Will a knot tied in a hanging, frictionless rope slip out under the force of gravity?Do the composition of...
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Will a knot tied in a hanging, frictionless rope slip out under the force of gravity?
Do the composition of two Knots always yield a distinct knot (ignoring orientation)?Spectra of periodic Schrödinger equationsAre there enough knots to cover $mathbb{R}^3$?What curved ramp transports a ball from (1,1) to (0,0) most quickly, under the acceleration of gravity, with no friction or air resistance?Unit and dimension of anglesFind the energy for which the motion under the central force is circularCyclic coverings of knotsDoes the monoid of knots form a ring?Rutherford's $alpha$-Particle Scattering Experiment
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I am overall just curious about what keeps knots where they are in a rope. Another related question you might be able to answer is: What happens if you tie a bowline on the bight in a frictionless rope and pulled on it, would it then slide together? I doubt it, but I'm not quite sure.
For clarity, here's the bowline on a bight knot: https://www.netknots.com/rope_knots/bowline-on-a-bight/
mathematical-physics knot-theory
asked Jan 25 '14 at 7:36
KainuiKainui
504516
$endgroup$
add a comment |
$begingroup$
I am overall just curious about what keeps knots where they are in a rope. Another related question you might be able to answer is: What happens if you tie a bowline on the bight in a frictionless rope and pulled on it, would it then slide together? I doubt it, but I'm not quite sure.
For clarity, here's the bowline on a bight knot: https://www.netknots.com/rope_knots/bowline-on-a-bight/
mathematical-physics knot-theory
asked Jan 25 '14 at 7:36
KainuiKainui
504516
$endgroup$
$begingroup$
What keeps knots in a rope, either former with the rope itself or with another rope, is imo friction. A knot in a completely frictionless rope won't stand the slightest pull on one of its ends...
$endgroup$
– DonAntonio
Jan 25 '14 at 9:13
add a comment |
$begingroup$
I am overall just curious about what keeps knots where they are in a rope. Another related question you might be able to answer is: What happens if you tie a bowline on the bight in a frictionless rope and pulled on it, would it then slide together? I doubt it, but I'm not quite sure.
For clarity, here's the bowline on a bight knot: https://www.netknots.com/rope_knots/bowline-on-a-bight/
mathematical-physics knot-theory
asked Jan 25 '14 at 7:36
KainuiKainui
504516
$endgroup$
I am overall just curious about what keeps knots where they are in a rope. Another related question you might be able to answer is: What happens if you tie a bowline on the bight in a frictionless rope and pulled on it, would it then slide together? I doubt it, but I'm not quite sure.
For clarity, here's the bowline on a bight knot: https://www.netknots.com/rope_knots/bowline-on-a-bight/
mathematical-physics knot-theory
mathematical-physics knot-theory
asked Jan 25 '14 at 7:36
KainuiKainui
504516
asked Jan 25 '14 at 7:36
KainuiKainui
504516
asked Jan 25 '14 at 7:36
KainuiKainui
504516
asked Jan 25 '14 at 7:36
KainuiKainui
504516
asked Jan 25 '14 at 7:36
KainuiKainui
504516
504516
$begingroup$
What keeps knots in a rope, either former with the rope itself or with another rope, is imo friction. A knot in a completely frictionless rope won't stand the slightest pull on one of its ends...
$endgroup$
– DonAntonio
Jan 25 '14 at 9:13
add a comment |
$begingroup$
What keeps knots in a rope, either former with the rope itself or with another rope, is imo friction. A knot in a completely frictionless rope won't stand the slightest pull on one of its ends...
$endgroup$
– DonAntonio
Jan 25 '14 at 9:13
$begingroup$
What keeps knots in a rope, either former with the rope itself or with another rope, is imo friction. A knot in a completely frictionless rope won't stand the slightest pull on one of its ends...
$endgroup$
– DonAntonio
Jan 25 '14 at 9:13
$begingroup$
What keeps knots in a rope, either former with the rope itself or with another rope, is imo friction. A knot in a completely frictionless rope won't stand the slightest pull on one of its ends...
