Why is the tension in a massive cable always tangent to its portion?

[Karol]

Homework Statement

Why is the tension always tangent to the portion of the cable?

The correct situation is on points A and B.
Why can't be like on C and D?

Homework Equations

Newton's force-mass: $F=ma$

The Attempt at a Solution

If i move the portion of the cable and examine, for equilibrium, portion CD, then what about portion AC, does it pull up now, while in position A the portion of the cable on the left of A, if the tension wouldn't be only along the cable, would have pulled down since it was part of the examined cable.
So the tension is only along the cable from symmetry only. is it true? i sense it's not the single reasoning.

 

[mfb]

Forces in other directions are only possible if your cable is stiff. Which can happen, but that is not the typical rope problem.

 

[haruspex]

Consider a very short elemental section of the cable. It is almost straight. The forces on it are a small gravitational force and the two much larger tensions. The tensions won't be quite tangential, but most be close to it. They must be almost equal and opposite, and if not almost tangential they would apply a net torque. In the limit, as the length and mass of the element tend to zero, the tensions are tangential.

 

[yucheng]

Consider a very short elemental section of the cable. It is almost straight. The forces on it are a small gravitational force and the two much larger tensions. The tensions won't be quite tangential, but most be close to it. They must be almost equal and opposite, and if not almost tangential they would apply a net torque. In the limit, as the length and mass of the element tend to zero, the tensions are tangential.

But won't this same argument work for a copper rod bent into a catenary shape?

I thought it's related to material properties, that ideal ropes cannot exert an internal shear, so the only forces to consider are stretching forces (tangential to this tiny segment) (think a spring)

 

[haruspex]

But won't this same argument work for a copper rod bent into a catenary shape?

No, because the neighbouring parts of the rod can apply torques to the element. The tensions will still be nearly equal and opposite as vectors, but can be at an angle to the tangent, resulting in an opposing torque.

 

[Chestermiller]

Copper tubing has substantial bending stiffness, which is not present in a thin rope or wire.