Why Is the Tension in a Rope Not Zero When Pulled Equally from Both Ends?

[DrummingAtom]

Homework Statement

If the two ends of a rope in equilibrium are pulled with forces of equal magnitude and opposite direction, why is the total tension in the *not* zero?

Homework Equations

The Attempt at a Solution

If there are two vectors with same magnitude and opposite direction their sum is zero. So, at the middle point that has to be in equilibrium. Are they asking what the sum is at the end of one side of the rope? Should I be taking the absolute value of each vector and adding them to equal a non zero number?

 

[Doc Al]

If there are two vectors with same magnitude and opposite direction their sum is zero. So, at the middle point that has to be in equilibrium. Are they asking what the sum is at the end of one side of the rope?

No, they are checking that you understand what 'tension in a rope' means. That the net force on the rope is zero happens to be true, but that's not what defines tension.

Should I be taking the absolute value of each vector and adding them to equal a non zero number?

Nothing like that. You just have to know that tension is the force that the rope exerts. If each end is pulled with a force F, what's the tension in the rope?

 

[DrummingAtom]

nothing like that. You just have to know that tension is the force that the rope exerts. If each end is pulled with a force f, what's the tension in the rope?

T = F₁+F₂ Right?

 

[Doc Al]

T = F₁+F₂ Right?

Nope!

What force does the rope exert? You and I are playing tug of war. We both pull our ends with a force F. What force does the rope exert on each of us? (Recall Newton's 3rd law.)

 

[DrummingAtom]

Nope!

What force does the rope exert? You and I are playing tug of war. We both pull our ends with a force F. What force does the rope exert on each of us? (Recall Newton's 3rd law.)

The rope would be an equal opposite force on us, a force toward the center.

 

[Doc Al]

The rope would be an equal opposite force on us, a force toward the center.

Right. So what will be the tension in the rope in terms of F?

 

[DrummingAtom]

Right. So what will be the tension in the rope in terms of F?

Hmm, well:

If T₂= -F₂ and T₁ = -F₁.

And T_total = T₂ + T₁ then

T_total = -F₂ + -F₁

How's that?

 

[Doc Al]

Hmm, well:

If T₂= -F₂ and T₁ = -F₁.

And T_total = T₂ + T₁ then

T_total = -F₂ + -F₁

How's that?

Not quite it yet. For one, there's no T₁ and T₂ (at least for the usual 'massless' rope) or T_total, there's just a single tension. And there's nothing to calculate. If I pull on the rope with a force F, then the rope pulls back on me with a force F. And thus the tension in the rope is simply F.

What's confusing to most--and is the point of this question--is that in order for a tension to be created in the rope both ends must be pulled with the same force. And the tension created is just F, not 2F.

 

[DrummingAtom]

Not quite it yet. For one, there's no T₁ and T₂ (at least for the usual 'massless' rope) or T_total, there's just a single tension. And there's nothing to calculate. If I pull on the rope with a force F, then the rope pulls back on me with a force F. And thus the tension in the rope is simply F.

What's confusing to most--and is the point of this question--is that in order for a tension to be created in the rope both ends must be pulled with the same force. And the tension created is just F, not 2F.

Ahh, it's making more sense now. That explanation helps a lot. Thank you.