$endgroup$
– DonAntonio
Jan 25 '14 at 9:13
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let's consider the following (simplified) free body diagram with no friction force:
Since a rope is a single body, the force $vec F$ you use for pulling the rope to tighten it is offset by the tension $vec T$. If we suppose a frictionless rope and the situation in which you are not acting over the rope, it turns out that the entire $vec F$ is done by the gravity force and, in some moment, the rope reaches an equilibrium, that is, the vector sum of all the forces acting over it is zero. What happened? It is true that there was not friction force (apparently), but if you look at the yellow region, it is necessarily performing a force upward since the green region is applying a force downward (its weight).
Finally, the rope does not slide since the pink region is applying a force against the section of rope which contains the green region (due to $vec T$), since the green region is doing the same against the region of rope which contains the pink region (due to $vec F$), and since the yellow region is supporting the weight of the green region.
More: I would say that it doesn't make sense to consider a frictionless rope, at least in this situation, because as I have pointed out, we have three contact points... but what does that "contact points" really mean? Friction force is an electromagnetic phenomenon in which the particles repel to each other, and macroscopically we can only see a kind of resistance between two bodies when they contact in some way (i.e. when rubbing to each other).
It's like to ask: if we drop an apple from the air and we consider no friction forces between it and the ground, will the apple reach the center of the Earth? Well, the apple has atoms, the Earth has atoms, so the electromagnetic force must exist between them, otherwise the ground would not "stop" the apple when they were at the same level, and the apple would keep on falling until it reached the center of the Earth due to the gravity force it experiments (the unique one).
edited Mar 18 at 20:13
Glorfindel
3,41581830
answered Jan 25 '14 at 9:39
Daniel Muñoz ParsapoormoghadamDaniel Muñoz Parsapoormoghadam
22316
$endgroup$
$begingroup$
I think I am mostly convinced. However, I'm not sure if I quite agree with the last part. The idea of frictionless surfaces always comes up with regard to things rubbing against each other and I'm not so sure that this is really fair to dismiss like this. The reason being, suppose you make the rope heavier and heavier and oilier and oilier with actual, physical materials, you might be able to do something that approaches what I describe. The problem I have with the body diagram is that although the forces cancel out, I still feel like they are almost "magically" suspended there.
$endgroup$
– Kainui
Jan 25 '14 at 10:13
$begingroup$
I'll think about your answer and see if I can't try to glean something from it that convinces my intuition, even though logically it seems fairly acceptable. Thanks. =)
$endgroup$
– Kainui
Jan 25 '14 at 10:17
$begingroup$
You are right, but observe that although we consider frictionless surfaces in the sense you are talking about (a force that is perpendicular to the motion), and the rope slide at the beginning, it will reach an equilibrum anyway due to the yellow region: it is stopping the motion of the rope because "it's there", blocking the advance of the green region (which was sliding downward before the equilibrium).
$endgroup$
– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:29
$begingroup$
In a way, I guess it's like trying to push something through a hole, but the hole gets smaller the harder you push until you can't possibly squeeze through it anymore. I think that makes sense now.
$endgroup$
– Kainui
Jan 25 '14 at 10:39
$begingroup$
Yeah, that's a good analogy :)
$endgroup$
– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:48
add a comment |
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1 Answer
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1 Answer
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oldest
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$begingroup$
Let's consider the following (simplified) free body diagram with no friction force:
Since a rope is a single body, the force $vec F$ you use for pulling the rope to tighten it is offset by the tension $vec T$. If we suppose a frictionless rope and the situation in which you are not acting over the rope, it turns out that the entire $vec F$ is done by the gravity force and, in some moment, the rope reaches an equilibrium, that is, the vector sum of all the forces acting over it is zero. What happened? It is true that there was not friction force (apparently), but if you look at the yellow region, it is necessarily performing a force upward since the green region is applying a force downward (its weight).
