Why the earphones in the pocket always form knots? How to explain this problem in Maths? And how to build a model to study of it?
 A: When you are pulling/inserting your earphones in your pocket, you are 


*

*folding them up, usually asymmetrically, i.e randomly folding them up instead of a proper arrangement.

*pushing/ pulling in such a way that different parts of the earphones experience different forces, for example, by the virtue of the insertion,  the outer edges experience more contact with the pocket rather than the inner edges

*the end points experience a different force which is distinct from what is experienced by the remaining wire, even if you are pulling/ pushing the earphones with a symmetrical force.


A mathematical model needs to take into account such characteristics by making appropriate free body diagrams. Usually a knot forms while you pull out your earphone when subject to the above features. In other words, a more balanced pulling out of your earphones will result in a lower probability of forming knots although there will still be a possibility.
For a simple model, you may assume the earphones to be a finite helix with a constant linear density. The ends of the helix are connected to different masses. A knot will form by the virtue of the above mentioned processes


*

*if the point masses leave their place at the end and get entangled between the helix.

*a ring of the helix overlaps another ring and then shortens around while the earphone is pulled out
A: Number of ways in which your earphones can form somethin like knots is much greater than the number of ways in which it wont. I would not say that there are actually complicated knots that form I would say that there are mainly some simple loops that interweave...Next thing, often the final touch to this mess is your atemmpt to unfold them when you get them out of your pocket. We mainly do this carelessly without thinking. Mathemtical model would be the same as something like a canonical ansamble I guess. But this is just a guess. I once had to solve something like finding the entropy of a chain molecule by using statistical methods of thermodynamics...earphones act in a similar way. Entropy is always the answer.
