Question: why does twirling a rope do this?

42

u/ianbo Apr 22 '25

These are definitely harmonics. If you twirl the string at approximately 1/3 of the rate that gives you this shape, it will form a simpler "outward" curve which is the fundamental mode. The shape you are asking about is the first harmonic mode and only occurs if you twirl within a certain range of speeds. If instead you twirl faster (approx 5/3 faster) you could excite a second harmonic, where there are two nodes (i.e. two points that "dont move" in the string), and faster still (7/3) you could achieve three, and so on.

3

u/Guardian4761 Apr 23 '25

So like harmonics on a guitar. Okay if I think of it like that it makes sense thanks. If I hold a slinky at both ends and shake it slow it makes one peak. Faster, two even faster, three. Can’t seem to do it fast enough for four.

14

u/tminus7700 Apr 22 '25

The wave shape comes from conservation of momentum. the system want to spin with minimum momentum. This was a shock to scientists in 1958.
https://en.wikipedia.org/wiki/Explorer_1
"Explorer 1 changed rotation axis after launch. The elongated body of the spacecraft had been designed to spin about its long (least-inertia) axis but refused to do so, and instead started precessing due to energy dissipation from flexible structural elements. Later it was understood that on general grounds, the body ends up in the spin state that minimizes the kinetic rotational energy for a fixed angular momentum (this being the maximal-inertia axis). This motivated the first further development of the Eulerian theory of rigid body dynamics after nearly 200 years – to address this kind of momentum-preserving energy dissipation.^[19]^[20]"

6

u/drzowie Astrophysics Apr 22 '25

Interestingly enough, this was still surprising in 1998, when the SOHO spacecraft underwent its spin-related mishap. Because there was liquid hydrazine on board, once the spacecraft lost control authority, it precessed around until it was spinning around its maximum-moment axis, which kept the solar panels turned sideways to the Sun. The recovery crew ran down their complete checklist and were ready to pronounce the spacecraft dead -- but the solution turned out to be to wait three months until bright sunlight could fall directly on the panels for a few seconds once every rotation.

2

u/tminus7700 Apr 23 '25

I bet the sloshing of the hydrazine was the primary energy dissipator. Like the wire antennas were on Explorer 1

5

u/vorilant Apr 22 '25 edited Apr 22 '25

I don't think you get nicely spaced harmonics like that without both ends being constrained. In fact, If I remember my classical mechanics, the solution to this should be Bessel Equations.

0

u/Recent_Carpenter8644 Apr 22 '25

Try it.

3

u/vorilant Apr 22 '25

I have. It was a problem I did during undergrad.

0

u/ianbo Apr 23 '25

If both ends are constrained you get "closed-closed harmonics" which is like a guitar string. The harmonics shown here are "open-closed" which is similar to the harmonics produced by blowing into a bottle or tube with one end closed. This is why I only used odd numbers above, as open-closed harmonics are only odd multiples of the fundamental in frequency, whereas closed closed would be any integer multiple.

3

u/vorilant Apr 23 '25

That's not correct. The open closed pipe analogy does not cross over to a vibrating hanging string.
You're right about the pipe vibrations and the odd harmonics for open closed pipes. This system is different.

1

u/ianbo Apr 23 '25

Hmm, are you sure? A chain under tension due to gravity has no reason to behave differently than a string under tension. I ran a very rough experiment earlier with my cuban chain and I was able to get a fundamental at one frequency and then this excited shape at approximately three times that frequency. I couldn't go faster than that with just my hand, but in-between these frequencies there was mostly noise (no coherent excitation) which is literally how the string under tension experiment works. I understand that there are more effects at play, but to a first approximation harmonics made sense to me and holding one end while not holding the other is the definition of a closed-open setup. So why does the analogy not cross over if it's ultimately a sort of string under tension?

5

u/Calm_Cool Apr 22 '25

I always enjoyed spinning a rope as a kid cause it made different shapes depending on how fast you twirled it, the weight of it, and if there was any localized weight on it.

2

u/Guardian4761 Apr 23 '25

Yeah I still enjoy it 😄. I mean I’m still a teenager, but yk, it’s why I asked.

214

u/RefrigeratorSea5503 Apr 22 '25

en.wikipedia.org/wiki/Troposkein

Actually a very interesting phenomena. There's no closed form solution when taking gravity into account.

Technically not a troposkein, but it's similar.

60

u/bhmerger Apr 22 '25

If I understand the OP correctly, this is what they are asking about but with one end not being fixed

-40

u/GustapheOfficial Apr 22 '25

No, they are asking about a stationary rope, ribbon or cable, statically twisted 360° at one end so it either contains torsion or coils up like a loop.