Finally, the rope does not slide since the pink region is applying a force against the section of rope which contains the green region (due to $vec T$), since the green region is doing the same against the region of rope which contains the pink region (due to $vec F$), and since the yellow region is supporting the weight of the green region.
More: I would say that it doesn't make sense to consider a frictionless rope, at least in this situation, because as I have pointed out, we have three contact points... but what does that "contact points" really mean? Friction force is an electromagnetic phenomenon in which the particles repel to each other, and macroscopically we can only see a kind of resistance between two bodies when they contact in some way (i.e. when rubbing to each other).
It's like to ask: if we drop an apple from the air and we consider no friction forces between it and the ground, will the apple reach the center of the Earth? Well, the apple has atoms, the Earth has atoms, so the electromagnetic force must exist between them, otherwise the ground would not "stop" the apple when they were at the same level, and the apple would keep on falling until it reached the center of the Earth due to the gravity force it experiments (the unique one).
edited Mar 18 at 20:13
Glorfindel
3,41581830
answered Jan 25 '14 at 9:39
Daniel Muñoz ParsapoormoghadamDaniel Muñoz Parsapoormoghadam
22316
$endgroup$
$begingroup$
I think I am mostly convinced. However, I'm not sure if I quite agree with the last part. The idea of frictionless surfaces always comes up with regard to things rubbing against each other and I'm not so sure that this is really fair to dismiss like this. The reason being, suppose you make the rope heavier and heavier and oilier and oilier with actual, physical materials, you might be able to do something that approaches what I describe. The problem I have with the body diagram is that although the forces cancel out, I still feel like they are almost "magically" suspended there.
$endgroup$
– Kainui
Jan 25 '14 at 10:13
$begingroup$
I'll think about your answer and see if I can't try to glean something from it that convinces my intuition, even though logically it seems fairly acceptable. Thanks. =)
$endgroup$
– Kainui
Jan 25 '14 at 10:17
$begingroup$
You are right, but observe that although we consider frictionless surfaces in the sense you are talking about (a force that is perpendicular to the motion), and the rope slide at the beginning, it will reach an equilibrum anyway due to the yellow region: it is stopping the motion of the rope because "it's there", blocking the advance of the green region (which was sliding downward before the equilibrium).
$endgroup$
– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:29
$begingroup$
In a way, I guess it's like trying to push something through a hole, but the hole gets smaller the harder you push until you can't possibly squeeze through it anymore. I think that makes sense now.
$endgroup$
– Kainui
Jan 25 '14 at 10:39
$begingroup$
Yeah, that's a good analogy :)
$endgroup$
– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:48
add a comment |
$begingroup$
Let's consider the following (simplified) free body diagram with no friction force:
Since a rope is a single body, the force $vec F$ you use for pulling the rope to tighten it is offset by the tension $vec T$. If we suppose a frictionless rope and the situation in which you are not acting over the rope, it turns out that the entire $vec F$ is done by the gravity force and, in some moment, the rope reaches an equilibrium, that is, the vector sum of all the forces acting over it is zero. What happened? It is true that there was not friction force (apparently), but if you look at the yellow region, it is necessarily performing a force upward since the green region is applying a force downward (its weight).
Finally, the rope does not slide since the pink region is applying a force against the section of rope which contains the green region (due to $vec T$), since the green region is doing the same against the region of rope which contains the pink region (due to $vec F$), and since the yellow region is supporting the weight of the green region.
More: I would say that it doesn't make sense to consider a frictionless rope, at least in this situation, because as I have pointed out, we have three contact points... but what does that "contact points" really mean? Friction force is an electromagnetic phenomenon in which the particles repel to each other, and macroscopically we can only see a kind of resistance between two bodies when they contact in some way (i.e. when rubbing to each other).
