# Can you tie a knot in a frictionless rope?

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## Main Question or Discussion Point

A question came up on a maths Q+A site about the possibility of wrapping a string around a sphere.
http://mathoverflow.net/questions/8091/is-it-possible-to-capture-a-sphere-in-a-knot

The obvious engineering solution is to just make a net and put the sphere in, and there are a bunch of photos of fishing floats where this is done.
But if the rope is frictionless you can always slide a knot along to make a hole big enough for the sphere to fall through

The answers descend into a lot of maths-speak I can't follow - but there is an interesting general question, can you tie any sort of knot in a frictionless rope?

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That depends, how well can you hold a frictionless rope?

Why waste time on theory? Just try it and see what happens.

CRGreathouse
Homework Helper
Why waste time on theory? Just try it and see what happens.
I'd be happy to try, if you'll lend me a frictionless rope.

That depends, how well can you hold a frictionless rope?
That was my thought. If you can hold onto the rope a lark's head or similar knot would suffice. If you can't hold onto it, I can't imagine any knot working.

In my high school physics book you can. We deal with frictionless, massless stuff all the time. It's really convenient.
If the cartoons in my book can do it, why can't you?

Pengwuino
Gold Member
I don't really think you can. At some point, you have to create a tension to tighten it and the only way to do that is eventually using friction. That is, unless you cheat and... i suppose "weld" the rope on both ends to something that does have friction to create a method of making tension :rofl:

Gold Member
I don't understand how one would even be able to pick up the rope without friction coming into play. One might be able to manipulate the rope by pushing it around while it's setting on the ground, but I don't think you'd be able to grip it at all.

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Borek
Mentor
Is the rope free-ended, or is it a loop?

There was some trick with clock, stick and piece of rope that was based on the friction, but I can't remember details.

Is the rope free-ended, or is it a loop?

There was some trick with clock, stick and piece of rope that was based on the friction, but I can't remember details.
It has to be a finite length of your choosing, and if the ends touch eachother, they become welded together. The rope does not stretch, the sphere and the rope experience no friction.

At least that is what the problem states.

russ_watters
Mentor
I don't really think you can. At some point, you have to create a tension to tighten it and the only way to do that is eventually using friction.
A great many types of ropes can be fused to what is holding them. In addition, a knob on the end of the rope will keep it from slipping through the knot.

These are, of course, not specifically prohibited by the question, but are probably against the spirit of it.

Homework Helper
I think it's the normal physics-lab frictionless rope, so you can hold it to tie the knot - and according to the question you can join the ends into a loop.

But from the point of view of making a bag, with a knot in a frictionless rope - surely you can always push the knot along to the end where it will disapear?

Well, according to a paper from Eilsenhause(sp?) university in Germany, a completely frictionless rope can maintain a knot IF the knot is tied using a "seven-fold" method, whatever that is.
Came across that several weeks ago, will try to get the link if anyone is interested.

Moonbear
Staff Emeritus
Gold Member
How do you even get a frictionless rope? Wouldn't it just be frictionless strands of fibers? The strands wouldn't all stay intertwined to form a rope without some friction, right?

Darn, I don't have any beer in the fridge...this sort of conversation really needs to be held after a few beers.

How do you even get a frictionless rope? Wouldn't it just be frictionless strands of fibers? The strands wouldn't all stay intertwined to form a rope without some friction, right?
Picky, picky, picky...
I will now crack open another beer and bow my head in utter contempt for my intelligence.
BTW, I have 2 beers. Want one?

Moonbear
Staff Emeritus
Gold Member
Picky, picky, picky...
I will now crack open another beer and bow my head in utter contempt for my intelligence.
BTW, I have 2 beers. Want one?
Thanks, I will! It's been a long week.

Know what you mean. Sometimes I have long days.

Anyway, I recall buying some synthetic rope that was a nightmare.
It was too slippery for one.
Then when you cut it, the ends frayed immediately.
And, if you held it in the middle with both hands(separated) and pushed toward each other, the rope would start to "bulge"

I doubt I could have made very secure knots. I threw it out.

Is this now the general rope and cord discussion thread?

I have around 1-2k yards of paracord hanging around my shack, my car, and my house. 550 Paracord is the greatest thing since sliced bread.

But if two things are frictionless, how does one tie a knot when the knot wont hold?

BobG
Homework Helper
The answers descend into a lot of maths-speak I can't follow - but there is an interesting general question, can you tie any sort of knot in a frictionless rope?
No. The secret of a knot is that, when you try to pull one end of the rope out, the pressure (and friction) increases proportionally on another piece of the rope, preventing it from coming out.

Russ's idea of tying another knot would seem to work. A knot that would be pushed instead of pulled when it runs into the knot in question. It's hard to push a rope.

Except, even though the knots are pushing against each other, the rope itself would still be pulling through the second knot.

Pythagorean
Gold Member
Yes, as long as it's not a friction not. It also depend on the ductility of the rope. If it has no restoring force, it will stay in a knot. If the rope has a restoring force ( it's stiff and wants to straighten out) then a friction knot will come undone.

Others here have alluded to non-friction knots that rely on electron repulsion (the force keeping you from falling through the floor)

-ex commercial fishermen

Is it even like, physically possible to have a frictionless rope?

This problem sounds a lot like that of the matchmakers of the community I came from. Untying the knot is where finding friction and tension not a problem.
Perhaps you should look at the problem backwards.

I believe that in math-speak, a generic knot is a smooth, closed curve embedded in three dimensional space. In manipulating the knot, it is not allowed to pass through itself.

As far as mgb_phys' question goes, you can apply normal forces to a knot to change it shape.

But what happens if you assume it has non vanishing uniform mass density, subjected to a uniform gravitational field, say. Can it be held up without oozing out when constrained by a finite number of normal forces, or does it need to be craddled?

A closed loop can be hung over a hook without going anywhere, but what about a finite length, open curve? It can be balanced over various hooks. It seems it would be stable if hung in just the right places, but unstable due to any perturbation. I think a small imbalance would cause the whole thing slither out of it's constraints.

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DaveC426913
Gold Member
I would point out that a frictionless rope is not enough to prevent knotability.

I posit a frictionless rope that is quite stiff; so stiff that it doesn't make sharp bends. It should be possible to tie it in a knot that is held, not by friction, but by simple geometry i.e. you cannot fold a loop small enough to go through another loop to untie it.

Borek
Mentor
Good point, I like it

No. The secret of a knot is that, when you try to pull one end of the rope out, the pressure (and friction) increases proportionally on another piece of the rope, preventing it from coming out.

Russ's idea of tying another knot would seem to work. A knot that would be pushed instead of pulled when it runs into the knot in question. It's hard to push a rope.

Except, even though the knots are pushing against each other, the rope itself would still be pulling through the second knot.
Pushing would still require friction - if not at the point of contact, then elsewhere (or the entire rope would move).