Trouble with the concept of tension....

[Achintya]

Homework Statement

Two persons A & B are pulling the two ends of a rope (As people do in tug war).Person A pulls rope with a force=10N and the person B is pulling another end of the rope with a force =20N. According to Newton's 3rd, rope will also pull person A with force=10N and B with a force=20N. It means tension at one end of the rope is 10N and at the other end tension is 20N(Because of the definition of tension).Is it correct? If yes ,then pulling force on A by the rope will only depend on force exerted by A on rope and will not depend upon the force applied by B on the rope. In that case , if B is pulling the rope with a force=1000N and A is pulling the other end of rope with a force=10N, then the tension at that end(where A is applying force) will be only 10N.It means the force by which rope is pulling A will be only 10N. Is it possible?

Relevant Equations

Newton's laws and fundamental of mechanics

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[PeroK]

This is very confused. There can only be one tension along the rope between the two people. If there is an unbalanced force on the rope, then the rope will accelerate (in which case the tension will vary slightly along the length of the rope depending on its mass). If we assume the mass of the rope is small compared to the forces involved, then the tension is still approximately constant.

In a tug of war, the two people are also pushing on the ground.

 

[Achintya]

This is very confused. There can only be one tension along the rope between the two people. If there is an unbalanced force on the rope, then the rope will accelerate (in which case the tension will vary slightly along the length of the rope depending on its mass). If we assume the mass of the rope is small compared to the forces involved, then the tension is still approximately constant.

In a tug of war, the two people are also pushing on the ground.

even if the rope is light(but not completely massless), then what forces will A experience...will it be just the Newton's 3rd law reaction pair force of his own force or the force exerted by B from the other side...pleasez it will be really helpful if you could explain with help of an example and on the basis of Newtonic mechanics...
i would really appreciate the help!

 

[archaic]

 

[Adesh]

A will experience the reaction force + the force applied by B (in the direction opposite to which A pulls)

 

[PeroK]

even if the rope is light(but not completely massless), then what forces will A experience...will it be just the Newton's 3rd law reaction pair force of his own force or the force exerted by B from the other side...pleasez it will be really helpful if you could explain with help of an example and on the basis of Newtonic mechanics...
i would really appreciate the help!

The simplest way to think about it is that there is a tension $T$ in the rope. And both people are pulling and being pulled with the same force $T$. For an idealised, massless rope there can be no unbalanced force. The forces at either end must be the same.

The difference in a tug-of-war comes from the forces exerted on the ground.

Alternatively, analyse a tug-of-war in space.

 

[Achintya]

A will experience the reaction force + the force applied by B (in the direction opposite to which A pulls)

then what will be the tension in the rope? Because we know that for a massive rope the tension won't be constant throughout the string rather it would be a function of the distance x from the end of the rope...by Newton's 3rd law if A pulls on the rope with 30 N , the rope will also pull A with 30 N ...so the tension of the rope segment connected to A will be 30N , then what about the force that B is exerting on the rope ...will it be of no significance to A?

 

[hutchphd]

If the rope has no mass then then tension in the rope will be the same everywhere (regardless of its state of acceleration). The force exerted by the rope at each end will be T in the obvious directions. If the rope has mass then the tension will depend upon the acceleration of the system of rope + people and will vary according to mass distribution in the rope (linear in a uniform rope) .
PLEASE ask one particular question at a time...not an incoherent jumble.

 

[PeroK]

then what will be the tension in the rope? Because we know that for a massive rope the tension won't be constant throughout the string rather it would be a function of the distance x from the end of the rope...by Newton's 3rd law if A pulls on the rope with 30 N , the rope will also pull A with 30 N ...so the tension of the rope segment connected to A will be 30N , then what about the force that B is exerting on the rope ...will it be of no significance to A?

The two ends of the rope cannot be pulled with different forces without the rope itself accelerating. And, if the rope is light, the difference in forces (tension) at each end will be small.

In an ideal analysis, you have $T$ constant throughout the rope.

If you add the mass of the rope, then you have $T + am$ at one end and $T$ at the other. The difference in tension along the rope is only $am$, where $a$ is the acceleration of the rope/system and $m$ is the mass of the rope. In the idealised scenario, you neglect the $am$ term, as it is small.

Your scenario of an approximately static, light rope pulled by $30N$ at one end and $20N$ at the other is physically impossible. That isn't a valid free-body diagram for a stationary rope.

 

[Adesh]

then what will be the tension in the rope? Because we know that for a massive rope the tension won't be constant throughout the string rather it would be a function of the distance x from the end of the rope...by Newton's 3rd law if A pulls on the rope with 30 N , the rope will also pull A with 30 N ...so the tension of the rope segment connected to A will be 30N , then what about the force that B is exerting on the rope ...will it be of no significance to A?

If the forces on the rope don’t cancel each other then the rope will accelerate. Tension exists in the static situation .

 

[hutchphd]

If the forces on the rope don’t cancel each other then the rope will accelerate. Tension exists in the static situation .

Let us be precise. Tension will exist for any (non-slack) rope. It will be uniform for a massive rope only if not accelerated.

 

[Achintya]

The two ends of the rope cannot be pulled with different forces without the rope itself accelerating. And, if the rope is light, the difference in forces (tension) at each end will be small.

In an ideal analysis, you have $T$ constant throughout the rope.

