10. Conservation of Energy

Force & Potential Energy

# Potential Energy Graphs

Patrick Ford

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Hey guys. So occasionally in some problems, you have to calculate things like speeds and energies, but instead of using your energy conservation equations, you're actually gonna have to use these things called potential energy graphs. So I'm gonna show you how to use and sort of read these potential energy grass. We're gonna work out this problem together. Let's check this out. So the whole idea here guys is that potential energy graphs will graft the potential energy of an object and the Y axis versus the position of that object in the X axis. And these can actually tell us some pretty interesting things about the motion of an object without having to get your crazy work energy equations. So let's go ahead and get to the problem here, we have a marble that is following this potential energy graph and we're going to release the marble from rest at X equals one. So, but I locate X equals one on my diagram. Figure out where that point is, I'm going to call this point A. And we're releasing this marble right here with the speed of zero. In this first part, we want to calculate the total mechanical energy of the marble. We know exactly how to do that. The mechanical energy right at point A. Is just gonna be K. Plus you. So it's just gonna be the Connecticut A. Plus the potential at A. So how do we calculate this? Will remember kinetic is always related to speed and we just said that the speed at A is going to be zero. So there actually is no kinetic energy here at point A. So all of the mechanical energy is really just going to be the potential energy. Now, what I want to point out here is that this potential energy actually isn't necessarily gravitational potential, It's not spring potential energy. It's kind of just some vague potential energy that I don't really know what it's from. So we actually don't use an equation to calculate this. We're just going to look at this, we're just gonna look at the value on the graph at point a to figure this out. So here at .8 the Y value is equal to eight jewels. And that's our potential energy. So really our total mechanical energy is actually equal to eight jewels. That's the answer. So what ends up happening is that in general the mechanical energy at any point is going to be the K. Plus you at that point. Now, you only have to solve this once in a problem because we know the mechanical energy is always going to be conserved if the work done by non conservative forces is zero. And that's always going to be the case in these problems. So, what I like to do is I like to draw a little horizontal line once I've calculated the mechanical energy and I say that this is the mechanical energy of this object. Throughout the entire motion, it always has to equal eight jewels. This marble, no matter where it is on the graph always has to have a jewels of mechanical energy. So let's take a look at part B. Now part B. Were asked to calculate the kinetic energy at X equals three, so X equals three is right over here. So I'm gonna call this part point B. And before we actually go ahead and calculate this, I kind of want to use a roller coaster analogy. I like to think of this marble is kind of like a roller coaster cart that's traveling on some tracks. The potential energy graph is basically the roller coaster track. So as we're going from A To B. We're gonna be going downhill and therefore we're going to gain some speed and therefore some kinetic energy. That's what I want to figure out. So how do I figure out KB. Well remember the whole idea with these problems is that the mechanical energy is conserved. So that I can say that the mechanical energy at B. Is just gonna be K. Plus you. This is gonna be K. B. Plus you be. Now we actually know what the mechanical energy is because we said it always has to equal eight jewels no matter what. So we know this is eight. So all we have to do is just figure out what the potential energy the potential energy is that be in order to figure out what KB is. So we've got this eight jewels. This equals K. B. Plus. And then we're gonna do exactly what we didn't part A and part B at point B. Here, your potential energy really is just gonna be the Y. Value here, which is just too jules. So you have eight equals K. B. Plus two. So you have eight minus two equals K. B. And therefore you get six jewels. All right, so going back to our roller coaster analogy, this makes sense you're going from A and as you're going from a down to be, you're losing potential energy and therefore you have to gain kinetic energy for your total energy to be eight. So really the kinetic energy is just gonna be the difference between the mechanical energy and the potential at any specific points. So one way I like to visualize this is by basically looking at the potential energy graph. Right? So here part A. All of my mechanical energy really was just the eight jewels of potential energy. So here I had just eight jewels of only potential energy here at part B. I know that I've lost a potential and gains of kinetic. So what happens here is that my potential energy here at Part B is still equal to two jewels. And the kinetic energy is really just going to be the difference between where I am on the graph and my line of eight jewels. So this vertical line here really just represents my kinetic energy and I know that this is equal to six. Right? So let's move on to part C. Now part C. We're supposed to figure out the speed of the marble at X equals four. So here and back up to X equals four. So this is gonna be my point C. And this is actually very straightforward. We want to figure out the speed at V. C. We're really just going to use our roller coaster analogy, right? As we're going downhill, we're picking up speed. But then if you're going uphill, you're actually going to lose that speed again. So what ends up happening is if I started from rest here at point A and then I'm basically back up to the same heights, if you will, then the speed here at zero at sea also has to be zero. You can actually end up going any higher than the initial height from which you started from, unless you actually had some initial energy or initial speed, which you didn't in this case, So your speed here at C is going to be zero. And again, this makes sense because basically, you can't go anywhere above this eight jewels of energy. All right, it's actually kind of brings me to an important conceptual point, which is you're gonna have a speed here at zero and you're gonna have a speed here at zero. And basically what's gonna happen is at this point you're gonna go downhill at this point, you're gonna go downhill again. So without any additional energy that's added into this problem or removed, the objects are always going to remain stuck underneath this line. They're always going to remain stuck underneath my mechanical energy line between these two points right here, my VCR my age and my C. So these are actually called turning points because there are places where the marble is just gonna keep turning around forever. Unless it's given some additional energy, it's never gonna be able to escape this little sort of well, that it's fallen into. All right, So now let's move on to part D and part D. We're going to figure out without touching the marble again, right without actually inputting any energy into the system. Can it ever reach X. Equals five. So X. Equals five is right here. So I'm gonna call this point right here, Point D. So what we know here is that the mechanical energy for this marble throughout the entire problem has always been eight jewels. If we look at the energy that you would require to be at part D. We look across, We looked at sorry horizontally. And the potential energy you would need is 10 jewels. So what ends up happening here is that your mechanical energy? Your h jewels of mechanical energy will always be so always be less than the 10 joules of energy you would need. So the answer to this problem is actually no, you could never actually reach point D. One way I like to think about this also is if you have zero velocity at speed here, a point C, and you turned around and went back down the hill again, there's no way you can actually continue upwards and actually arrive at point D. All right, so that's if this one guys let me know if you have any questions.

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