# Why does a chain or rope move the way it does when suspended and rotated on a vertical axis?

I have always been interested in why objects like chains, ropes, etc. move the way they do when "rotated" around a vertical axis while being held only where it is suspended. It forms a shape if you will, resemblant of a "C" or a wave depending on the length. I am curious to know what laws and effects of physics are at play.

• Do you mean holding the rope by two ends and rotating it? Do you hold one still and rotate the other or do you rotate both simultaneously?
– user137661
Oct 30, 2019 at 19:58
• Only one end, and allowing it dangle freely, then rotating the end held around a vertical axis. Oct 30, 2019 at 20:50
• I tried to attach an image of the effect in action in the original post, but the site told me the img was too large, sorry Oct 30, 2019 at 20:51
• You should edit your question because it says "horizontal axis". As for the image you could use the "snipping tool" (if you use windows) to get it. Saving the image directly from there should make it small enough to post.
– user137661
Oct 30, 2019 at 20:53
• Duely noted, I managed to add one, just edited it to a smaller size, and used a screenshot of it. Greatly reduced the file size, its a bit blurry as I used my phone camera. Didnt think to use a video and select a frame that wasn't as bad until now. I might change it if this doesn't work for you. Oct 30, 2019 at 20:59

Not a real answer but I'd like to note the model used while studying these phenomena. The shape of rotating chains is modeled by the Bessel functions. Precisely speaking, the radial displacement function $$x$$ is given as:

$$r(x) = J_o ( \frac{2 \omega \sqrt{x}}{\sqrt{g}})$$

Where, $$\omega_o$$ is the angular frequency at which rope is spun, $$g$$ is gravitational constant and $$x$$ is the distance from the bottom of the chain

Where,

$$J_o(x) = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k}}{(k!)^2}$$

A better presentation is done by professor Ghrist in this video

• The omega is up to the person. I theorize there is a natural human response to alter that omega until there is a C shape by the bessel Aug 17, 2021 at 4:37
• Great answer! but your first equation can't be an equality (lhs is a distance and rhs is nondimentional). Furthermore, I think you mean that the distance is $x$ not $\sqrt{x}$. Aug 17, 2021 at 4:59
• I suppose the units got put outside, if it were inside, the series would violate the principle of only things of similar dimensions being added Aug 17, 2021 at 5:14

If i understand correctly you spin the chain on an vertical axis with angular speed $$w$$. The 3-dimensional shape of the spinning chain is rather complex, but a 2-dimensional projection of this shape on a vertical plane resembles a standing wave. Where the nodes are the points where the chain is kept in the middle. If you increase the length of the chain you will have to adjust the angular speed, but more nodes will appear. And i guess that these nodes have the same relation with the length of the chain and $$w$$ as a simple 2-dimensional standing wave

• Yes, the chain is only held at one point, and my hand quickly rotates around the vertical axis. Where most expect the chain to simply swing out and spin like a fan, it forms that wave pattern. Greatly appreciate the answers, this question has been in the back of my mind for almost 21 years (i first realised this phenomenon at the age of 5) none of my science or physics teachers have ever been able to offer a definitive answer. I'll continue to pursue a better understanding of what's at play. Nov 4, 2019 at 18:59
• Yes node. 👍🏻 Great. Why does it seem so often that it acts like one node, ie a half of a wavelength? Do you think it is the natural response of the person adjusting their omega to make it that way? Aug 17, 2021 at 4:33

I think it is hard to study your problem, however, I try to justify why a chain/rope hanging from the ceiling makes a $$C$$ shape as it dangles freely. For a better perception, simulate each ring of the chain as a separate tiny ball which tends to move on a circular frictionless surface. Due to different radii of these surfaces, the period of oscillation would be different for each of the balls. $$(T\approx 2\pi \sqrt{r/g})$$ That is, the farther balls need more time to reach the lower part of their assigned circles because farther surfaces have greater radii. This difference in periods causes the distant balls (rings of the chain) from the center (pivot) to lag behind the nearer ones making a C shape.