How do constraint equations in mechanics work?

[Ashu2912]

How do constraint equations in mechanics work?

Hi, friends! I'm having some trouble understanding the constraint equations:-

(1) How do they relate the length of the string to the position of the block attached to it? The position of the block must be a vector and it must be differentiated to get the velocity and then the acceleration. However in my book, they have just differentiated the length of the string, which they have taken as the position of the block, to get the velocity and then the acceleration...

(2) Are they scalar or vector relations?

(3) Are the with always derived with respect to some fixed frame of reference?

(4) If no in question (3), do we consider relative velocities/accelerations in moving pulleys, like acceleration of block with respect to pulley, and then use the relative velocity/acceleration equation?

(6) Are they applicable to all kinds of pulley-string-block arrangements, because I was under the impression that it is applicable only if the blocks move in opposite directions...

(7) Are constraint equations coordinate system and origin (fixed) dependent, or are they general relations applicable in all (fixed) coordinate systems?

 

[tiny-tim]

Hi Ashu2912!

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 https://www.physicsforums.com/library.php?do=view_item&itemid=632" 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)

if that's not clear, can you ask about a constraint equation for a specific system?

 

[darkxponent]

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 don't know what you mean