# Virtual Work in a Pulley System

1. Aug 6, 2015

### lowea001

1. The problem statement, all variables and given/known data
The system shown in Fig. 2-6 is in static equilibrium. Use the principle of virtual work to find the weights A and B. Neglect the weight of the strings and the friction in the pulleys.

Quick aside: This is straight out of the Exercises for the Feynman Lectures on Physics, in which he introduces virtual work as an imaginary displacement that let's us find the weight or height of objects in what he calls a "reversible machine".

I know how to do this equilibrium problem using force vectors, but am not sure where to begin with virtual work. Can anyone help me to solve the question using this method?

2. Relevant equations

Ug = mgh
Total Ug = 0

3. The attempt at a solution
Right off the bat I know the answer to this question, I'm desperate for a solution/explanation.

I begin by imagining the 1kg weight falls 1 unit, which means that weight A must rise 1/2 a unit based on the angle of the string. Then I don't know whether or not I should include weight B in this displacement, or if it should fall, or if the entire right side of the system should rise according to the displacement of weight A in order to preserve the geometry of the system. After a bit of easy mathy stuff you can get the equation 0 = -1(1) + 1/2(A) + (whatever B does)B. I've tried many different imaginary displacements for B, but can't seem to get the right answer. This isn't very difficult using force vectors, but the question specifically asks for it to be solved using the method of virtual work, which I am admittedly unfamiliar with. What do I have to imagine the system does here? Maybe my original guess that weight A should rise a distance of Sin30 is wrong as well.

2. Aug 6, 2015

### TSny

Did you try virtual displacements of the knot where the three strings meet?

If the knot is moved horizontally by an infinitesimal amount $\delta x$, what are the vertical displacements of the three masses to first order in $\delta x$?

Similarly, what happens if the knot is displaced vertically by $\delta y$?

What do you mean when you state that "Total Ug = 0"? How do you use that?

3. Aug 6, 2015

### lowea001

Sorry, I meant to say that the total Wg = 0, not the energy. So if I move the weight A down by an infinitesimally small 1 unit, then the 1kg weight must move up 2 units and weight B must move up √2 units. This gives me A = 2 + √2(B). My mistake is somewhere here. If I perform a horizontal displacement of 1 unit right, weight B must move down √2 units and the 1kg weight must move up (doing the math) 2/√3 units. Weight A is horizontally displaced so it remains at the same height. This gives me B = 2/√6, which I know is incorrect. Where did I go wrong? Am I doing virtual work incorrectly? Thanks for your help so far. :)

4. Aug 6, 2015

### TSny

You have the right idea that the total work done by gravitational forces will be zero for a virtual displacement.

However, your calculations of the displacements of B and the 1 kg mass are not correct. Can you explain how you got these answers?

5. Aug 6, 2015

### lowea001

To be honest, not very well. I may have made many false assumptions as the whole concept of imaginary work confuses me at the moment. I know that weight A can be said to move down 1 really small unit. From there, I need to know the changes in heights of the other weights, and multiply all the weights by their respective changes in heights and equate them to zero. I know that if weight A moves down 1 unit and the angle 30° is preserved, the string must have been translated a distance of 2 units because 1/Sin30 = 2 = hyp = distance the string moved. Is this a false assumption? Or is the assumption that the angles are preserved false? I think I'm getting warmer. Kinda.

6. Aug 6, 2015

### lowea001

Or maybe I need to use Pythagorean theorem.

7. Aug 6, 2015

### TSny

Suppose the knot is moved vertically upward a distance dy from point a to point a' as shown below. The portion of the string that goes from the knot to the right pulley changes length from d to d'. Can you work out how much shorter d' is than d? You just need to get an expression that is accurate to first order in dy.

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8. Aug 7, 2015

### haruspex

The method I use is to think in terms of displacement (or velocity) components. If the knot above A moves straight down at some rate, what is the component of that in the direction of string going up to the left hand pulley? What is its component perpendicular to that string? Which of these will turn the pulley?

9. Aug 7, 2015

### TSny

Yes, that's a nice way to think about it.

10. Aug 7, 2015

### lowea001

Yay I finally solved the question, thanks a lot guys. By doing what haruspex suggested, I found the components of the virtual velocities/displacement vectors that ran in the same direction as the pulley strings. Moving the knot horizontally means that the height of A doesn't change and you can just solve for B based on what the 1kg weight does and equating the total work done to 0. Then, I can substitute the value for B into the equation you get for the total work done from a vertical virtual velocity/displacement. Basically, it wasn't too tough after all. Thanks for your guidance.

Just out of curiosity Tsny, how would you find out how much shorter d' was than d in your diagram? Would you connect d to a' with a line perpendicular to d and treat the base of that triangle as the distance you're looking for?

11. Aug 7, 2015

### TSny

Yes! Since dy is infinitesimal, the lines d and d' will be approximately parallel to the accuracy we need. So, dropping that perpendicular line as you suggest will give the the difference between d and d'. (See figure below.) d-d' will be how much mass B moves down.

#### Attached Files:

• ###### virtual dy blowup.png
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Last edited: Aug 7, 2015