# The exact definition of tension

## Main Question or Discussion Point

When people refer to the tension in a rope, string, etc...are they referring to the magnitude of the tensile force? But isn't tension a vector. So tension in general is a vector, but when referring to the tension in a rope they are talking about the magnitude?

Please clarify..

edit: btw, i'm talking about rope or string with no mass, if that makes any difference.

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## Answers and Replies

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It's a vector whose direction can change from point to point, if you have a pulley, while the magnitude is the same on both sides of the pulley. If you pull down on a rope that is wrapped over a pulley, and the other end of a rope pulls up a mass, draw the direction of the tension vector downward on your side of the pulley and upward on the other side.

stewartcs
Science Advisor
When people refer to the tension in a rope, string, etc...are they referring to the magnitude of the tensile force? But isn't tension a vector. So tension in general is a vector, but when referring to the tension in a rope they are talking about the magnitude?

Please clarify..

edit: btw, i'm talking about rope or string with no mass, if that makes any difference.
Yes, tension is a vector.

When a rope (cable, string, etc) is attached to a body and pulled taut, the rope pulls on the body with a force $$\vec{T}$$ directed away from the body and along the rope. Therefore it has a direction (and obviously a magnitude). The force is often called a tension force because the rope is said to be in a state of tension. The tension in the rope is just the magnitude $$T$$ of the force on the body.

Hope that helps.

CS

When people refer to the tension in a rope, string, etc...are they referring to the magnitude of the tensile force? But isn't tension a vector. So tension in general is a vector, but when referring to the tension in a rope they are talking about the magnitude?

Please clarify..

edit: btw, i'm talking about rope or string with no mass, if that makes any difference.
Tension is a tricky thing. If you're working with ropes and pulleys then there are instances where you can define a vector. E.g. consider a rope with a weight on the end. The tension in the rope due to gravity is in the direction of the rope in the direction of the force of gravity. The magnitude is force per unit area. However consider the situation where you have two people pulling on the ends of the same rope and exerting the same force. If one asks what the tension in the rope is at the mid point then there is no unique answer to this since it represents a completely symmetric situation. But there is one thing that is common in all cases of tension and that is that the magnitude is force per unit area and the forces are acting to pull the rope apart. Stress, on the other hand, is when you try to squeeze a rod. The stress is forces squeezing something together while the tensions is the forces trying to rip something apart. This is important when you're constructing the stress tensor. A postive sign means that there is a compression going on while a negative sign indicates that there is pulling apart going on.

Pete

engineers when working with constructions and stuff like that they use to define the stress tensor and its magnitude along each axis!!!!
and sometimes when the geometry is esay we can digonalized it and get vector fields radially or in some others configurations..... it depends on what we are working on!!!!
regards
marco

Tension is a tricky thing. If you're working with ropes and pulleys then there are instances where you can define a vector. E.g. consider a rope with a weight on the end. The tension in the rope due to gravity is in the direction of the rope in the direction of the force of gravity. The magnitude is force per unit area. However consider the situation where you have two people pulling on the ends of the same rope and exerting the same force. If one asks what the tension in the rope is at the mid point then there is no unique answer to this since it represents a completely symmetric situation. But there is one thing that is common in all cases of tension and that is that the magnitude is force per unit area and the forces are acting to pull the rope apart. Stress, on the other hand, is when you try to squeeze a rod. The stress is forces squeezing something together while the tensions is the forces trying to rip something apart. This is important when you're constructing the stress tensor. A postive sign means that there is a compression going on while a negative sign indicates that there is pulling apart going on.

Pete

I think thats a bit sloppy, but just a tad. Stress is not squeezing something. Its just a force per unit area, thats it. It can be normal or in shear, tensile or compressive. Also, I dont see how two people pulling on each side of a rope wrapped around a pulley has no unique answer because its symmetric. The answer is that the tension is the force each person is pulling down with.

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Thanks stewart, that cleared it up...I think.

I think thats a bit sloppy, but just a tad. Stress is not squeezing something. Its just a force per unit area, thats it.
Yep. You're correct. Thank you for pointing that out.

What I should have wrote was that tension is when something is being pulled apart and compression (not stress) is when something is being squeezed. This determines the sign in the stress tensor.
Also, I dont see how two people pulling on each side of a rope wrapped around a pulley has no unique answer because its symmetric.
I was referring to the same thing the other person said when they wrote The tension in the rope is just the magnitude of the force on the body. Since in this case there is only a magnitude if follows that there is no direction related to it.
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I didn't intend to imply that a rope around a pully has no unique answer. In fact I never made such a comment. Please re-read that part. Here is what I said However consider the situation where you have two people pulling on the ends of the same rope and exerting the same force. (opposite directions of course). The tension in the rope is scalar quantity and refers to the magnitude of the force divided by the cross sectional area of the rope. alStress, on the other hand, is neither a scalar or a vecor quantity. It is a tensor quantity. The sign of the components are dependant on whether there is tension or compression.
The answer is that the tension is the force each person is pulling down with.
I was referring to the difference between tension and compression. This is an important distinction to make since it determines the sign in the stress tensor.

Pete

I didn't intend to imply that a rope around a pully has no unique answer. In fact I never made such a comment. Please re-read that part. Here is what I said However consider the situation where you have two people pulling on the ends of the same rope and exerting the same force. (opposite directions of course). The tension in the rope is scalar quantity and refers to the magnitude of the force divided by the cross sectional area of the rope. alStress, on the other hand, is neither a scalar or a vecor quantity. It is a tensor quantity. The sign of the components are dependant on whether there is tension or compression.
I was referring to the difference between tension and compression. This is an important distinction to make since it determines the sign in the stress tensor.

Pete
Ah, yes. I agree with this.

I would simply say that stress is a tensor. In plane terms, it depends on the orientation of the plane you are looking at.