Potential energy and conservative forces (part 1) | AP Physics | Khan Academy

[00:00:00]Let's talk about conservative forces and potential energy in this video. Let's start with an example. Suppose I release a box from here. We know it's going to slide down and it's probably going to fall off. But let's pause the animation at this point. Okay. The question is, what is the work done by gravity from here to here on this box? There might be other forces acting on it, but I'm only uh interested in the work done by gravity. What would that be? Well, we know how to calculate the work done by any force. It's the line integral of the force dot dr. Since we are interested in the work done by gravity, we have to consider the force of gravity. What's the force of gravity here? We know the force of gravity close to earth is mg.

[00:00:41]It's directed downwards. But how do I write that vectorally?

[00:00:45]Well, what we can do over here is we can choose a unit vector to represent direction. And usually the unit vector along the horizontal we call it ihat.

[00:00:55]And here I've chosen right direction to be positive. It's completely your choice. Okay. And in the vertical, the unit vector is often called jhat. And again over here, I've chosen upwards to be positive. It's a choice. Now, according to this convention, how do we write down the force of gravity? We can say it's negative mg jhat. The negative because this is in the opposite direction of our defined positive direction. So, we can say the force of gravity is negative mg jhat. Remember jhat does not add anything to the magnitude because magnitude of jhat is just one. It's a unit vector. So it's only useful in representing the direction. So that's the force of gravity. Okay. What about dr? That is the infinite decimal displacement. How do we write that vectorally? Well, let's draw over here. We can write this dr vectorally as a sum of these two vectors. Right? We can write this as a sum of dy vector and the dx vector. And I can write this vector as dx * ihat and this vector as dy * jhat. So we can write d r as dx ihat plus dy jhat. Now at this point you may be like wait a second wait a second shouldn't it be negative dyj jhat because it's in the downward direction like we did for the force of gravity. Well this can be a this can be confusing. Okay, remember that when it comes to your displacement, the direction, the signs, all of that is taken care by the bounds of the integral. And that's why I'm not adding any signs over here. It's the bounds that will take care of the sign automatically. Okay, another question we could have is, hey, this is in two dimensions. What if this object was moving in three dimensions? Well then for the third dimension the zaxis we can define one more unit vector k hat and again we can choose it to be in the positive direction when it's coming out.

[00:02:48]Okay. And so now what will be dr in three dimensions? It'll be dx ihat plus dy jhat plus dz khat. So let's consider the threedimensional case because it will be the most general case that we can think about. Okay. So now what will be the work done by the force of gravity? Well, it's going to be the force of gravity dot the displacement vector. And that'll be a great idea to pause and see if you can simplify this yourself. Okay, remember that the dot product has a cos theta in in it. So if theta is 90°, then cos 90 is zero, that means the dot product vanishes. In other words, if two vectors are perpendicular to each other, the dot product becomes zero. And notice I had jhat and khat are all mutually perpendicular to each other. Therefore, jhat do ihat would be zero. So that term would manage vanish.

[00:03:39]jhat dot khat would also vanish. So the only term that sures is jhat dot jhat.

[00:03:46]So we can now write the work done would be minus mg dy time jhat dojhat. What's jhat dot jhat? Well, it's the magnitude of jhat, which is just one because remember jhat is a unit vector. Okay, by definition it magnitude is one times the measure of the other jhat which is again 1 time cos of the angle between the two.

[00:04:07]What's the angle between jhat and jhat?

[00:04:08]Well, it's the two same vectors, right?

[00:04:10]So the angle is 0. cos 0 is 1, which means this term is 1 * 1 * 1, which is just one. So we are only left with this part. And now I can pull out the constant minus mg is a constant. And so if I pull it out, I get minus mg * the integral of just 1 with respect to y.

[00:04:29]And since we are integrating only with respect to the y direction over here, our bounds are from y a to yb. So let me just change those bounds from ya to yb.

[00:04:38]Okay. So what's this integral? Well, that's just y. So I get minus mg y. And then I have the bounds. And if we plug in the bounds, we first substitute the upper bound minus the lower bound. I get minus mg yb minus y a. And if I simplify, I'll get the work done to be mg y a minus mg yb. This is the work done by gravity. Now here's a question.

[00:05:02]What if instead of going along this particular path, this object went along a completely different path altogether?

