Potential energy and conservative forces (part 4) | AP Physics | Khan Academy

[00:00:00]The change in potential energy of any system equals the negative work done by the conservative force in going from one configuration to another. For example, if you have two masses interacting gravitationally, then we can define the change in potential energy, the change in gravitational potential energy of this system as the negative work done by the gravitational force in going from one position to another or one configuration to another. Similarly, if we had, say, a system involving springs, then we can say the change in the spring potential energy equals the negative work done by the spring force in going from one position to another.

[00:00:39]So, look, if we know the expression for the conservative force, we can find the expression for potential energy. So, the question we want to try and explore in this video is, can we do the reverse? If we knew the expression for potential energy, how do we recover the expression for the underlying force? That's what we're going to try and figure out in this video. So, let's begin.

[00:01:04]All right. So, how do we do this? Well, we can do a guesswork. If we zoom out, we notice that potential energy equals negative integral of the force.

[00:01:14]What's the reverse of integral? That's a derivative, right? So, we can guess that the force should equal the negative derivative of the potential energy.

[00:01:25]But, of course, that is just a guesswork.

[00:01:28]Let's see if we can confirm this mathematically. And here's how I like to think about it. Suppose we have a system of particles interacting with whatever conservative force we want. It could be a system of springs, you know, gravity, or some other conservative force. Let's think about this in general. Let's pick one of those particles, of, you know, which is part of that system, and let's say that the force acting on it, this is a conservative force, and let's say that the force acting on it is F. We're only going to consider a one-dimensional case, so let's consider this as our X direction or X dimension. And so we can call this as Fx. But the strength of this force can depend on the position of this particle. So we should say that this force is a function of X.

[00:02:14]So now the question is, how do we connect this to potential? In fact, what we're trying to do is we're trying to confirm that can we say that this force equals negative derivative of the potential? That's what we're going to try and check. So how do we do that?

[00:02:26]Well, here's what we what we can do.

[00:02:28]Let's say that the force displaces this particle by a tiny amount dx.

[00:02:35]Dx is an infinitesimal displacement. You can imagine that the displacement is insanely tiny, so tiny that over the displacement the force virtually stays the same. That's the whole point of considering a very tiny displacement so that I can assume the force over that displacement stays the same. It hasn't changed much.

[00:02:54]All right. So now because of the displacement, because the position of the particle has changed, the potential energy also changes ever so slightly.

[00:03:02]So we can now write the change in potential energy, which is dU. Again, d because it's a tiny change in potential energy. And again, the potential energy is a function of X, right? It's a function of position. Depending on the position of the particle, the potential energy changes. That's why we have um you know, we have we have a function of X over here.

[00:03:20]So the change in potential energy, what is that equal to? Well, that equals the negative work done by the conservative force. So it'll be the negative What's the work done by the force over here?

[00:03:31]Well, since we're only considering a very tiny infinitesimal displacement, we can assume over the displacement the force is pretty much a constant. So if the force is a constant, then the work done would be just the product of the force and the displacement, right? So it would be the product of force and displacement. Now, if you look at this and this, you can see that they're pretty much the same thing. They're both saying that the change in potential energy equals the negative work done by the conservative force. The difference is on the right-hand side, we're considering the work done over a tiny infinitesimal displacement. But the left-hand side represents a summation over many such infinitesimal intervals.

[00:04:09]That summation is what we call an integral.

[00:04:12]Anyways, if we rearrange for force, what do we get? We will get negative d u over d x. So, indeed, the force equals negative derivative of the potential energy with respect to position. Now, if we were to be mathematically rigorous, we should have actually started with delta x and then written limit as delta x tends to zero, but then it eventually would yield the same result. And look at how beautiful this is.

[00:04:41]If you take the negative derivative of potential energy with respect to position, you get the force. If you take the negative integral of the force with respect to position, in other words, the negative line integral of the force, you get the potential energy. So, one is the reverse of the other. It's pretty cool if you think about it. Okay, now let's put this to action. Let's apply this to some very familiar systems. The first one is we raise a tiny ball or a tiny stone or something. Um we raise it to some height. When we do that, the change in the gravitational potential energy of this ball-Earth system is given as m g delta y, where m is the mass of this ball or stone, g is the acceleration due to gravity, and delta y is this height.

[00:05:24]So, what we want to do is use this potential energy function to arrive at the expression for the underlying conservative force over here.

[00:05:34]Now, we already know that we're dealing with gravity. And since it's a case where we're very close to a massive planet or a star, for that matter, we know that the force of gravity is just m g.

[00:05:44]But we want to use this to get the same result.

[00:05:48]So, we just have to differentiate the potential energy function. So, for that I have to first write this as, you know, as a as a function of position. How do I do that? Well, we'll choose some reference point as our zero.

[00:06:03]If this is Y = 0 and we call this configuration as potential energy zero, then we can call this point as some Y and let's call this point as having the potential energy U of Y. Then notice the height becomes just Y. So, this part becomes Y and the change in potential energy is U of Y - 0, which is just U of Y. And boom, I now have the potential energy function. So, how do I get the force? Well, that equals negative derivative of this with respect to not X, Y over here.

