Potential energy and conservative forces (part 5)

0:00- [Instructor] When you take the negative derivative of potential energy with respect to position, you get the expression for the underlying conservative force. The goal of this video is to see how incredibly powerful this equation is, and to develop a brand-new intuition behind it.

0:14So let's begin. All right, the first question for ourselves is, what exactly is this equation telling us? It's saying that the force equals a negative rate of change of potential energy with respect to position, which means if we can visualize a potential energy graph, then the force has something to do with the slope of that graph.

0:32But let's dig deeper. Let's think about the magnitude and the direction of the force. If you only concentrate on the magnitude of the force, the strength of this force, then we can ignore the negative sign, that equals what is dU over dx? Well, that represents the slope, right?

0:48But slope of what? It's a slope of the graph of potential energy versus position. We just call that as potential energy graph. So our first insight is that the magnitude of the force is given by the magnitude of the slope of the potential energy graph. Okay, now let's talk about the direction.

1:08What is this negative sign saying? Well, think about it this way. If dU/dx is positive, that means the change in potential energy is positive, that means that the potential energy is increasing. What is negative of that? Well, the potential energy is decreasing. (laughing) So this is telling us, this question is telling us that the direction of the force is in the direction where the potential energy is decreasing.

1:32So our second insight is that the direction of the force, well, it's in the direction where the potential energy is decreasing. So now let's put it in action. In fact, let's consider an arbitrary system. Let's not worry about whether we're dealing with system of springs, or, you know, gravitational force, or let's not worry about which conservative force we're dealing with, but let's just consider a random system, okay?

1:55Let's say that the potential energy graph looks like this, some random graph we have. What is this graph trying to tell us? Well, what this is saying is that in that entire system, whichever system we're dealing with, if you pick one particle, okay, then how does the potential energy of this system change as the position of that one single particle changes?

2:20So when the particle is at origin, the potential energy is this much. When the particle comes from here to here, the potential energy reduces, then the potential energy increases, then the potential energy reduces, then the potential energy increases. So you can see it's a very, very complicated system.

2:34So let's not worry about which system we're dealing with, but the question now is, can we gain some insights about the force that is acting on that particle? So let's think about its magnitude and direction separately. So what we see is that the magnitude equals the slope of the PE graph.

2:52So let's analyze the slope. As I start from here, and I go towards the right, notice from here to here, the potential energy is reducing. I don't care about that, what I care about is the slope. The slope is pretty much a constant. So I can say that the force, the underlying force acting on my particle is pretty much a constant from here to here.

3:15As I go from here to here, the force is pretty much a constant. But then what happens to the slope as I go from here to here? The slope, look, it's becoming less and less steep. It's becoming flatter and flatter and flatter. And eventually at this point, the slope is zero.

3:30That means at this point, the force or at least the x component of the force, because you're only considering one dimension, okay? So the force in the x direction, that goes to zero. This means if that particle is over here, remember, this is where the particle is allowed to move, it's one dimensional case, okay?

3:50So if the particle is over here, then the force acting on that is zero. In other words, our particle will be in equilibrium. So now we have some insight about how the magnitude or the strength of that force changes, varies, as we go from here to here. It's pretty much a constant, and then it reduces to zero.

4:09So let me use some dots to represent it. So the size of the dots is, you know, representing the strength of the force. So it's pretty much a constant, and then goes to zero. It's not very quantitative, it's very qualitative, ikay? All right, now what happens as I go from here to here?

4:25It'll be a great idea to pause the video and think if you can comment on the strength of the force for the rest of the graph. All right, so as I go from here to here, the potential energy increases, but I don't care about the value of potential energy, I care about the slope.

4:41That's what this, you know, the force depends on. So let's look at the slope. You know, it starts from zero at equilibrium point, and then the slope increases, it's becoming steeper and steeper, steeper, and then the slope decreases and you go back to zero. So you get another equilibrium point over here because the slope is zero.

4:58So how does the force varies? Well, it starts from zero, increases, and then decreases back to zero, okay? And then what happens when I go from here to here? Well, again, look, the slope increases. Ooh, it becomes very steep at this point. So the magnitude of the force will be very strong.

5:13And then again, goes to zero, you find another equilibrium point. So we get a very strong force at this point. The force value is very strong at this particular position. And then finally, as I go from here to here, what happens? Well, again, the force, the strength of the force increases, increases drastically.

5:29Very strong slope over here, so very high force over here, strength of the force is very high. And then again, eventually decreases to zero. And then look, the slope is pretty much zero because it's flat. So from here to here, the force increases, decreases, and goes back to zero, and then stays zero.

5:51Isn't this wonderful? I mean, think about it. This is some complicated system, I have no idea, and yet, just by looking at the potential energy, I can comment on what's happening to my force. That is the power of potential energy graphs. But this is just the magnitude of the force.

6:06We can now also comment on the direction. Okay, let's think about this. Remember, the direction of the force is always in the direction where the potential energy is decreasing. So let's consider this section over here. As I go to the right, the potential energy is decreasing.

6:21If I go to the left, the potential energy increases. So which direction should be the force? The direction which the potential energy is decreasing. So the force in this particular region is to the right. Now, this is not representing the magnitude, this is only useful for direction.

