Virtual work in Atwood's machine

In summary, the first chapter of Goldstein's Classical Mechanics discusses Lagrange's equations and provides three examples of how to apply them to simple problems. The second example focuses on the Atwood's machine, where the tension of the rope is ignored due to the constraint being holonomic. Despite the constraining forces doing virtual work, their sum is zero according to the constraint equation. The motion in this problem is one-dimensional and the only external force is gravity, which is conservative. The pulley in the problem serves to physically justify the constraint, but can be ignored for the purposes of the Lagrangian formulation.
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kiuhnm
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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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The constraint is holonomic: forces do work, but the sum of the work is zero by virtue of the constraint equation.
 
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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.
 
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BvU said:
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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1. What is virtual work in Atwood's machine?

Virtual work in Atwood's machine refers to the work done by the tension force in the string connecting the two masses in the machine. It is called "virtual" because the string does not actually move, but it is used to calculate the work done by the force.

2. How is virtual work calculated in Atwood's machine?

Virtual work in Atwood's machine is calculated using the formula W = Fd, where W is the work done, F is the tension force, and d is the displacement of the string. This formula assumes that the tension force remains constant throughout the displacement.

3. What is the principle of virtual work in Atwood's machine?

The principle of virtual work in Atwood's machine states that the work done by the tension force in the string connecting the two masses is equal to the change in potential energy of the system. This principle is based on the conservation of energy and can be used to solve problems involving Atwood's machine.

4. How is virtual work related to mechanical advantage in Atwood's machine?

In Atwood's machine, the mechanical advantage is equal to the ratio of the weight of the heavier mass to the weight of the lighter mass. This is because the tension force in the string is equal to the difference between the two masses multiplied by the acceleration due to gravity. Therefore, the greater the mechanical advantage, the greater the virtual work done by the tension force.

5. What are some real-world applications of virtual work in Atwood's machine?

Virtual work in Atwood's machine can be applied in various situations, such as in elevators, cranes, and pulley systems. It is also used in engineering and physics to analyze and design mechanical systems. Additionally, it can be used to understand the concept of work and energy in a simple and practical way.

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