What's my mistake in this problem in dynamics involving pulleys?

[haruspex]

Accelerations of block A and block B.

So what equation do you get by carrying out the differentiation in post #16?

 

[Lnewqban]

...
Please explain me as I have no reason to believe that block A and B will have same acceleration.

Another way to look at this problem:
A moveable pulley can be considered as a second class lever with mechanical advantage of 2.

 

[Adesh]

So what equation do you get by carrying out the differentiation in post #16?

Acceleration of the blocks.

 

[Adesh]

So what equation do you get by carrying out the differentiation in post #16?

But there is something that is troubling me, $(-l_1, 0)$ is the current position of the block A and $(0, -l_2)$ is the current position of block B, but acceleration is not the second derivative of fixed positions (because in that case it will always come out to be zero). We should say, that at $t=0$ the positions of the blocks were so and so.

 

[haruspex]

But there is something that is troubling me, $(-l_1, 0)$ is the current position of the block A and $(0, -l_2)$ is the current position of block B, but acceleration is not the second derivative of fixed positions (because in that case it will always come out to be zero). We should say, that at $t=0$ the positions of the blocks were so and so.

You did not define l₁ etc. as merely initial positions. Why should they not mean positions at time t, i.e. define l₁ = l₁(t) etc.
Also, you don't need to use vector representations. Just define each displacement in the direction that suits it.

 

[Adesh]

You did not define l₁ etc. as merely initial positions. Why should they not mean positions at time t, i.e. define l₁ = l₁(t) etc.
Also, you don't need to use vector representations. Just define each displacement in the direction that suits it.

Yes, this advice and way of solving it is helping me very much. Now, I’m able to solve almost all problems of pulleys and strings, credit goes to you.

I think these pulleys and strings problems are archetypical, textbooks on Newtonian Mechanics (like French’s Mechanics, Univeristy Physics) don’t actaully teach it exclusively.

Now, I’m into elevator problems .