How do constraint equations in mechanics work?

In summary, constraint equations in mechanics are used to relate the length of a string to the position of a block attached to it. This allows us to differentiate the length of the string to obtain the velocity and acceleration of the block. These equations are scalar and can be applied to all kinds of pulley-string-block arrangements, regardless of the direction of motion. They are also independent of the coordinate system and origin, making them applicable in all fixed coordinate systems.
  • #1
Ashu2912
107
1
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?
 
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  • #2
Hi Ashu2912! :smile:

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". :wink:

(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?
 
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  • #3


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
 

1. How do constraint equations affect the motion of a system?

Constraint equations in mechanics describe the limitations on the movement of a system. These equations restrict the degrees of freedom of the system, which in turn affects the motion of the system. In simpler terms, constraint equations control how a system can move.

2. What is the purpose of using constraint equations in mechanics?

The use of constraint equations in mechanics is essential in accurately describing the motion of a system. Without these equations, the behavior of a system can be unpredictable and difficult to analyze. Constraint equations help simplify the analysis of a system's motion and provide a more complete understanding of its behavior.

3. How are constraint equations derived?

Constraint equations are derived using the principles of mechanics, such as Newton's laws of motion and the principle of virtual work. These equations are formulated based on the restrictions or limitations placed on the motion of a system. The derivation process can vary depending on the specific constraints present in the system.

4. What happens when a system violates a constraint equation?

If a system violates a constraint equation, it means that the system has moved in a way that is not allowed by the equation. This can lead to inaccurate predictions and analysis of the system's motion. In some cases, the violation of a constraint equation may also result in the system being unstable or in a state of self-collision.

5. Can constraint equations be solved for unknown variables?

Yes, constraint equations can be solved for unknown variables using mathematical techniques such as Lagrange multipliers or the method of virtual work. These methods allow for the determination of unknown variables while still satisfying the constraints placed on the system. This is important in accurately predicting the behavior of a system in real-world situations.

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