how to find the constraint equations?

The only difference between a constraint equation and eg a conservation equation is that a conservation equation is physics, but a constraint equation is geometry.

With eg a system of three pulleys at heights p q and r, we get physics equations (usually F = ma) for each pulley,

but the "a" in F = ma is different for each pulley

(in fact, it's p'' q'' and r'' respectively)

so we need a geometric equation relating p q and r …

usually this simply tells us the length of the string in terms of p q and r …

(we can also have eg a rolling constraint equation, relating the linear speed and angualr speed of a rolling object)

since we know that that length is constant, we can differentiate once (or twice) to get a neat "constraint equation". 

(which will be a scalar equation, applicable to all kinds of pulley-string-block arrangements, and since it depends on the length of the string it's the same in all coordinate systems)

1) in constraint relationship we diffrentiate length of string only(though i might be wrong but i have encounterd only such ques)

2)these are vectors(as diff gives negative and positive velocities)

3,4) as it is pure kinematics u can take ant frame of refrence inertial or non inertial

6) can be applied to any question even simple questions can be done through this

7) i dont know what you mean

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