# What cancels out tension in hanging chain?

### Diagram

A chain with mass $$m$$ is fixed on two points $$A$$ and $$B$$ making an angle $$\theta$$ with tension $$T_1$$ and $$T_2$$.

### Question

So I'm told that $$T_1 \cdot \cos \theta$$ is cancelled out by $$T_2 \cdot \cos \theta$$. But imagine for a second that point $$B$$ is not fixed and a force equal to $$T_2 \cdot \cos \theta$$ is applied on point $$B$$ but in opposite direction (or towards $$T_1$$) what would happen then? will point $$A$$ start moving since the counteracting force that was stopping it has now been already cancelled out? Basically I am not convinced how can point B's horizontal force cancel out A's horizontal force?

In that case the chain would have an unbalanced net force and would accelerate. The catenary equation is based on the assumption that the chain is stationary, so it would no longer hang in a catenary shape. The exact details of the shape and how it evolves over time are very complicated and depend on the stiffness, the mass, and the speed of sound in the chain

• I understand that the shape would change, presumably more and more like a parabola from a more spherical shape. But I'm more interested in what would happen to point A, it should start accelerating in the same direction as the newly applied force but from intuition I think it would stay right there and point B would come closer and closer to point A right? If that's true what is the mysterious force keeping point A fixed? Aug 18, 2022 at 11:58
• In the problem description you said that A was fixed. So it cannot move by definition. There is no mysterious force here. Whatever method you used to fix A provides the force
– Dale
Aug 18, 2022 at 12:08
• But by similar reasoning shouldn't the method used to fix point A also be the one to cancel out $T_1 \cos \theta$ rather than $T_2 \cos \theta$ ? Aug 18, 2022 at 12:21
• In the original scenario the chain is not accelerating so you know that the external forces sum to 0 by Newton's 2nd law. In your revised scenario the chain is accelerating, so $T_2 \cos \theta$ is not canceled out at all. I don't think that your revised scenario is helping you understand the original scenario, if that was the goal.
– Dale
Aug 18, 2022 at 12:29
• Sure. That works. Even if they are fixed on different buildings, the ground between the buildings still acts as a rod in the sense you need here
– Dale
Aug 18, 2022 at 13:05