Double Atwood Machine: relation between the contraints & the variables ?

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SUMMARY

The discussion focuses on the Double Atwood Machine and the relationship between constraints and variables in the context of analytical mechanics. The key equations derived are (x2 + x1) - l = 0 and (2x1 + x2 + x3) - (2l + l') = 0, which relate the positions of the masses to the lengths of the strings. The Lagrangian, defined as L = T - V, is central to solving the problem, with the assumption that the radii of the pulleys are negligible compared to the string lengths. The user seeks clarification on the addition of the term πa, where a is the radius of the pulley, and its relevance in the equations.

PREREQUISITES
  • Understanding of Lagrangian mechanics
  • Familiarity with the concepts of kinetic energy (T) and potential energy (V)
  • Knowledge of constraints in mechanical systems
  • Basic proficiency in algebra and geometry
NEXT STEPS
  • Study the derivation of the Lagrangian for multi-body systems
  • Explore the implications of negligible pulley radius in mechanical systems
  • Investigate the role of constraints in the equations of motion
  • Learn about the principles of energy conservation in mechanical systems
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This discussion is beneficial for students of analytical mechanics, mechanical engineers, and anyone interested in understanding the dynamics of multi-body systems like the Double Atwood Machine.

bobmerhebi
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Double Atwood Machine: relation between the contraints & the variables ?!

Hello. I am taking an analytical mechanics course & there's 2 "simple" equations relating the constraints to the variables. The problem is actually a class example. Here is it

1. The figure of the example is attached.

We are supposed to find the Lagrangian L = T-V, but I was stuck at correctly proving the following (for which I am asking for help in proving). I should note that I was able to find L & the equations of motion by altering the figure & making use of the height of the masses relative to (my) chosen reference.:

(x2 + x1) - l =0 && (2x1 + x2 + x3) - (2l + l') =0

2. NO Relevant equations: simple arithmetic



3. The Attempt at a Solution :

Apparently l = xp + x (though another element should be added -- can you tell me why this cannot be added ? -- & that is: \pi a, where a is the radius of the pulley).

Moreover,

x3 = (l' - x') + xp = (l + l') - (x1 + x')
 

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well i now know how this turns out to be. simple said, the radii of the pulleys are assumed to be negligible with respect to the length of the strings.
 

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