>> Hello class. Professor Anderson here. Welcome back to the learning glass lectures on physics. We're still working on the name, but I kind of like that, the learning pool. Alright? Let's talk a little bit more about energy conservation, and if you guys have any questions as we go, just chime in and I'd be happy to talk about whatever you want to talk about. So, energy conservation is really not so bad. It's sort of given away by the words. Energy is conserved. Alright? Applying this is sometimes problematic, but let's see if we can simplify it a little bit. What we really mean is, the energy initially in the system has to be there finally in the system. The trick is keeping track of all the energy terms. Okay? Like we said, energy is not created or destroyed, it's just transferred. It's moved from different places to different places, and so we just have to be careful about keeping track of it. Alright. What do we have? Well, in the initial we maybe have kinetic, and maybe have gravitational. We maybe have a spring. In our final we might have kinetic. Still have some gravitational. Still have some spring. And, if we have friction involved then we know we're going to heat things up, and that's going to show up as thermal energy, okay? Heat. So, let's take a look at some of these terms in detail. We know what kinetic energy is. Kinetic energy is one-half mV squared. We also know what gravitational potential energy is, it's mgh, or mgy, let's put a variable there so we can think about height above our coordinate system. Spring potential energy is one-half kx squared, and where x is the compression or the stretch of the spring, how far has it compressed or stretched from its equilibrium position. Okay, if the spring is just sitting there at rest, then there's no potential energy in the spring. If you compress it, the distance x, then you put potential energy in to the spring. Okay? What about the right side of this equation? Well, we've got kinetic energy final which is one-half mVf squared. We've got potential energy due to gravity, final, which is mgy final. We've got spring potential energy, one-half kxf squared, and then we still have thermal energy hanging out. Okay, I like to call this the big energy equation, because this is going to help us solve a large number of problems. Okay, if these are the forces that are interacting with your system; gravity, springs, friction, then this is going to take care of the energy conservation that we're worried about. Alright, with this in mind, let's take a look at a roller coaster problem. Okay, let's take a look at one of the classic problems in physics which is the roller coaster problem, and we know what a roller coaster is. You start up here at some height, h, you coast down the first hill and then after that you are just coasting, hence the name roller coaster. Right? There's no x or a mechanism to drive the thing other than its initial potential energy. So, right off the bat that should tell you that the height of any hill after the first one, including a loop of radius R, none of that can be higher than the initial hill. Alright? Because, if you had a bigger hill, you wouldn't be able to get all the way up it if you are truly coasting. Okay, so think back to when you were a kid and you were riding on roller coasters that first hill that you go up, it's always the biggest. Okay? Let's see if we can figure out how big it needs to be in order for this roller coaster to just make it around the loop. Okay, this is an excellent problem, because it's going to incorporate some earlier stuff that we talked about with regards to circular motion. So, let's find the minimum. h, [writing] such that it still makes it around the loop. Alright. If the loop is radius R, then it is height two R, the diameter of the thing, and right off the bat we know that h has to be bigger than two R. Okay, that's pretty clear. But, if h is equal to two R, does it make it around the loop? What do you guys think? Who's got the microphone? And what's your name. >> (student speaking) Nick. >> Nick. What do you think? If I set h equal to two R, is my roller coaster going to make it around the loop? >> (student speaking) No, I don't think so. >> Why not? >> (student speaking) Umm, I don't know. I just, a guess. >> Okay. You've been on roller coasters before? >> (student speaking) Yeah. >> You know from experience that when you get to the top of the hill and you look out across the rest of the roller coaster, there's nothing as high as that initial hill, right? >> (student speaking) Right. >> Even any loop after that. >> (student speaking) Hmm mmm. >> And, what's going to happen is, if I start at height h and I end up back here at height h, I come to a stop, and then I fall. >> (student speaking) Right. >> And so, that's not going to work. We're pretending the roller coaster is not clung to the track. It doesn't have wheels on the underside of the track. It's only on the top side, so if it comes to a stop it's going to fall off. So, it's got to be bigger than that. So, let's take a guess that h has to be bigger than two R, and let's see if we can figure out what the minimum is. So, how about we go to conservation of energy and see if we can make some sense of this. Conservation of energy says that energy initial is equal to energy final. Initially, what is my energy? Nick, what is it? I'm starting up here. This is my initial position. Okay, and we're going to say that we are starting at rest. >> (student speaking) So, that'd be the potential energy, right? >> Good. So, it's just potential energy. We know what that is, mgh. Now, in this final position here what do I have? >> (student speaking) One-half mV squared. >> One-half mV squared, because I'm moving. Anything else? >> (student speaking) And there is still potential energy, right? Because you're -- yeah. >> Because we are up here at height two R and so we have to include that. Okay, and so that looks like a nice equation and we can simplify this quite a bit and solve it for h, right? Cross out all the m's. Now, I get h equals, I'm going to divide by g. So, I get one over two g times V squared, plus, when I divide out the g there that goes away. I have to add two R. So, that looks pretty good, except we don't know this. Right? We're given this, but this guy right here we don't know yet. How are we going to get V? Who has a thought how we can get V? Conner? You want to hand the mic to Conner? So, Conner we're looking for V, that's the thing that we don't know in this equation. Do you have any ideas on how we can get that? >> (student speaking) You could find the sum of the forces in the radial direction. >> Okay. Excellent. So, let's go back to our free body diagram for this roller coaster. We'll draw it right here, and what are the forces that are acting on the roller coaster? >> (student speaking) There's mg going towards the center. >> Okay. >> (student speaking) And, also the normal force going towards the center. >> Good. Okay, those are the two forces that are acting on the roller coaster at that point, and now we can do the sum of the forces in the radial direction. As Conner said, they're both towards the center of the circle, so I make those plus. And, when we did Newton's Laws in circular motion we know that they have to add up to mv squared over R. In this case our R is that capital R. Now, Conner if it is just making it, that means it is just about to lose contact with the track, what's the normal force equal to? >> (student speaking) Zero. >> Correct. Any time two surfaces are just losing contact with each other there can be no normal force anymore, right? If it's not actually on the track, there's no normal force that's pushing on it. And so, look what happens, we get a nice little expression now for V and we can plug it in to this equation. So, let's make some room and we'll do that. Okay, let's rewrite this equation up here. We have mV squared over R is now going to equal mg, and I need to plug in V squared here, so let's just solve this thing for V squared. I can cross out the m's right away, and I get V squared is equal to gR. And now, we can plug that in to this equation.