# Why I think tension should be twice the force in a tug of war

I'm going to provide my argument for why I think the tension in a rope should be twice the force exerted on either side of it.

First, let's consider a different example. Say, there is a person named A and a block in space. A pushes on the block with a force of 100 N. Then, the block will also push A with a force of 100 N by Newton's third law. Now, consider the case where instead of the block, there is a person B who is also pushing on A with a force of 100 N while A is pushing on him. A will experience a force of 100 N because he pushed on B, AND another 100 N because he is being pushed by B. Hence he will experience a force of 200 N. Similarly, B also experiences 200 N of force.

Now, back to the original problem. There are two people A and B in space with a taut rope (no tension currently) in between them. If only A is pulling and B is not, then I agree that the tension is equal to the force A exerts. This situation (in my opinion) becomes analogous to the above if B is also pulling. So, say both of them pull from either side with a force of 100 N. Then the rope at the end of B will pull B with a force of 100 N (this pull is caused by A). By Newton's third law, the rope will experience a pull of 100 N. But B is also pulling his end of the rope with 100 N. Therefore, the tension should be 200 N. Similarly, the end of the rope at A must pull A with 100 N of force (because B is pulling from the other side) and hence experience a force of 100 N itself by Newton's third law plus another 100 N because A is pulling on the rope.

Apparently, the answer is not this (according to my searching on the web). So, could anyone tell me why this reasoning is wrong? Thanks.

EDIT : So apparently people don't agree with my first example, leave alone the second. This is to the downvoters and the upvoters of the highest-rated answer: You all agree that if only A pushes B with a force of 100 N, then A and B both will get pushed by a force of 100 N in opposite directions, right? Then, in the case where B is also pushing with a force of 100 N, it doesn't make sense that the answer would be exactly the same. It doesn't seem right that no matter what B does, B and A will always experience the same force as they would have if B hadn't applied any force.

EDIT 2 : I'm going to provide here a link to a question that I posted: Two people pushing off each other According to the answer and the comments there, the reason as to why my first example is incorrect is different to the one provided here. So maybe you should all read the answer and the comments provided by the person and reconsider what you think.

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Imagine 1Kg-block hanging on the ceiling. Each hook (the one in the ceiling and the one in the 1Kg-block) pull with 100N still you would not get the Idea that the tension would be 200N. In the rope example the ground takes care of the other 100N. – miceterminator Oct 20 '12 at 14:42
@Alaxrite: I suggest you get two spring scales and a rope and perform this experiment yourself. Your example is wrong. The action/reaction pair of A and the block has nothing to do with what B is doing. You can make this really clear by drawing free body diagrams for the three objects. – Jerry Schirmer Oct 20 '12 at 18:27
@Alraxite: you have the block, person A and person B. Or the rope, person A and person B. As for your tension example, if you pull a rope and move it with no one pulling back on the other end, the tension will be zero. Pick up a string or an extension cord and do it yourself. It will flip about all floppy-like. No tension. In both cases, person A does not feel force from person B. Person A feels a force from the rope, ONLY. – Jerry Schirmer Oct 20 '12 at 19:08
A number of comments that were contributing to personalizing the discussion deleted. – dmckee Oct 20 '12 at 21:01
By the way, downvoters are of course entitled to their opinion, but I do think this is a good question because it asks about a conceptual problem, and a somewhat subtle one at that. The fact that it's based on a misconception is not a problem IMO, and in fact such questions often turn out to prompt insightful answers. – David Z Oct 21 '12 at 10:01

It is always best to draw a diagram to convince yourself of things in a case like this.

This is intended to represent a steady state situation: nobody is moving / winning. As you can see, there are two horizontal forces on A: the floor (pushing with 100N) and the rope (pulling with 100N). There will be two vertical forces (gravity pulling down on center of mass, and ground pulling up) to balance the torques - I did not show them because they are not relevant to the answer.

Now I drew a dotted line between A and B. Consider this a curtain. A cannot see whether the rope is attached to B (an opponent) or a wall. A can measure the tension in the rope by looking (for instance) at the speed at which a wave travels along the rope - or by including a spring gage.

Now ask yourself this question: if A feels a tension of 100N in the rope (this is the definition of the force on A), and can confirm (by looking at the gage) that the tension is 100 N, but he cannot see whether the rope is attached to a ring or to an opponent, then how can the tension be 200N? If I pull on a gage with a force of 100N, it will read 100N - it cannot read anything else (in a static situation, and where the gage is massless, ... )

I think I understand the source of your confusion based on the earlier q/a that you referenced - so let me draw another diagram:

In this diagram, I have move the point of attachment of the rope with which A pulls B away from B's hands, to his waist. Similarly, the rope with which B pulls on A is moved to A's waist.