If you add the mass of the rope, then you have $T + am$ at one end and $T$ at the other. The difference in tension along the rope is only $am$, where $a$ is the acceleration of the rope/system and $m$ is the mass of the rope. In the idealised scenario, you neglect the $am$ term, as it is small.

Your scenario of an approximately static, light rope pulled by $30N$ at one end and $20N$ at the other is physically impossible. That isn't a valid free-body diagram for a stationary rope.

But i never said that the rope is stationary...it is obvious that if there are unbalanced forces on a system then the system would accelerate according to Newton's 2nd law

 

[Achintya]

If the forces on the rope don’t cancel each other then the rope will accelerate. Tension exists in the static situation .

so you mean once the system accelerates there will be no tension in the rope?

 

[Adesh]

so you mean once the system accelerates there will be no tension in the rope?

Take a rope which is not attached to any thing, if you pull it then it will certainly going to move and the stretch in the rope will be due to molecular attraction/repulsion.

 

[PeroK]

But i never said that the rope is stationary...it is obvious that if there are unbalanced forces on a system then the system would accelerate according to Newton's 2nd law

Okay, but what about the case where two people are anchored to the ground and one pulls the rope with $30N$ and the other pulls the rope with $20N$?

That's impossible.

It's also invalid for a massless rope, since any unbalanced force would lead to an infinite acceleration of the rope.

It's also invalid for a light rope ($1kg$, say), unless the rope and the whole system is accelerating at $10m/s^2$.

Fundamentally, you cannot have different forces pulling two ends of a rope without the rope accelerating under those forces. If there are other objects in the system stopping it from accelerating, then those cannot be the forces on the rope.

 

[Achintya]

Let us be precise. Tension will exist for any (non-slack) rope. It will be uniform for a massive rope only if not accelerated.

yes i have understood the fact that if system is in static equilibrium(0 acceleration), then tension would be constant throughout the string whether it be massless or massive...but what happens when the system accelerates like in my original question(20N and 30N forces are acting at each end of the string, making the system accelerate)...i hope i am clear.

 

[PeroK]

yes i have understood the fact that if system is in static equilibrium(0 acceleration), then tension would be constant throughout the string whether it be massless or massive...but what happens when the system accelerates like in my original question(20N and 30N forces are acting at each end of the string, making the system accelerate)...i hope i am clear.

That's physically impossible.

 

[Achintya]

Take a rope which is not attached to any thing, if you pull it then it will certainly going to move and the stretch in the rope will be due to molecular attraction/repulsion.

yes i could feel this

 

[Achintya]

That's physically impossible.

it would be really helpful if you can explain me why is it physically impossible.

 

[PeroK]

it would be really helpful if you can explain me why is it physically impossible.

Newton's second law: $F = ma$.

The net force on the rope (tension at one end minus tension at the other) must equal the mass of the rope times its acceleration. In this case:

$F = T_2 - T_1 = ma \approx 0$

 

[Adesh]

yes i could feel this

Consider this scenario:

We got a sponge, I push it from east and you push it from west with same force. Would the sponge going to be squeezed?

 

[hutchphd]

Are the people in your original question rigidly attached to the earth? Are they holding the ends of a fixed length string? So they must also be accelerating. Please succinctly rephrase your question because it is inconsistent as stated.

 

[PeroK]

Consider this scenario:

We got a sponge, I push it from east and you push it from west with same force. Would the sponge going to be squeezed?

I fail to see the relevance of this.

 

[Adesh]

I fail to see the relevance of this.

I’m trying to explain to him with that sponge example because deformation in sponge is very intuitive. If we have unequal forces on sponge it will move with some net deformation in it.

 

[hutchphd]

unless your rope stretches I don't see the relevance. Stick to original

 

[Adesh]

unless your rope stretches I don't see the relevance. Stick to original

Tension will be caused only when there is a stretch, in my opinion.

 

[hutchphd]

Your opinion is incorrect. Tension implies nothing about stretch inherently. An "ideal" rope is absolutely inexstensible but still has tension. Not a contradiction. In practice all things stretch a little (although Carbon Fiber is remarkably resistant) but the approximation is fine. Rigid bodies aren't absolutely rigid either!
So restate your original question in a concise and precise way please, and it will be addressed..

 

[Adesh]

Your opinion is incorrect. Tension implies nothing about stretch inherently.

It seems that you’re using this definition of tension :
The tension force is the force that is transmitted through a cable, rope, wire or string when it is pulled tight by forces acting from opposite ends. It is directed along the length of the cable and pulls equally on the objects on the opposite ends of the wire..

What I meant by the tension was the restoring force that the rubber band exerts when we pull (keeping one it’s ends fixed).

 

[hutchphd]

The two are not mutually exclusive. You may use what you like but the rest of the world uses the more generic one you quote. If you wish to describe an elastic rope then you need to indicate that and characterize it as to details. Otherwise everyone will be confused as you have witnessed. Hopefully lesson learned!

 

[Adesh]

The two are not mutually exclusive. You may use what you like but the rest of the world uses the more generic one you quote. If you wish to describe an elastic rope then you need to indicate that and characterize it as to details. Otherwise everyone will be confused as you have witnessed. Hopefully lesson learned!

Yes! I should use those things the way everyone uses it (if I’m posting it in public).