1. The problem statement, all variables and given/known data All in the pic below. Part of the solution presented. Didn't present the whole thing as that would clutter the page. I just want to know how they set up the x coordinate for the particle. 2. Relevant equations This problem is just about using the lagragian. So my issue is with the first line in the solution. x=R*theta+R*sin(phi) where x is presumably the position of the particle relative to the lab frame. This is just using gallian relativity. 3. The attempt at a solution Shouldn't it be x=R*theta--R*sin(phi) instead? We have to minus the x position of the particle relative to the hoop frame from the x position of the hoop itself since we know that they are in opposite directions. As the particle slides down from the right moving to the left, the CM of the hoop must move right.