[00:06:34]So, the underlying force, which is going to be a function of Y coordinate over here, is going to be negative derivative of this. What is the derivative of this function? Well, it's just going to be mg into derivative of Y, which is just one.

[00:06:48]So, this is going to be mg, which means I'll get the answer to be negative mg and that's exactly the force of gravity.

[00:06:54]You know, close to Earth the force of gravity is just mg and the negative sign is saying that the force is in the opposite direction of our chosen positive direction. We've chosen upwards as our positive Y.

[00:07:06]And so, the force is downwards. So, we are getting the same result that we know. So, this is working out. All right, now if you're very curious, you might have one question. You could say, "Hey, but this is the expression for potential energy provided we choose this as our reference, we choose this as our Y = 0 and U = 0. But what if we change this? What if we chose this as Y is equal to, I don't know, maybe some Y not and U is equal to some U not?"

[00:07:33]Then the potential energy function changes, right? Yes, there will be some U nots over here and there will be some Y nots over here. But when you take the derivative, the derivatives of the constants anyways goes to zero. So, even then, the final answer would stay the same. That is very reassuring because we don't want our force [snorts] expression to change based on what we consider as our reference, where we consider as our y equals zero. So, that is truly reassuring that it doesn't really depend on our zero reference.

[00:08:09]All right. Let's take the next system that we are familiar with, where you have two masses gravitationally interacting, but they're very far apart.

[00:08:17]Now, if we choose the the potential energy to be zero when they're infinitely far apart, that's our reference, then the potential energy of this system is given as -Gmm by r. Now, from this, can we recover the underlying um conservative force? Again, we might know that the underlying conservative force is the force of gravity, and in general, we can use Newton's law of universal gravity to figure that out.

[00:08:41]But again, can we use this to figure it out? It'll be a great idea to pause the video and see if you can try this yourself now.

[00:08:47]All right. So, again, to recover the underlying force, we have to find the negative derivative of the potential energy with respect to position. This time we're using r to represent the position. Whether you use r, whether you call this as x, it doesn't matter. As long as you're in one dimension, this works. Okay? So, the underlying force, which is now a function of r, is going to be negative derivative of the potential energy with respect to r.

[00:09:12]And so, if we plug in over here, the negative negative sign cancels, and so you get the derivative of Gmm divided by r.

[00:09:19]And I can take Gmm, which are constants, I can pull them out, and then I'll have to find the derivative of 1 over r.

[00:09:26]What's the derivative of 1 over r? It's -1 over r squared. So, if you put it together, we get the underlying force as -Gmm by r squared, and that is the force we get from Newton's universal law of gravity. It's the inverse square law.

[00:09:43]And again, what is the minus sign say?

[00:09:46]It's saying that the force that's acting over here is in the opposite direction of our positive r. Our positive r is this way, so the force must be downwards, and we know that that should be the case. The force is attractive.

[00:10:00]So, notice this again worked. But, remember, even though we're using the general position variable r over here, we're still sticking to one dimension.

[00:10:09]We're assuming that the object is moving only in one dimension. This expression only works for one dimension. So, what if the objects are moving on in two or three dimensions? Well, we can extend this expression to two or three dimensions by using something called the gradient. But, let's not worry too much about that because it's conceptually the same. So, that's why we're just going to stick to one dimension. So, remember, all of these are just one-dimensional cases.

[00:10:36]Let's do one last system that we might be familiar with, and that is a system of springs. If we stretch a spring by some amount delta x, then we know the potential energy stored in the spring, block, and the wall system is given by this expression.

[00:10:54]Now, again, it'll be a great idea to see if you can use this to find out the expression for the underlying conservative force, which is the spring force over here. Pause the video and try this on your own.

[00:11:09]All right. The first step is to write this as a function of x. How do we do that? Well, again, we can choose this as our x equals zero, and we can call this as some x. Then, this change um in the position or the extension is basically x. So, I can now write the potential energy as a function of x equals half k x squared. Okay, this is ready for differentiation. So, the underlying force equals negative dU over dX.

[00:11:37]What is the derivative of this part over here?

[00:11:40]Well, half k is a constant, so I can pull that out.

[00:11:45]And derivative of x squared is just 2x.

[00:11:49]And so, the two and two cancels, so I get the underlying force as minus kx.

[00:11:54]Hey, that is the expression for the spring force. Again, what is the minus sign saying? It's saying that the direction of the force is in the opposite direction of our chosen positive, which was in right side to be positive, and so the force must be towards the left. And yes, if you stretch the spring, it'll try to pull itself back together, so the force will be in the opposite direction of the displacement. This is what we call Hooke's law. So, we have recovered our force. So, long story short, if you know the expression for potential energy of any system, then you can recover the underlying force by just finding the negative derivative of that potential energy.