It's like to ask: if we drop an apple from the air and we consider no friction forces between it and the ground, will the apple reach the center of the Earth? Well, the apple has atoms, the Earth has atoms, so the electromagnetic force must exist between them, otherwise the ground would not "stop" the apple when they were at the same level, and the apple would keep on falling until it reached the center of the Earth due to the gravity force it experiments (the unique one).
edited Mar 18 at 20:13
Glorfindel
3,41581830
answered Jan 25 '14 at 9:39
Daniel Muñoz ParsapoormoghadamDaniel Muñoz Parsapoormoghadam
22316
$endgroup$
$begingroup$
I think I am mostly convinced. However, I'm not sure if I quite agree with the last part. The idea of frictionless surfaces always comes up with regard to things rubbing against each other and I'm not so sure that this is really fair to dismiss like this. The reason being, suppose you make the rope heavier and heavier and oilier and oilier with actual, physical materials, you might be able to do something that approaches what I describe. The problem I have with the body diagram is that although the forces cancel out, I still feel like they are almost "magically" suspended there.
$endgroup$
– Kainui
Jan 25 '14 at 10:13
$begingroup$
I'll think about your answer and see if I can't try to glean something from it that convinces my intuition, even though logically it seems fairly acceptable. Thanks. =)
$endgroup$
– Kainui
Jan 25 '14 at 10:17
$begingroup$
You are right, but observe that although we consider frictionless surfaces in the sense you are talking about (a force that is perpendicular to the motion), and the rope slide at the beginning, it will reach an equilibrum anyway due to the yellow region: it is stopping the motion of the rope because "it's there", blocking the advance of the green region (which was sliding downward before the equilibrium).
$endgroup$
– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:29
$begingroup$
In a way, I guess it's like trying to push something through a hole, but the hole gets smaller the harder you push until you can't possibly squeeze through it anymore. I think that makes sense now.
$endgroup$
– Kainui
Jan 25 '14 at 10:39
$begingroup$
Yeah, that's a good analogy :)
$endgroup$
– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:48
add a comment |
$begingroup$
Let's consider the following (simplified) free body diagram with no friction force:
Since a rope is a single body, the force $vec F$ you use for pulling the rope to tighten it is offset by the tension $vec T$. If we suppose a frictionless rope and the situation in which you are not acting over the rope, it turns out that the entire $vec F$ is done by the gravity force and, in some moment, the rope reaches an equilibrium, that is, the vector sum of all the forces acting over it is zero. What happened? It is true that there was not friction force (apparently), but if you look at the yellow region, it is necessarily performing a force upward since the green region is applying a force downward (its weight).
Finally, the rope does not slide since the pink region is applying a force against the section of rope which contains the green region (due to $vec T$), since the green region is doing the same against the region of rope which contains the pink region (due to $vec F$), and since the yellow region is supporting the weight of the green region.
More: I would say that it doesn't make sense to consider a frictionless rope, at least in this situation, because as I have pointed out, we have three contact points... but what does that "contact points" really mean? Friction force is an electromagnetic phenomenon in which the particles repel to each other, and macroscopically we can only see a kind of resistance between two bodies when they contact in some way (i.e. when rubbing to each other).
It's like to ask: if we drop an apple from the air and we consider no friction forces between it and the ground, will the apple reach the center of the Earth? Well, the apple has atoms, the Earth has atoms, so the electromagnetic force must exist between them, otherwise the ground would not "stop" the apple when they were at the same level, and the apple would keep on falling until it reached the center of the Earth due to the gravity force it experiments (the unique one).
edited Mar 18 at 20:13
Glorfindel
3,41581830
answered Jan 25 '14 at 9:39
Daniel Muñoz ParsapoormoghadamDaniel Muñoz Parsapoormoghadam
22316
$endgroup$
Let's consider the following (simplified) free body diagram with no friction force:
Since a rope is a single body, the force $vec F$ you use for pulling the rope to tighten it is offset by the tension $vec T$. If we suppose a frictionless rope and the situation in which you are not acting over the rope, it turns out that the entire $vec F$ is done by the gravity force and, in some moment, the rope reaches an equilibrium, that is, the vector sum of all the forces acting over it is zero. What happened? It is true that there was not friction force (apparently), but if you look at the yellow region, it is necessarily performing a force upward since the green region is applying a force downward (its weight).
