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Why do we define tension only in strings and ropes and rods and such? Shouldn't every object experience tension force? Like when you pull a paper from opposite sides, it gets taut, and experiences what seems like a state of tension. If every object does experience tension, can you define tension?

Tension is not defined only for strings.

However, the unique thing about ideal strings is that they can ONLY experience tension, whereas rigid bodies can experience tensions and compression. Ideal strings would collapse.

Hope this helps

You can think of tension as a force that helps maintain structural integrity under a tensile 'stretching' force (being quite vague here, but hold on).

Imagine a solid cylindrical rod. You can compress its ends, and there is a normal reaction you have to overcome. This normal reaction is trying to preserve the structural integrity of the rod. Tension is, in a sense, the equivalent of normal reaction but in the opposite direction - if instead of compressing, you try to pull on both ends of the rod, you have to overcome another force trying preserve the structure of your rod (by preventing it from being ripped apart). This force is what one normally calls 'tension'.

I have stressed the similarity between the normal reaction and the tension forces because they have the same origin- they arise from intermolecular interactions that describe the structure of your object. For all practical purposes, you can call normal reaction a 'compressive tension'. The point of this discussion is that, objects try to maintain their structural integrity, and it's a matter of semantics to call this 'restoring' force a normal reaction or a tension or whatever. The molecular origin of these forces are identical.

That said, sometimes it is useful to keep this distinction for intuitive clarity. An object that is hard to compress, for example a string along its length, will provide almost no normal reaction (it is very easy to squish a string), but stretching it certainly induces a tension that one must overcome before it breaks (i.e. loses it's integrity).

"Tension" is a simple special case of the state of stress in any solid object.

But you usually start learning about mechanics using simple situations where common-sense ideas like "tension" are all you need, rather than starting by learning about stress and strain tensors and constitutive equations in continuum mechanics!

If you are thinking something really fundamental, there's only four acknowledged fundamental force, and the "tension"(of something like a string) was usual treated as a result of electromagnetic interaction(electromagnetic force) from the atoms or particles.

However, tension in sub particle level can also resulted from strong force, like protons inside a nuclei.

Depends, tension is just a more formal wording of the usual "stretch", it's basically a word commonly used by physicists to mention in a way that physicists can easily understand.

Pick up a brick. Try pulling it apart. What stops you from pulling it apart? It's the tension in the brick. Where does the tension in the brick come from? It comes from the inter-molecular forces inside the brick.

The same goes for strings. Here's a second example. If you look closely at a thick rope, it's made of many small strings wrapped around each other. Why is this? So that the friction in between the strings increases the available tension in the entire rope. In other words, the inter-molecular forces (in this case, friction) increase within the rope, making the rope "want" to stay together more; the more a rope wants to stay together, the more force it can handle when you try to pull it apart.

It's just like how normal force is the force from electron shells repelling each other (due to the Pauli Exclusion Principle), tension comes from a micro-scale force as well.