ï»¿ Hello, class. Professor Anderson here. Let's talk about the box on the frictionless incline problem. This is one of these things that's going to come back to haunt you again and again in physics, so it'd be good to get a handle on it right now. So, what does an incline look like? Incline looks like a triangle, okay. This is how we always draw inclines, and we always reference that angle right there, okay, not always but 95% of the time that's the angle that we give you. What is the angle of the incline relative to the horizontal? Now, if I take a box, and I put it on this incline, what is the box going to do? >> Slide It's going to slide. If there's no friction, if this thing is covered with ice, it is going to slide. What direction is it going to slide? Obviously, it's going to slide downhill, okay. Everybody knows this. Now, we need to figure out what this acceleration is down the incline. So, what do we know about acceleration? What we know is that if something falls straight down, then the acceleration is negative g, negative 9.8 meters per second squared, and that would be an a sub y if we're talking about the vertical direction. What about the a that we have here? Is our a equal to g, bigger than g, or smaller than g? >> Smaller than g. Okay. Raise your hand and say it out loud. >> Smaller than g. Why? Why would it be smaller than g? >> Because it's not going in the y direction; >> it's going in the x direction for the acceleration Okay. And probably just experience thinking about sliding things on inclines, they don't fall as fast, right. The rate that they fall does not match gravity, right. Everybody agree that it's got to be smaller than g? Okay. So, we're going to guess that it's got to be this. All right. So, I need to multiply g by something that is smaller than 1. Now, whenever you see a theta, what do you think about in terms of other functions? Whenever you see a triangle like this, particularly a right triangle, what do you think about when you see theta? Yeah? >> Sine, cosine, and tangent. Sine, cosine, and tangent, right. Sine is opposite over hypotenuse; cosine is adjacent over hypotenuse; tangent is opposite over adjacent. Good old sohcahtoa. You guys remember the other moniker I told you to remember, right, some old hippie caught another hippie tripping on acid, right. That's how you remember sines and cosines. So, anytime you see this, anytime you see a triangle, you should immediately think, oh, there's got to be some sines and cosines involved here, absolutely. And in fact, that's good because if I write sines, cosines, or tangents, I can maybe make some sense of this. So, let's try a couple. Which one of these are likely to be the right answer? Well, we know that it's got to be less than g. We said from experience it has to be less than 1 that we're multiplying it. Is tangent less than 1? Is tangent less than 1 for all angles of theta? No, it's not, right. In fact, tangent goes to infinity for certain angles of theta. So, it can't be that one. That one's gone. G cosine theta, is that a possibility? >> Yes. Yeah. Cosine theta is always less than 1. Sine theta is also a possibility because sine theta is always less than 1. All right. So, now we're down to these two choices, and this is the way you remember how to do this. This is one of the ways. Let's look at the limits. Let's let theta go to 0 degrees. So, if theta equals 0 degrees, what do we get for these accelerations? Well, if a is equal to g cosine of 0 degrees, what's cosine 0 degrees? >> Zero. No. >> One One, okay, cosine 0 degrees is 1. What about sine of 0 degrees? That's the one that's 0, right. So, we get 0. So, now all we got to do is figure out which one is the right limit. If theta equals 0, should we have an acceleration that's equal to gravity, or should we have an acceleration that's equal to 0? And remember, we define the acceleration down the incline, okay. So, let's look at that. If theta equals 0 degrees, or we'll say it's approaching 0 degrees, then what does my triangle look like? It looks like nearly a flat surface okay. And if it's a flat surface, what do you think the acceleration should be in that direction? What does it have to be? (Students) Zero. Zero, right. If I put an object on a level surface, it's not going to suddenly rush to the right or to the left. It's got to be 0. And so immediately, you know which. The answer is acceleration down the incline is equal to g sine theta. Okay. Hopefully, that's clear. If not, come see me in office hours. Cheers.