Suppose there is a rod hung vertically from the ceiling. It experiences gravitational force W (its weight).It's Young's Modulus is $Y$. Now I know that the end connected to the ceiling experiences a lot more force as it has to bear the entire rod. Whereas the bottom doesn't have to bear anything. So clearly there is non uniformity in stress. In increases gradually $(\propto length)$ from the bottom. So the net strain should be $$ \delta L=WL/(2AY)$$
There is a rod kept on a smooth horizontal surface and is pulled by a Force $F$ on one end. The other end is free. My book says the tension experienced by a differential length at a distance $l$ from the pulled end decreases linearly. The reasoning is that the differential experiences a force experienced by the complete right hand part $l$. Treating this as a statistical phenomena only, the net elongation is $$\delta L=FL/(2AY)$$
Next the book says that if we pull by an equal force F from the other side too, the elongation would be double. so now the elongation is $$\delta L=FL/(AY)$$
My issue is, isn't Case 1 a particular case of the modified case 2 (pulled by 2 equal forces F). Here F=W. I say this because the rod is hung. It is clearly in equilibrium. The ceiling too exerts a reaction force equal to W. So the hung rod is also pulled by equal forces on 2 sides. Using the general formula explained in case 2, the elongation should be double of the one written.
I am very uncertain about Case 2. I think it is conceptually very clumsy. But it's a general formula mentioned by the book.
I don't want to learn a wrong concept so please clear this confusion. Either I am missing something or the book is wrong.