Finally, the rope does not slide since the pink region is applying a force against the section of rope which contains the green region (due to $vec T$), since the green region is doing the same against the region of rope which contains the pink region (due to $vec F$), and since the yellow region is supporting the weight of the green region.
More: I would say that it doesn't make sense to consider a frictionless rope, at least in this situation, because as I have pointed out, we have three contact points... but what does that "contact points" really mean? Friction force is an electromagnetic phenomenon in which the particles repel to each other, and macroscopically we can only see a kind of resistance between two bodies when they contact in some way (i.e. when rubbing to each other).
It's like to ask: if we drop an apple from the air and we consider no friction forces between it and the ground, will the apple reach the center of the Earth? Well, the apple has atoms, the Earth has atoms, so the electromagnetic force must exist between them, otherwise the ground would not "stop" the apple when they were at the same level, and the apple would keep on falling until it reached the center of the Earth due to the gravity force it experiments (the unique one).
edited Mar 18 at 20:13
Glorfindel
3,41581830
answered Jan 25 '14 at 9:39
Daniel Muñoz ParsapoormoghadamDaniel Muñoz Parsapoormoghadam
22316
edited Mar 18 at 20:13
Glorfindel
3,41581830
edited Mar 18 at 20:13
Glorfindel
3,41581830
edited Mar 18 at 20:13
Glorfindel
3,41581830
3,41581830
answered Jan 25 '14 at 9:39
Daniel Muñoz ParsapoormoghadamDaniel Muñoz Parsapoormoghadam
22316
answered Jan 25 '14 at 9:39
Daniel Muñoz ParsapoormoghadamDaniel Muñoz Parsapoormoghadam
22316
answered Jan 25 '14 at 9:39
Daniel Muñoz ParsapoormoghadamDaniel Muñoz Parsapoormoghadam
22316
22316
$begingroup$
I think I am mostly convinced. However, I'm not sure if I quite agree with the last part. The idea of frictionless surfaces always comes up with regard to things rubbing against each other and I'm not so sure that this is really fair to dismiss like this. The reason being, suppose you make the rope heavier and heavier and oilier and oilier with actual, physical materials, you might be able to do something that approaches what I describe. The problem I have with the body diagram is that although the forces cancel out, I still feel like they are almost "magically" suspended there.
$endgroup$
– Kainui
Jan 25 '14 at 10:13
$begingroup$
I'll think about your answer and see if I can't try to glean something from it that convinces my intuition, even though logically it seems fairly acceptable. Thanks. =)
$endgroup$
– Kainui
Jan 25 '14 at 10:17
$begingroup$
You are right, but observe that although we consider frictionless surfaces in the sense you are talking about (a force that is perpendicular to the motion), and the rope slide at the beginning, it will reach an equilibrum anyway due to the yellow region: it is stopping the motion of the rope because "it's there", blocking the advance of the green region (which was sliding downward before the equilibrium).
$endgroup$
– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:29
$begingroup$
In a way, I guess it's like trying to push something through a hole, but the hole gets smaller the harder you push until you can't possibly squeeze through it anymore. I think that makes sense now.
$endgroup$
– Kainui
Jan 25 '14 at 10:39
$begingroup$
Yeah, that's a good analogy :)
$endgroup$
– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:48
add a comment |
$begingroup$
I think I am mostly convinced. However, I'm not sure if I quite agree with the last part. The idea of frictionless surfaces always comes up with regard to things rubbing against each other and I'm not so sure that this is really fair to dismiss like this. The reason being, suppose you make the rope heavier and heavier and oilier and oilier with actual, physical materials, you might be able to do something that approaches what I describe. The problem I have with the body diagram is that although the forces cancel out, I still feel like they are almost "magically" suspended there.
