10. Conservation of Energy
Force & Potential Energy
Potential Energy Graphs
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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.
A particle with a mass of 0.1kg moves according to the Potential Energy graph shown. What minimum speed does the particle need at Point A to reach Point B?
Forces & Equilibrium Positions
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Hey guys, so now that we've been introduced to potential energy graphs, there's a couple more conceptual points that you'll need to know about forces and equilibrium positions by using these graphs. So we're gonna work out this example together, let's check this out. The idea here is that in potential energy graphs you can get some information about the sign of the force by looking at the slope of the graph, the sign of f is going to be the opposite sign of the slope. What do I mean by that? Well, let's take a look at our example here, we have a ball that's obeying or following this potential energy graph. And we've got these four points of interest that are labeled here in part A we're going to figure out the sign of the force whether it's just positive, negative or zero by looking at these four points here. So we're gonna do is have A, B, C, and D. And to figure this out. I'm just gonna look at the slopes at each one of these points here. So let's take a look at a here at a the tangent line or the slope of the graph is sort of downwards like this. It doesn't have to be perfect. Just, you know, it's really just conceptual. So the idea here is that the slope of the graph is downwards and whatever you have downward sloping potential energies and therefore the slope is negative, the force is going to have the opposite sign of that. So the force is going to be positive in this case. So here we have a negative slope and so we have a positive force. That's the rule. Let's take a look at the second parts in point B, we're gonna have the sort of bottom of this little valley like this. And remember at the bottoms of the valleys and the tops of the hills, your slope is actually gonna be a flat line. So here we have a flat or horizontal slope. And whenever this happens, whenever the slope is flat or horizontal, a horizontal slope means a slope of zero, so therefore your force is going to be equal to zero. So here we have zero slope, so therefore f is equal to zero here. All right, so that's the answer. So here we've got a positive here, we've got negative. Let's move on to part C parts C is basically the opposite of a So your point, see, your your slope is actually gonna be upwards like this or positive. So if your slope is upwards and positive, the force is going to have the opposite sign of that, and it's going to be negative. So here we have a positive slope. Therefore we're gonna have a negative force. Now, finally, Point D. Is going to be basically the same thing as Point B. We have the top of a hill, so therefore your your slope here is gonna be flat like this. If you have a flat slope, it's basically you're just gonna be a zero force. So we're just gonna copy this thing over like this and that's gonna be your force. All right, so let's take a look now at points B and C. We're gonna figure out the positions of stable and unstable equilibrium. What does that mean? We'll remember that when you're force is equal to zero. We have a special name for that. That was called equilibrium. So just by looking at the potential energy graph here, we can actually get some information about when the force is equal to zero for an object and when it's at equilibrium. And depending on what the potential energy graph is doing at these points, these equilibrium is actually fall into two different categories. So there's two different types. The first one is called a stable equilibrium. This happens whenever you have the potential energy graph, which is at a minimum. So it's basically going to be right over here. So here this potential energy graph sort of like dips down like this and so therefore it's going to have a minimum value right here. So one way I like to think about this is that these minimums happen whenever the potential energy graph is curving up. So an unstable equilibrium is actually the opposite of this. An unstable equilibrium happens whenever the potential energy graph has a maximum, like it does in this point over here. So this happens whenever the potential energy is curving down. So one way I like to think about this is that if you're a stable person, you're likely pretty happy all the time. So this happens whenever you have sort of a smiley in the potential energy graph. If you're an unstable person, you're generally frowning, you're probably frowning all the time. And that's you know, usually what's going to happen here. So you can have a frowny face in the potential energy graph at unstable equilibrium. All right, So the reason these are called stable and unstable, it has to do with what happens when you have objects that are actually at these equilibrium points. So what I like to do is I kind of like to think about a marble that's sitting in a bowl right here at point B. So imagine you had a little bowl, right? And you put a marble inside of it and eventually it's gonna settle down towards the bottom. So this we know that the at the bottom here, the marble is going to be in equilibrium. What happens if you move it from either one of those from that position? If you move it to the left or to the right, What happens is the marble always wants to return back down to the bottom of the bowl. So the reason that stable is because if it's ever nudged from this position, objects are always going to return, so they're always going to return back to this position here. An unstable equilibrium is going to be like if you actually flip the bowl upside down right, you have to flip, you flip the bowl upside down and then you put the marble right on top. If you are able to perfectly balance the marble on top, the marble is going to be at equilibrium here. But what happens if you nudge it from that position? Well if you nudge it, then the marble just goes flying off the box or off the bowl like this, and it never can get back up to the top. So what happens is objects will never return back to the equilibrium positions once they are displaced or nudged from those places. That's basically what stable versus unstable means. So, to solve parts B and C really quickly here, your positions of stable equilibrium is going to be part point B because the F is equal to zero and we have the curving up and you're unstable equilibrium happens whenever your F. Is zero and you're curving down. So that's going to be here point D. So here point the your F. Is equal to zero, that's an equilibrium. But your potential energy graph is curving down like this. So that's really all there is to this uh for this one guys, let's move on.
Potential Energy Graph of a Marble
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Alright folks. So in this problem we have a marble and it's moving according to this potential energy function or this potential energy graph. And let's take a look at the first part here, which is we have to figure out the labeled points where the force on the marble is equal to zero. Now remember the big idea here guys is that the force is equal to zero when the potential energy graph, when you have X. Is flat. So that's the most important thing that you need to remember that there's no force when the potential energy graph is flat. So if you look at this at this graph here, this is not flat. This is not flat. There's only two points that actually are flat over here too with the labeled points which is point B and D. So that's the point in which momentarily the graph actually flattens out as it sort of curves as it changes from curving up to down. So in other words, the points in which the force is zero are going to be points B and D. Those are your two points here. Right. So it's going to be these two. All right. Now, let's take a look at the second part here. 2nd part asks us to calculate well which one of these labeled coordinates is a position of stable equilibrium. Remember we have stable and unstable equilibrium. And remember the rule you use a smiley face, if you're smiling, that means that you're stable. And if you have a sad face, that means that you are unstable. Right? So you want to be stable. right? That's that means good. So we're looking for the happy faces, right? So that means that the um it's gonna be point B. Alright, so that is point B. This is also just the one that is stable. This one is unstable, which means um that if you had a particle, when you remember one way you can think about this is if you had a little ball that was at the top of this hill, if it moved like a tiny bit to the left or right, it would fall down the hill. But if you had a ball over here at the bottom of this bowl, then it would just stay there, right. It would always just stay there unless something jostled it out of place. Alright guys, that's it for this one.