# Problem in constraint equations

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?

This isn't really a homework problem, more of a conceptual one (in any case, I did solve it; just that I developed a new kind of doubt when I solved this. So this question is more of an example to explain my question; and hence, I am not tagging it as 'homework' :).

I find a conceptual error in the statement

if we look closely, the, lower end of string is FIXED to the pulley A, and hence, it should have zero acceleration

fact is that you have been told that on differentiating the length twice we get acceleration which is not wrong but lacks a bit of concept.

think about such a setup. You would find the mass M (hanging) falling down. Forget everything except the string just coming from the pulley. Apparently it appears that length increase. Finding rate of " increment in the length" will give you velocity of any particle on the rope( or string.)

But that velocity is changing. Since there is some force( doesn't matter which one now"). So again find the rate at which velocity is changing( or in complicated way of saying rate of "rate of " increment in the length""[ its not that i have written "rate of" twice by mistake]. So now it gives me the acceleration of any particle in the rope. I hope you were thinking only about the rope whose length was increasing. now rate at which the length increases is equivalent to the velocity of the mass. And the rate of velocity(rate at which the length increases ) will be the acceleration of the mass itself.

So even if one end APPEARS to be fixed it doesn't matter since the length is always increasing.

Above wherever it i have used "rate of" it means Differentiating. So if you differentiate twice it will give you the acceleration.

I tried to make you understand why we differentiate which should answer your question( problem) . If it doesn't yet let me know , I would be pledged to give another try.

• thanks a lot for the answer...i realised the mistake almost as soon as I posted the question =P....upvoted and accepted nevertheless. Commented Feb 14, 2017 at 15:48

the differentiation of lower length of string is not zero, it's change is "supplied" by the mass M. Just because one end of a string is fixed doesn't imply that its "acceleration" is zero as we can have two acceleration of opposite side acting on parts of string just like described above, in pulley method, both upper and lower strings have same acceleration and importantly, the above string does not have same acceleration as M, it is half of M as the rest half is lost in the lower string to keep the strings parallel