Tension in the Wire- Thermodynamics Question

[Cookiey]

Homework Statement

consider a horizontal cylindrical tube of cross sectional area A fitted with two frictionless pistons. The piston are connected to each other by an inextensible wire. Initially, the temperature of the gas is To and its pressure is Po which equals the atmospheric pressure. Find the tension in the wire if the temperature is raised to 3To.

Homework Equations

(P1V1)/T1=(P2V2)/T2

The Attempt at a Solution

Using the above equation; as volume is constant, new pressure P'=3Po. So, force applied on one piston=3PoA

Equating with the force from atmospheric pressure outside the tube,
3PoA=PoA+T
So T=2PoA ; which is the required answer.

My question is, isn't this the force on one cross section of the wire? So why isn't this answer multiplied by two?

Thank you for your time!

 

[haruspex]

It's a common mistake to think that if there's a pull F on each end of the wire then the tension must be 2F.
You need to understand how tension is defined. If we say that the tension is T at some point in a wire (need not be the same all along the wire) we mean that if you were to insert a block at that point in the wire the block would experience a pull T on each side. Thus, if the tension is the same all the way along then it is also equal to the force exerted at each end.
Is that your question?

 

[Cookiey]

It's a common mistake to think that if there's a pull F on each end of the wire then the tension must be 2F.
You need to understand how tension is defined. If we say that the tension is T at some point in a wire (need not be the same all along the wire) we mean that if you were to insert a block at that point in the wire the block would experience a pull T on each side. Thus, if the tension is the same all the way along then it is also equal to the force exerted at each end.
Is that your question?

Yes, thank you very much! So if there /is/ such a wire which is experiencing a force F on both sides, and assuming it experiences equal tension everywhere; the tension would be F?

 

[haruspex]

Yes, thank you very much! So if there /is/ such a wire which is experiencing a force F on both sides, and assuming it experiences equal tension everywhere; the tension would be F?

Yes.

 

[Cookiey]

And if one end of this wire was fixed to a wall and force F was applied on the free end, it would still experience the same tension F as in the earlier case because of reaction force from the wall?

 

[Chestermiller]

And if one end of this wire was fixed to a wall and force F was applied on the free end, it would still experience the same tension F as in the earlier case because of reaction force from the wall?

Yes.

 

[Cookiey]

Yes.

All right, thank you very much!