So, what we're going to do next is uh inclines or ramps. I mean, they're the same thing, but we usually call them inclines. So, if you have an incline or ramp, the way we approach these problems is a little bit different than you're used to, what we've seen so far in the two examples we've done. So, if you have something on an incline like this, the first step you got to do is draw all the forces acting on the mass. But the next thing you're going to do is use a weird coordinate system which is not like anything you've seen before. Maybe unlike anything you've seen before in your life. So normally the uh x direction is parallel to the ground uh you know right or left and then the y direction is perpendicular to the ground like up or down. When you use when you see something at an incline actually the best way to do it is not doing it like that that coordinate system. And what you do is this weird coordinate system, which is hard at first when you first see it, but after you see it a few times, it's actually not so bad. Instead of um saying the x direction is parallel to the ground, what we do is say it's parallel to the incline. Not only that, the x direction is parallel to the incline but down the incline. So inverse a force is going down the incline, it's in the positive x direction. It's going up the incline, it's going in the negative x direction. uh which is sometimes confusing because if you uh you know if you look at this here the x direction would be exactly as they drew it here but uh some people think that's going to left because it's kind of going left but it doesn't matter I mean the the idea here is the positive x direction is down the incline basically the negative x direction is up the incline that's always true no matter which way the incline's pointing right for our new coordinate system the y direction is perpendicular to the incline away from the incline. So, positive y is perpendicularly away from the incline. Negative y is perpendicular but into the incline like like this. So, using that coordinate system is weird at first, but it actually allows you to do the set up the equations easier in most cases. In fact, just about every place it's easier. So when you do it during using this coordinate system right fn is defined to be perpendicular to the surface right perpendicular away from the surface. So actually the definition of fn is in the y direction and normally there's a lot of forces either parallel or parallel down the incline and parallel up the incline. However the weight force is always going down right the weight force always goes straight down because it goes straight down in this new coordinate system. It's not in the y direction completely. There's a part of in the x direction, there's a part of in the y direction. It's kind of written the triangle messed up, but I'll draw it in the notes when we do it in the next example. It's easier to see when I draw it. But because the weight's going straight down has a part that's perpendicular into the incline and a part that's parallel to the incline. And you have to find the x and y components of the weight in this new coordinate system. So you have this right triangle here. It's like I said, a little bit hard to see, but you'll see it u more clearly when I do it in the notes. But there's going to be a a triangle formed by the weight and the angle between the weight which is pointing straight down and actually the y part of the weight is the same as the angle in the incline. So the inclination angle how much the angle you know the ramp is raised this angle here and these this angle here is the same for every single case right it's just a geometric sort of derivation that I'm not going to do but it's the same angle so this always confuses people too so if you drew this there's a right triangle here right see this is the x part right because it's going down the incline see is the x part opposite of this angle or adjacent to the angle Here's the angle, right? Is it opposite or adjacent? Opposite. It's got to be sign, not cosine. Everyone's used to using cosine with x. It's going to be sign for the x. It's also going down the incline, so it's going to be positive. The y part is actually adjacent, so it's going to be coine. It's going in, so it's going to be negative. So that's where these two things come from. The y part of the weight is minus w cosine theta. The x part is w sin theta. Right? Any it makes more sense when I do an example. So let's do an example. So this is the last example.