Virtual work in Atwood's machine

  • #1
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The first chapter in Goldstein's Classical Mechanics ends with 3 examples about how to apply Lagrange's eqs. to simple problems. The second example is about the Atwood's machine. The book says that the tension of the rope can be ignored, but I don't understand why. The two masses can move vertically and the constraining forces (one on each mass) have the same vertical direction so shouldn't they do virtual work?
 

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  • #2
BvU
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The constraint is holonomic: forces do work, but the sum of the work is zero by virtue of the constraint equation.
 
  • #3
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In the case of the Atwood's machine the motion is one-dimensional and you have only one coordinate, which by the way, is your generalized coordinate.

The only external force is gravity, which is conservative. There follows the calculations.
 
  • #4
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The constraint is holonomic: forces do work, but the sum of the work is zero by virtue of the constraint equation.

OK, thanks. I was confused by the remark "This trivial problem emphasizes that the forces of constraint--here the tension in the rope--appear nowhere in the Lagrangian formulation."
Let's say I want to be extremely formal. How would I proceed? The constraint is ##x_1+x_2=l##, where ##x_i## is the position of the ##i##-th mass. How do I deduce from that, analytically, that the sum of the constraint forces is ##0##?

edit: On second thought, I think the pulley is there just to physically justify the constraint. The pulley is massless and frictionless so the text of the problem is basically saying to ignore it and identify it with the constraint.
 
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