What happens? Now there are two distinct points where A experiences a force of 100 N: one, his hands (where he is pulling on the rope attached to B's waits); and another where the rope that B is pulling on is tied around his waist.

The results is that there are two ropes with a tension of 100N each, that together result in a force of 200N on A (two ropes) offset by a force of 200N from the floor, etc.

This is NOT the same thing as the first diagram, where the point on which B's rope is attached is the hands of A - there is only a single line connecting A and B with a tension of 100 N in that case.

As was pointed out in comments, you can put a spring gauge in series with your rope to measure the tension in it; and now the difference between "a single person pulling on a rope attached to a ring at the wall (taken to be the dotted line) and two people pulling across a curtain (so they cannot see what they are doing) is that in one case, a single spring (with spring constant $k$) expands by a length $l$, while in the second case you find a spring that's twice as long, with constant $k/2$), expanding by $2l$.

These are all different ways to look at the same thing.

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I love everything except the final paragraph. If two people push off each other, it should be indistinguishable from a single person pushing off a wall. As an entertaining demonstration of this, with cars instead of people, we have mythbusters to thank. – alemi Aug 7 '14 at 1:10
@alemi No, it would not be the same as one person pushing off the wall. Suppose the people use springs to push off instead of their arms. Person A compresses a spring until it pushes back with a force F and applies it to person B. Person B feels a force F. Now let persons A and B each prepare springs the same way person A did. Person A and B now apply these springs so that they touch. Persons A and B will now each feel a force of 2F. – abalter Aug 7 '14 at 6:09
A simpler way to measure the tension, that I think is more intuitive, is to simply cut the rope and insert a spring scale. That being said, I think your method is very clever, and I would not have thought of it myself. – abalter Aug 7 '14 at 6:11
@Floris When they have their springs touch, it is as if it were a single spring twice as long deformed twice as much. But this will give the same force, as the force constant for a spring is inversely related to its number of coils. – alemi Aug 7 '14 at 6:17
@alemi - indeed two people pushing off each other is the same as one person pushing off a wall - but NOT (and that was my point) the same as one person pushing off another (who is not also pushing - so the velocity wrt the center of gravity is half). Sorry if that wasn't clear – Floris Aug 7 '14 at 6:58

Your first example is facetious. If each is providing 100N then each is feeling 100N, period. In order to feel 200N, each would have to provide 200N. This is what Newton's Laws of Motion are all about; one does not feel their own force, only external forces, or when their own force comes into contact with an external body.

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I absolutely don't follow. Your wording is too vague. Look at this way: You agree that by the third law, forces always come in action-reaction pairs, right? Now, consider the same situation where the two people are pushing on each other. Let's consider these two forces: the force exerted by 'A' on 'B' and the force exerted by 'B' on 'A'. Note here, that I'm talking about the deliberate forces that they exert on each other. These do not form the third law's action-reaction pair. It's entirely B's choice that he is pushing. continued – Alraxite Oct 20 '12 at 15:52
Since these two do not form an action-reaction pair of each other, there must be two other reaction forces for these two. So now I ask, what are the they? Think about it and then reply. – Alraxite Oct 20 '12 at 15:52
I keep trying to explain why, but you say "no, I think I'll ignore the math". You can't ignore the math. You can't trust intuition, or common sense, or the fuzzy-wuzzies when it comes to physics. You must listen to the math; it is the only thing that has proven itself correct time and time again. – Ignacio Vazquez-Abrams Oct 20 '12 at 19:05
You are just not giving me a direct explanation! I just gave you a reason why they can't be action-reaction pairs and now you have brought maths into this. Your explanation didn't include any maths to begin with. – Alraxite Oct 20 '12 at 19:11
-1: Alraxite makes reasonable arguments (he confused me), and the responses to his questions are flippant and wrong. If you have two astronauts carrying nearly infinitely heavy backpacks, with ropes, one attached from A to B's chest and the other from B to A's chest, you wouldn't get confused that the pulls of A and B add up to determine how fast they approach each other. So why is it confusing when they are both pulling on the same rope? To be honest, I agree that it is confusing, but I haven't sorted out why. There is some failure of intuition here among the trained, not among the untrained. – Ron Maimon Oct 31 '12 at 4:18

No.

Your mistake is that the block also pushes back on A, with the same 100 N that A is pushing with, by Newton's third law. Assuming both weigh 100 kg, both the block and the person will accelerate at 1 m/s$^2$ for the duration of the push. If there's two people pushing on each other, then clearly they will also accelerate at 1 m/s$^2$ so the forces must be the same. (If this isn't obvious, consider placing a very massive, very thin, very strong wall between them, which can't alter the physics. Then it's just two 100 kg people pushing off a stiff wall with 100 N.)