$endgroup$
– Kainui
Jan 25 '14 at 10:13
$begingroup$
I'll think about your answer and see if I can't try to glean something from it that convinces my intuition, even though logically it seems fairly acceptable. Thanks. =)
$endgroup$
– Kainui
Jan 25 '14 at 10:17
$begingroup$
You are right, but observe that although we consider frictionless surfaces in the sense you are talking about (a force that is perpendicular to the motion), and the rope slide at the beginning, it will reach an equilibrum anyway due to the yellow region: it is stopping the motion of the rope because "it's there", blocking the advance of the green region (which was sliding downward before the equilibrium).
$endgroup$
– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:29
$begingroup$
In a way, I guess it's like trying to push something through a hole, but the hole gets smaller the harder you push until you can't possibly squeeze through it anymore. I think that makes sense now.
$endgroup$
– Kainui
Jan 25 '14 at 10:39
$begingroup$
Yeah, that's a good analogy :)
$endgroup$
– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:48
$begingroup$
I think I am mostly convinced. However, I'm not sure if I quite agree with the last part. The idea of frictionless surfaces always comes up with regard to things rubbing against each other and I'm not so sure that this is really fair to dismiss like this. The reason being, suppose you make the rope heavier and heavier and oilier and oilier with actual, physical materials, you might be able to do something that approaches what I describe. The problem I have with the body diagram is that although the forces cancel out, I still feel like they are almost "magically" suspended there.
$endgroup$
– Kainui
Jan 25 '14 at 10:13
$begingroup$
I think I am mostly convinced. However, I'm not sure if I quite agree with the last part. The idea of frictionless surfaces always comes up with regard to things rubbing against each other and I'm not so sure that this is really fair to dismiss like this. The reason being, suppose you make the rope heavier and heavier and oilier and oilier with actual, physical materials, you might be able to do something that approaches what I describe. The problem I have with the body diagram is that although the forces cancel out, I still feel like they are almost "magically" suspended there.
$endgroup$
– Kainui
Jan 25 '14 at 10:13
$begingroup$
I'll think about your answer and see if I can't try to glean something from it that convinces my intuition, even though logically it seems fairly acceptable. Thanks. =)
$endgroup$
– Kainui
Jan 25 '14 at 10:17
$begingroup$
I'll think about your answer and see if I can't try to glean something from it that convinces my intuition, even though logically it seems fairly acceptable. Thanks. =)
$endgroup$
– Kainui
Jan 25 '14 at 10:17
$begingroup$
You are right, but observe that although we consider frictionless surfaces in the sense you are talking about (a force that is perpendicular to the motion), and the rope slide at the beginning, it will reach an equilibrum anyway due to the yellow region: it is stopping the motion of the rope because "it's there", blocking the advance of the green region (which was sliding downward before the equilibrium).
$endgroup$
– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:29
$begingroup$
You are right, but observe that although we consider frictionless surfaces in the sense you are talking about (a force that is perpendicular to the motion), and the rope slide at the beginning, it will reach an equilibrum anyway due to the yellow region: it is stopping the motion of the rope because "it's there", blocking the advance of the green region (which was sliding downward before the equilibrium).
$endgroup$
– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:29
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In a way, I guess it's like trying to push something through a hole, but the hole gets smaller the harder you push until you can't possibly squeeze through it anymore. I think that makes sense now.
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– Kainui
Jan 25 '14 at 10:39
$begingroup$
In a way, I guess it's like trying to push something through a hole, but the hole gets smaller the harder you push until you can't possibly squeeze through it anymore. I think that makes sense now.
$endgroup$
– Kainui
Jan 25 '14 at 10:39
$begingroup$
Yeah, that's a good analogy :)
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– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:48
$begingroup$
Yeah, that's a good analogy :)
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– Daniel Muñoz Parsapoormoghadam
Jan 25 '14 at 10:48
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$begingroup$
What keeps knots in a rope, either former with the rope itself or with another rope, is imo friction. A knot in a completely frictionless rope won't stand the slightest pull on one of its ends...
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– DonAntonio
Jan 25 '14 at 9:13