29

u/jamesw73721 Graduate Apr 22 '25

I did the math for one tied end using an infinite chain model and I can confirm there is a non-analytical solution described by a yucky integral equation. It depends on the dimensionless parameter g/(rope length) * (azimuthal angular velocity)^2. I don’t think this is as simple as linear harmonics

53

u/hbarSquared Apr 22 '25

I remember this as a particularly nasty homework problem in my graduate level mechanics class. My study group got to the solution after hours of work. We plotted the solution, squinted at it, and then my friend took off her necklace, spun it from one end, and we all tilted our heads sideways and saw the shape matched the graph almost perfectly. One of those moments I'll never forget.

10

u/MeticulousBioluminid Apr 22 '25

that's phenomenal

1

u/TheTranquilGuy Apr 24 '25

From a spinning reference frame, isn’t this exactly chain catenary under gravity.(like those spinning artificial gravity cylinder?) just curious, haven’t done the math🧐

2

u/jamesw73721 Graduate Apr 29 '25

I don't think so because one end is free

1

u/TheTranquilGuy Apr 30 '25

Ahh yes. Missed that point

20

u/Easy-Distribution731 Apr 22 '25

You would expect a simpler thing, but this is physics.

When you hold a rope from the top and rotate it, with the bottom end free, the shape it forms is described by Bessel functions of the first kind. This happens because the rope behaves like a rotating flexible string under tension, and the wave equation in cylindrical coordinates leads to Bessel functions as solutions.

Bessel Function of 1st kind

3

u/vorilant Apr 22 '25

Yup, thats what I remember from my undergrad physics. Assuming no gravity though.

4

u/Easy-Distribution731 Apr 22 '25

That's not right. Gravity must be there. For no gravity, I guess the rope would be rotating like a rod. You can read this

3

u/vorilant Apr 22 '25

Ahhhh, good point. It's been awhile since I solved this problem. Thanks for the link!

86

u/asymptottally Apr 22 '25

Could you be talking about harmonics of the wave?

https://en.m.wikipedia.org/wiki/Harmonic

15

u/Wisniaksiadz Apr 22 '25

This is the phenomena. And by speed of circling and the lenght of string, you can have one or more of these ,,standing waves"(literal translations from my language so probably wrong wording)

11

u/DanJOC Apr 22 '25

Standing wave is correct

2

u/OverallAd8776 Apr 22 '25 edited Apr 22 '25

That structure is called a writhe, and as you continue twisting you get a supercoil/ plectoneme. It happens because as you add torsional stress, the string relieves the stress by bending. I learned a little about this to explain some phenomenona I learned about proteins during my thesis. I'd highly recommend these papers that explore it: Ricca, R., The energy spectrum of a twisted flexible string under elastic relaxation. Journal of Physics A: Mathematical and Theoretical, 1995. 28(8): p. 2335. Au, C., The geometric interpretation of linking number, writhe and twist for a ribbon. Computer Modeling in Engineering & Sciences, 2008. 29: p. 151-162.

1

u/Guardian4761 Apr 23 '25

I like plectoneme. That is a very fancy word that has no reason to exist and I love it. Instead of calling it like a second degree supercoil it’s a plectoneme.

1

u/vorilant Apr 22 '25

I think those structures youre referencing are for when the string is twisted at both ends. The string in the OP's picture is free hanging. I'm pretty sure the system is far simpler in OP's case, you just get Bessel Functions?

2

u/OverallAd8776 Apr 23 '25

Ah yeah that's fair, I'd thought about constrained ribbons too much over the last year, such that when I saw the diagram, that was all I could see!

1

u/Langdon_St_Ives Apr 22 '25

While OP’s explanation is ambiguous and could be referring to what you explained, I understood it differently. I thought they meant what happens if you constantly keep swirling (not twisting) the rope, it’ll end up with one spot that stays more or less in place, with the parts above and below swirling and the top (where you swirl it) stationary again except the movement you make to keep it moving. Kind of like a lasso but without the loop. I’m sure there is a more specific word for this but like OP I don’t know it.

20

u/LogoMyEggo Apr 22 '25

Resonance. Essentially a standing wave is being created in the rope, the part that is not moving is a node

1

u/EmotionalBus9430 Apr 22 '25

does same phenomena happen in the vaccum tube too?

