Chain slipping off a sphere - Mechanics problem A uniform flexible chain of length l rests on a fixed smooth sphere of radius R such that one end A of the chain is at top of the sphere while the other end B is hanging freely. The chain is held stationary by a horizontal thread PA as shown in the figure. Calculate the acceleration of the chain when the thread is burnt.

I have solved this and reached the correct answer, but I'm not sure whether the method I've used is 100% correct or not. Here it is:


Now I have a doubt:
Is it correct to integrate change in tension (dT) for each element this way, considering that the direction of dT is changing? Does it need to be further split into components? I was taught that integration should only be done for vectors that are unidirectional or be made unidirectional through components. Please help me with this.
Also, let me know if there is any other mistake I've made in this solution.
 A: What you're doing is perfectly fine! It is legitimate to integrate a force "in the chain direction", even when the chain changes direction. That's an example of a generalized coordinate. Instead of Cartesian coordinates, you're using "chain coordinates", which measure the overall displacement of the chain.
Generalized coordinates of this kind are properly justified in Lagrangian mechanics, but they're very useful even if you just know Newtonian mechanics. I give a few more examples of this kind of reasoning in this handout.
A: I'm also not sure about your $\int dT $ because the difference in tension at a point is already equal to the tangential force $(dm)gsin(\theta) $.
So I don't think you need to bother with the tension at all as it is already accounted for. The rest looks good though and maybe there is a good reason your T cancels in the end so I don't know... but to me the $\int_0^T dT $ just doesn't look right. Maybe someone else can explain if this makes sense or not.
Edit:
I also found a  video about a similar problem from a source which is probably not reliable, but they also don't bother with any tensions even if their reasoning is weird to me:
"There is tension but as we are calculating the total tangential force on complete chain, tension will be an internal force so it not needed here..."
If you calculate the difference in tension at a point it is already equal to $(dm)gsin(\theta) $, which is the expression you want to integrate.
