Here, I have a question. 2 persons are holding a rope of negligible weight tightly at its ends so that it is horizontal. A 30 kg weight is attached to the rope at the mid-point which now no longer remains horizontal. What is the minimum tension required to completely straighten the rope? What I got as the solution is that the tension would not be defined or lets say it would be infinite. Well this answer seems reasonable to me. What it actually means is that practically the rope will never straighten. It will be bent by some amount, however small, until there is some weight. But, my friends say that my answer is absurd, as one can see by experimenting that the rope will straighten. But I say that it's actually not happening. When we do experiments with normal rope and small weights, and apply the tension which seems to straighten the rope, the deformation is so small that naked eye cannot observe it. I have calculated this mathematically. But still my friends refuse to accept the answer. So, I need a little help with this. If I'm really wrong, can somebody let me know what's the mistake?
You are completely correct. I assumed you saw/derived the equation and saw the infinite result [T*cotan(theta), T*tan(theta)] - can you explain in words what it means?
This is where the real world is not like the theoretical world. The maths uses a rope that is a single uniform medium, whereas a string is made up of multiple fibres which will react slightly differently. I would say though that on the whole the maths holds true until you get close to the horizontal and then breaks down as the slight variances in the experimental setup (e.g. string) will begin to dominate.
The only place where the theoretical isn't like the real in this example is if the string is assumed to be massless (which the OP does not). It is actually not all that difficult to calculate the actual curvature of a real string (think: power lines).
But I say that the real string, of which you are mentioning, will still not completely straighten, as it will have its own mass and elastic properties which only add to the factors that prevent it from completely straightening.
This is the kind of situation where observable common sense and physics collide. I would ask your friend how a purely horizontal force can ever become a vertical one, which is exactly what would be required if the supposition that the rope could be made straight requires. Its the sort of question, like if a gumball is sent down a spiral chute, what will be its trajectory on departure? Many folk would say in a spiral.