The difference between the two scenarios is that in the first one the reaction force is provided by the block's structure, while in the second one by a work-performing human. In the first one the block goes away while in the second one A and B's palms stay stationary. Thus A is able to push for twice as long and therefore do ~twice the work, so they will - as intuition says - end up going faster.

This translates directly to the rope-pulling scenario, simply substituting tension for the compression force at the guys' palms.

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I'm not sure about the mistake you're referring here. I have mentioned in my question that the block pushes on A. Your answer then goes on to say that it is obvious in the case where B is a person who is also pushing that he will experience the same force as the block. You have given a reason with a wall which I don't quite understand. Apparently the rest of your answer is about the details of how they push each other and I guess it does get confusing if you think about them pushing directly at each other's hands. So let me alter the situation a little. continued – Alraxite Oct 20 '12 at 17:17
Firstly, to keep it simple, both of them are using only one hand of theirs. Secondly, each hand of theirs are resting on each others chest before they push. So, now it should be clear that when A pushes on B's chest, his chest provides an equal and opposite force, during which B's hands which are on A's chest provides another force. Thus twice the force. – Alraxite Oct 20 '12 at 17:18
The situation with pushing on each other's hands is different from the situation of pushing on each other's chests, it's serial vs. parallel springs. – Ron Maimon Oct 31 '12 at 5:47
Yes, if they push on each other's chests it's a different game (parallel instead of serial springs) and it cannot be translated to the rope example without introducing a second rope. – Emilio Pisanty Oct 31 '12 at 11:11
As for the wall, it's puzzling because the force is the same - but if you push a person (and they don't push back) they'll move and you'll only cover half the distance you would with a wall (or someone who pushes back) - thus you only do half as much work on yourself and get only half the kinetic energy. – Emilio Pisanty Oct 31 '12 at 11:13

I think the greatest confusion here is forgetting that there are no lone forces in the world, there are only third law pairs (as in Newton's third law). Yes the person pulls on the rope with 100N, but the string is pulling back with 100N. The floor pushes on the foot with 100N, and the foot pushes on the floor with 100N. In reality, the way to measure force is to insert a scale. This measures the force between two objects. Now just eliminate the rope altogether. First let person A and person B push on the same scale from opposite sides. When the scale reads 100N, they are each pushing with 100N. Now let them pull on the same spring scale (from opposite sides). When the spring reads 100N, they are each pulling with 100N. As @floris pointed out, as far as either of them are concerned, there could be another person pulling on the other end of the spring scale, or it could be attached to a wall. It matters not. And, we can always replace the spring scale with a rope. The spring scale just measures the tension

Maybe it's even more intuitive to make the scenario vertical. Person A holds onto a spring scale that hangs down. The scale has a 100N weight hanging from it, so the scale reads 100N. Clearly, person A is pulling up with 100N (of course person A and the spring are pulling on each other with 100N). Person A now closes her eyes. She can no longer tell if the weight is still there, or if person B is now pulling down on the spring with 100N of force. See? Persons A and B can each be pulling on the spring with 100N of force, and the scale just reads 100N. Take away the scale and insert a rope. The physics does not change.

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To keep things simple lets say the tug of war rope is stationary, both sides are pulling with equal force.

A and B are both people, A pulls left with 100N, B pulls right with 100N.

Now in your second example we replace B with C, your immovable weight.

A pulls left with 100N. Newtons law says that the block must be pulling right with 100N, otherwise the system would move.

The total strain on the rope is exactly the same in both cases.

To make it more complicated lets have A, B and C all in play:

A
C
B


The rope runs from A, to C, through a pulley and then on to B.

A and B both pull with 100N. C experiences a force similar to 200N (reduced a bit as A and B are in slightly different directions, however each length of the rope still experiences a force of 100N.

Now if you have a single length of rope, and both A and B pull on it now you finally have your 200N of force in the rope, because now A and B are pulling for a combined force of 200N so the counter-force from C increases to match it, otherwise the whole system will start moving.

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tension will be equal to force of one side. because wen a string is attach with a support, and other end is attached wit mass we take tension is equal to weight. do you think there are only two forces tension and weight? no, there are 3 forces, 1. weight downward tension upwards, another force of support upward, this is same as rope is pulling from either side by same forces. so tension is equal to 1 forces, if forces are same.

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## protected by Qmechanic♦Jul 24 '14 at 18:24

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