruler to try to understand the physics.

Suppose I have a plastic ruler that's not rigid but also not super easy to bend. Let it be totally straight. Now suppose I fix the bottom of the ruler and twist the top. The top twists fairly easily and when I let go it returns roughly to normal.

Now, let's suppose I contort the ruler so it forms a slight S shape along it's long axis. The idea is, now instead of being straight, the ruler has a node in it, kind of like a plane wave. Then I again hold the base of the ruler and I twist the top. This time a much stronger force arises that causes the bottom of the ruler to want to follow the top.

Why does the bottom of the ruler now experience more force when there is a node? Does it store potential energy? What kind of physics model would be similar to this so I could learn about the equations?

EDIT:

For this question

Why do we bend a book to keep it straight?

I think the first answer is excellent in that it addresses how intrinsic and extrinsic curvature are at play here, which is probably the meat of the mathematics I would imagine. However, I would like to understand how potential energy gets stored in the twist. I would also like an answer that addressed this in terms of nodes. A true spine has more than one curve and I would like the answer to also address how increasing the number of nodes affects the impact of twisting one of the endpoints of the plane while the other is curved. I don't feel that Gaussian curvature sufficiently explains that alone. I also think that, the second answer, while it does include nodes in the diagram, it doesn't discuss them and instead discuss corrugation, which I don't think has an analogue in this case.

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  • $\begingroup$ Does this answer your question? Why do we bend a book to keep it straight? $\endgroup$ – Sandejo Jul 14 '20 at 0:15
  • $\begingroup$ That answer is a good start. However, I would like to understand how potential energy gets stored in the twist. I would also like an answer that addressed this in terms of nodes. A true spine has more than one curve and I would like the answer to also address how increasing the number of nodes affects the impact of twisting one of the endpoints of the plane while the other is curved. I don't feel that Gaussian curvature sufficiently explains that alone. $\endgroup$ – Stan Shunpike Jul 14 '20 at 0:38
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I agree with the commenter above that Why do we bend a book to keep it straight? has a very good explanation with detailed background. The short version is that when you twist a straight ruler, it bends but does not stretch. However, in order to twist a curved ruler, you need to stretch or compress some parts of the surface.

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