1. The problem statement, all variables and given/known data A cart on an inclined plane is balanced by the weight w. All parts have negligible friction. Find the weight W of the cart using the principle of virtual work. Picture of set-up is attached. 2. Relevant equations [tex] ∫Fdh = 0 [/tex] for virtual displacement dh in statics [tex] F_p = W cos \theta [/tex] the part of the weight of the cart parallel to the slope [tex] ∑F = 0 [/tex] [tex] W = 4w/sin\theta [/tex] This is the given solution, by the book. 3. The attempt at a solution I tried two ways to get the solution: virtual work and forces but I got stuck in the virtual work solution, and I did not get the right answer with forces. So here is the force analysis first: There are two pulleys in the diagram: one attached to the slope at the bottom, holding w, and the other attached at the top holding W. Lets call the one holding W P, and the other p. The tension in P must be such that it balances the parallel component of W, as given by the formula in Relevant equations. Lets call that tension T. The perpendicular component of W can be ignored because it is balanced by the normal force. Because P supports p, the downwards force on p must be equal to T. I assumed that both p and P are massless. Then the force on p is w, plus the pull of the slope on the other side. Since the strings in p are not moving, the pull of the slope must equal that of w. The downwards force on p is the sum of the two, namely 2w. This must equal T, and since T is the parallel component of W, by replacing I end up with: [tex] 2w = W sin \theta [/tex] Which is missing a factor of 2 to get the correct answer. Where did I miss that factor? To apply the principle of virtual work, I let w fall a distance equal to the height of the slope, h. Then the point of the slope attached to p will rise by h, and the point attached to P rises by a h cos(theta). If P rises by h cos(theta), the string holding p will lengthen by h cos(theta), so the cart moves along the hypotenuse by this distance. The slope rotates around the point without any pulley by theta. I drew two triangles to represent the slope. In the rotated one, I drew another triangle to figure out the height difference, and it was proportional to h. I then used the change in height of the pulley to calculate the change in height due to the pulley pulling the cart. Or rather I tried. I have absolutely no idea of how to do that because it does not form a triangle. Of course, I could calculate the height change again using proportionality, but then I end up with two proportionality constants and I have no idea of how to find them. Anyone can help me? Is my approach correct, and if so how do I calculate the change in height for W? If not what would be the correct approach, that is, what do I vary to apply the virtual work? Also why is my Force analysis missing a factor of 2? Thank you for answering the questions.