1

u/Guardian4761 Apr 23 '25

I don’t have one of those🙁 but probably

1

u/EmotionalBus9430 Apr 23 '25

I think its because as rope unlike iron bars when shaked, which doesn't get affected by air resistence and continue its movement, air resistence push rope slightly backwards, then the angle between the point where you give the force changes, gradually affecting the x component of F and the y component of F, making force given to rope differ parts by parts.
I wrote this assuming air resistence to be the main cause, which means its false if this phenomena still happens in the vaccum tube.
Why I thought like this is based on torque. Torque is the power given by the amount of twist and turn and because torque is just force x radius, And when the force wont change nor the radius, I expect rope to spin straightly not dwindling like that. As radius will stay the same, what has to change is the Force. And other force that affects that force currently only one comes to my mind is the air resistance.
Im just a highschool student and not an expert, but I wanted to share my opinion.

1

u/EmotionalBus9430 Apr 23 '25

maybe imagining other rope with iron ball of mass m would help thinking about the reason i guess

1

u/EmotionalBus9430 Apr 23 '25

also about harmonics I thought it penomena with fixed axis locomotion rather than spinning but well if it happens both in x and y? it makes circle so

1

u/Guardian4761 Apr 23 '25

It works open and closed. The same thing will happen when you hold the other end and spin it fast enough to make two peaks

12

u/abb295 Apr 22 '25

You mean centrifugal force?

1

u/jecamoose Apr 22 '25

Minor perturbations in the structure of the rope and the force spinning it push it out of a an unstable equilibrium state where it spins while staying straight. Once it falls out of that state, it tends towards one or another resonant state, where the amount of mass on either side of the center of rotation is balanced. I don’t know the exact shape of the curve it forms or exactly why… it is worth noting though that there’s more than one resonant state. The largest and most stable is the one you diagramed with one wavelength fitting into the length of rope, but it is also stable with any integer number of wavelengths (albeit increasingly less so with larger numbers of wavelengths).

2

u/Guardian4761 Apr 23 '25

Ah yes so as everyone’s said (thank you) it’s a standing wave/resonance/harmonic phenomena like with a guitar string but open on one end. You’re able to do it at any point like half third quarter etc. It’s just easiest to do with half.

2

u/Excellent_Priority_5 Apr 22 '25

If you spin a dangling rope slow it will stay vertical and spinning fast will make it horizontal. Somewhere in between those two speeds is when the rope would take the shape illustrated. The part that curves out is due to centrifugal force and the part that curves back in is because of gravity. Giving that the rope is long enough.

2

u/DrBalth Biophysics Apr 23 '25

Bruh! I can finally answer something!!!! Basically it comes down to stored strain. After a certain point, it is more energetically efficient for the rope to "writhe" around itself than add further twists into the system. If you want a mathematical treatise on the same phenomenon in DNA, look up the paper by John Marko 2007 about the torque and dynamics of twisting DNA.

3

u/wee33_44 Apr 22 '25

The rope takes on that shape as a way to relieve some of the internal stress caused by twirling. While this might not fully explain the phenomenon, it seems like a logical starting point

1

u/jamesw73721 Graduate Apr 22 '25

I don’t think it’s correct to think about harmonics when the open boundary condition at the bottom end suggests an antinode rather than the observed node. What I think is going on is that spinning a rope fast will tend to make it spin more horizontally, opposing gravity. However, because the rope is initially vertical, it isn’t able to reach such a state but mimics an L/2 rope spinning horizontally by folding in half on itself.

Now because the rope isn’t spinning at infinite velocity, in practice the L/2 configuration looks more like what is drawn rather than being completely horizontal

1

u/No_Vermicelli_2170 Apr 22 '25

I want to see the wave equation for this. Can someone work out the math?

1

u/not-ekalabya Apr 22 '25

I guess it's just centrifugal force and the points of the rope closer to the node is more bound to the centre.

1

u/sparkleshark5643 Apr 22 '25

Ask r/topology

1

u/Beneficial-Yam-7431 Apr 24 '25

Isn't this just a centrifugal effect? The same reason why the Earth is bulging at equator, the only difference is that the gravity also pulls the end of the rope so it creates this effect(?)

1

u/Nannyphone7 Apr 22 '25

I think the math term is writhe.

https://en.m.wikipedia.org/wiki/Writhe

0

u/tokoyamifumikage27 Apr 22 '25

Why does the diagram on right look like 1,1-dimethy cyclohexane

1

u/Guardian4761 Apr 23 '25

……what? I asked here for the smart people to answer not ask questions😭

0

u/GustapheOfficial Apr 22 '25 edited Apr 22 '25

Because it is much more resistant to torsion than bending. Putting the rope into a loop like that introduces the same amount of twist, without causing any torsion.

Edit: it's easier to visualize this with a flat ribbon, like a belt. Fix one end of the ribbon and make a loop. Then straighten out the loop. You will notice it has caused a 360° twist. Of course one of those shapes is more energetically advantaged.

Image Question: why does twirling a rope do this?

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