The last thing to think about when we're talking about friction applications and bearings and axles and all of this rolling friction and wheels is what happens if you got more than one of them together. Remember when we were talking about rolling friction the whole goal of it was to simply shift the reaction a distance over so that your equilibrium could work so that the moments could sum to zero. It's kind of the same thing when we were dealing with friction circles. We had to actually shift the reaction so that there was something to balance the moment on the axle. What happens when we're dealing with a wheel that's rolling that has both axle friction and rolling resistance? Do both shift R over by some B? It still acts tensed into the friction circle. The goal of all of these friction applications is to show it as an example of equilibrium. So in all of these cases what you need to do is read the problem, draw free body diagram, and then write your equations of equilibrium. The only question is where the forces act. All of the formulas that we've derived are simply straight from either the geometry or the equations of equilibrium or friction. Where does it slip? Those are the three sources. Once you understand those you can put them all together. In this case here's my wheel. There's my friction circle R is tangent to the friction circle and does not act immediately under the weight but offset at the distance. Once you have those three facts you can derive this formula easily. Just look at this triangle. You've got two triangles in here this one and this little one down here, up here on the one over here because this is theta so is that and so is this. That means that this height is RF over sine theta and this little height is B over tan theta and the whole height is the radius of the wheel. So the radius of the wheel is this one plus that one. Again not going to suggest that you memorize that but that when you have a particular problem you go ahead and draw the free body diagram this appropriate. In this axle and rolling resistance problem this is the free body diagram you will most likely look at. Remember that if you have a wheel on a hill sometimes these two forces are actually both components of the weight. It can simply be as you're rolling down a hill that the driving force is the component along the hill and that this force is perpendicular to the hill. That's one of the ways that we can apply this. One of the reasons I don't want you memorizing formulas because I can give you a slightly different problem and your equations of equal and ring are slightly different. You need to learn how to set up the free body diagrams.