Constrained motion with movable pulley

[cr7einstein]

Homework Statement

In this, if I want the acceleration constraint between M and 2M, I write $$AM+2AB$$=LENGTH OF STRING, which on differentiating twice gives $$a_{m}=2a_{2m}$$(which turns out to be correct). However, if we look closely, the, lower end of string is FIXED to the pulley A, and hence, it should have zero acceleration. Indirectly, the differentiation of lower end of string must be $$0$$, which is possible only if the lower string is of constant length, which is obviously wrong, as the pulley rolls over it when pulled, and hence it is not constant. Now, the arithmetic mean of accelerations of 2 ends of a string on a pulley(signs included) gives the acceleration of the opposite end( a trivial result form constrained motion). If I try to use this here, the acceleration of upper string =$$a_m$$, as thy are directly connected, but as the other end is fixed, it should have a zero acceleration, and thus $$a_{2m}=(a_{m}+0)/2$$ , which again gives the same result. Now, the problem is, if the acceleration of this end is zero, it would mean that the displacement is zero ( because the same relation as the displacements exists between accelerations and velocities). However, when we pull the string, the pulley sort of 'rolls over' the lower string, so the length of string is NOT the same as the earlier length, as even though one end is fixed, the increase in length of other side must lead to a decrease on the other side. Hence, the length cannot be constant, so why must the acceleration be zero? I know it has to be, because one end is fixed, but then, when we derive constraint equations, we differentiate all the lengths which are variable. The only problem is, according to 'common sense', the acceleration must be zero, but as it turns out, the length is variable, and hence it should be nonzero. Obviously, I am missing something, can someone point out what exactly??

Thanks in advance!

Homework Equations

The Attempt at a Solution

 

[Simon Bridge]

However, if we look closely, the, lower end of string is FIXED to the pulley A, and hence, it should have zero acceleration. Indirectly, the differentiation of lower end of string must be
0
, which is possible only if the lower string is of constant length, which is obviously wrong, as the pulley rolls over it when pulled, and hence it is not constant.

An object can have zero acceleration and still change it's length (or anything you like about it's shape).
Be careful with your definitions though - you have tried to apply rigid-body reasoning to a non-rigid body (the string).
i.e. if the acceleration of the lower portion of the string is the rate of change of position of it's center of mass, then the acceleration won't be zero. Also take care you don't confuse zero acceleration with being stationary.

The rest has similar reasoning issues - rewrite it being more careful and see if you don't resolve much of your problem.

 

[cr7einstein]

So are my methods of deriving constraint equations correct?(both-the AM way and the differentiation)? Can you suggest a more rigorous and instructive approach please?

 

[Simon Bridge]

I didn't bother analysing that far - no point.

 

[cr7einstein]

No that's okay but is my method correct? Because it kind of raised plenty of doubts for me..

 

[Simon Bridge]

No.. it is not correct.
I cannot help you if you will not take advise.

 

[cr7einstein]

No I am ready to accept your advice, what exactly would you advise if I asked you to derive the constraint equations for the given system?