6:38We already have insights about the magnitude, okay? Okay, what is the direction of the force in this region? Well, let's see. If I go to the right, the potential energy increases. If I go to the left, it decreases. It's in the direction where the potential is decreasing.

6:55So at every point over here, the force would be to the left. Again, it'll be a great idea to, you know, think about the direction of the force in this region, this region, and this region. All right, let's do this. Over here, the right side is where the potential energy is decreasing.

7:09So the direction of the force is to the right. Over here, if you go to the right, the potential energy is increasing. If you go to the left, the potential is decreasing, so it's to the left. What is the direction of the force over here? Hey, (laughing) the force is zero, so it doesn't have a direction.

7:23And so what is so incredible about the potential energy graph is that not only do have insights about the magnitude, but I also know the direction of the force. So if I put it together, I can say, as I start from here, the direction of the force acting on this particle in this complicated system is to the right to begin with, okay?

7:43And then it pretty much stays a constant, and then it reduces, goes to zero. And then it flips over, and then again the strength increases, goes back to zero. And then again it flips, and then the strength increases drastically again, goes back to zero. (laughing) And again it flips, increases drastically, it goes back to zero, and then it stays zero.

8:01From here onwards, the particle does not experience any force. Isn't this credible? But there is more. Look at these equilibrium points, okay? If I were to keep my particle over here, then the force acting on my particle would be zero, it would stay put. But what would happen if I were to nudge it slightly to the right?

8:26Well, then the force is to the left, so it'll move back to the left. And then what if it gets nudged slightly to the left? Then the force is to the right, it moves back to the right. Notice that the system is trying to push this particle back towards the equilibrium position.

8:42Such equilibrium positions are called stable equilibrium. And it makes sense, right? If you nudge it a little bit, it'll tend to come back. The system drives it back to that equilibrium point. The same thing applies over here. Notice the force is driving it back, the particle back to the equilibrium position.

8:59So this is also a stable equilibrium. But what about this point? Even here, if you were to position the particle at this location, the force acting at that particular location is zero. But what happens if you nudge it a little bit? If you nudge it a little bit to the left?

9:16Look, the force is no longer driving it back to the equilibrium, it's taking it away. If you nudge it to the right, again, look, the force is taking it away from the equilibrium position. The nature is no longer driving it back. Instead, it's driving it away from the equilibrium position.

9:31So good luck trying to keep a particle, you know, stay put over here. Small nudges will take it away from the equilibrium position, and therefore this is called unstable equilibrium. So look, maximas in your potential energy graphs are unstable equilibriums, and minimas, local minimas, are stable equilibriums.

9:51Why? Well, because think of it this way. When you're at a minimum, on either sides, potential energy increases. And so system tries to drive it back to the decreasing direction, and so it tends to drive it back over here. But when you're at a maxima, on either sides, the potential energy decreases.

10:08So on either sides, the force is directed away. So notice forces tend to lower the potential energies. It's always directed in the direction of lowering the potential energies. So not only do I have insights about the magnitude and the direction of the forces, I also have insights about which points are stable and which points are unstable equilibrium.

10:32All right, but here's a last question for us. What if this entire graph was shifted upwards and the potential energy values everywhere has now increased? But does that change anything as far as your forces are concerned? No, it doesn't because the slopes stay exactly the same.

10:54And all I care about is the slope. I don't care about the actual value of the potential energy. And that is very, very reassuring because, remember, the actual value of the potential energy at any point is quite arbitrary. It depends upon where you choose your zero to be.

11:11So as you change your reference for zero potential energy, this graph will shift up or down, but it doesn't affect the underlying force. That's what we would expect. We wouldn't expect the force acting on a particle to change (laughing) based on what we choose as our zero reference point.

11:29So that is also pretty reassuring over here. So if you put it all together, look at what this graph looks like. It's as if that this is an actual hill or some kind of a racetrack, whatever you wanna think about it. And then if you were to put a marble over here, it would behave exactly the same as the actual particle.

11:56Remember, this is not a real hill, okay? It's a graph, all right? But if you were to put a marble over here, for example, look, it would roll down and the force at this point would be pretty much a constant because it's a constant slope. And then notice that if you were to put a marble over here, it would pretty much stay put at the equilibrium point.

12:18If you nudge it, if you nudge the marble, it'll come back over here. This is also an equilibrium point, but good luck trying to keep the marble at that point, because you know intuitively, if you nudge it, it'll go, (blowing) it goes away, you know, the marble will not spontaneously come back to this point.

12:34You have another equilibrium point over here. If the marble was at this location, ooh, we know it's so steep, the force acting on the marble will be very high. So what this all means is that when you look at a potential energy graph, you can assume that it's a hill or it's some kind of a racetrack, and you've put a marble over there.

12:55Whatever happens to that marble, the same thing will be happening to the actual particle. So it makes this analysis that much more intuitive. This is why we love potential energy graphs. Whenever we are dealing with very complicated forces like this, instead of writing an expression for the force or even trying to draw the graph of a force, it's much more useful to draw the potential energy graph, because not only does it tell us about the magnitude and the direction of the underlying force, it also helps us predict intuitively how the particle would move under such complicated forces.