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.