Guys, now that we've been introduced to potential energy graphs, there are a couple more conceptual points that you'll need to know about forces and equilibrium positions by using these graphs. So, we're going to work out this example together. Let's check this out. The idea here is that in potential energy graphs, you can gather 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 0 by looking at these four points here. So, what we're going to do is, I have a, b, c, and d, and to figure this out, I'm just going to 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; it just, you know, it's really just conceptual. So, the idea here is that the slope of the graph is downwards, and whenever 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 going to 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 going to 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 0. So therefore, your force is going to be equal to 0. So here we have 0 slope. So therefore, f is equal to 0 here. Alright, that's the answer. So here we've got positive; here we got negative. Let's move on to part c. Part c is basically the opposite of a. So here at point c, your slope is actually going to 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 going to 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 slope here is going to be flat like this. If you have a flat slope, it's basically you're just going to be a zero force. So we're just going to copy this thing over like this, and that's going to be your force. Alright? So let's take a look now at points b and c. We're going to figure out the positions of stable and unstable equilibrium. What does that mean? Well, remember that when your force is equal to 0, we had 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 0 for an object and when it's at equilibrium. And depending on what the potential energy graph is doing at these points, these equilibriums actually fall into 2 different categories. So there are 2 different types. The first one is called the 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 dips down like this. And so, therefore, it's going to have a minimum value right here. 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. 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 usually what's going to happen here. So, you're going to have a frowny face in the potential energy graph at unstable equilibriums. Alright. 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 going to settle down towards the bottom. So, this we know that at the bottom here, the marble's going to be at 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 it's 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 flip the bowl upside down, and then you put the marble right on top. If you're able to perfectly balance the marble on top, the marble's 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 these equilibrium positions once they're displaced or nudged from those places. So, that's basically what stable versus unstable means. So, to solve parts b and c really quickly here, your positions of stable equilibrium are going to be part point b. Because the f is equal to 0 and we have the curving up. And your unstable equilibrium happens whenever your f is 0 and you're curving down, so that's going to be here point D. So here at point D, your f is equal to 0, that's an equilibrium, but your potential energy graph is curving down like this. So that's really all there is to it for this one, guys. Let's move on.
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10. Conservation of Energy
Force & Potential Energy
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