[00:05:10]What would now be the work done by gravity? It would be a great idea to pause the video and think about whether you'd get the same answer or a different answer.

[00:05:18]All right, my initial reaction to this would be okay, we have to repeat the whole thing because we have a different path all together. But wait a second.

[00:05:26]The force of gravity stays exactly the same. So that doesn't change. But what about my dr vector? Hey, I consider the dr vector in the most general case. So even over here, the dr vector would stay exactly the same. And so everything what follows stays the same, which means the work done by gravity stays exactly the same.

[00:05:47]This is interesting. This means it doesn't matter what path is taken. As long as we're going from this point to this point, the work done by gravity will be exactly the same. And so we see a feature of the work done by gravity.

[00:06:02]It does not depend on the path. It only depends upon the initial and the final position. Such forces are called conservative forces. Conservative forces are the forces whose work done are path independent. it they only depend upon the initial and the final path. Now I'm sure at this point you have a lot of questions but before addressing them let's take another example. This time we're seeing a top view of a person who's pushing a box on a rough floor.

[00:06:29]And again the question is what is the work done this time by the force of friction? Because it's a rough floor there will be some frictional force over here. How do we figure that out? Well let's think about the force of friction.

[00:06:42]We know the magnitude of the sliding friction equals mu k * n. n is a normal force. Now mu k is the same everywhere.

[00:06:50]N is also the same everywhere. The normal force is the same everywhere. And therefore the magnitude of the frictional force stays the same. Let's just call this the magnitude of the frictional force. It's going to be the same everywhere. But what about the direction of the frictional force? Well, that is always in the opposite direction in which the object is sliding. Now over here it's sliding this way. So the frictional force is in this direction.

[00:07:08]Over here it's sliding this way. So the frictional force is in this direction.

[00:07:12]So look in general the frictional force its magnitude is the same but its direction is changing. So the vector the the frictional force vector that is not a constant that keeps changing. So that's slightly different than what we got earlier um in the case of gravity.

[00:07:28]So how do we do this? Well we don't have to worry too much about it because let's look at the dot product over here. Let's just consider the work done by friction in this tiny section. That would be the magnitude of the force of friction FF times the magnitude of the displacement that's dr times cos of the angle between the two. Well, they're in the opposite direction. The angle is 180°. So, it's going to be minus1. So, it's going to be minus ffdr. And guess what? The angle stays the same. It's 180 everywhere. So, the work done in every tiny section stays the same. In other words, the dot product stays the same everywhere.

[00:08:04]Therefore, we can we don't really have to formally integrate this. We can just do this do this in our head. If you to integrate this, well, the force of friction comes out and there's a negative sign that also comes out. Um, and we only integrate dr. And what's the integral of dr over here? Well, integral of dr is basically summing all of these tiny tiny dr up. That will give me the total path length. And so, the work done by friction is going to be minus the force of friction times the path length.

[00:08:33]And now we have the same question. Do you think the work done by friction along this path is the same as along this path? Again, great idea to pause the video and think about it. All right, we are seeing over here that the work done by friction depends on the path length. If the path length is different, which is clearly the case over here, then the work done would be different.

[00:08:54]And therefore, we'll call friction a non-conservative force because the work done by it depends on the path. So non-conservative forces are the forces whose work done depends on the path taken. All right. Now let's consider some questions. First of all, why do we care whether the work done by a force depends on the path or not? And secondly, why do we use the term conservative? Where does that term come from? Let's go back over here and now ask another question. How do we calculate the kinetic energy of the object? Let's say over here. Well, we can use the work energy theorem which says that change in kinetic energy equals the total work done on that particular object. The change in kinetic energy is the final kinetic energy minus the initial kinetic kinetic energy. And what is the total work done on this object? If we ignore friction and air resistance, then gravity is the only force doing work. I mean there is a normal force acting but it's perpendicular to the direction of the motion. So it will not be doing any work. So since gravity is the only force doing work, this is the total work done.

[00:10:00]So I can just plug that in over here.

[00:10:03]And if I rearrange to get all the B terms on one side and the A terms on the other, then I get something really interesting. I get KB plus MG YB equals K A + MG YA.

[00:10:14]Why is this interesting you ask? Well, first of all, let's think about this term. What exactly is this term? Well, it must have the units of energy, right?

[00:10:23]Because we're adding it to kinetic energy. We give it a name. We call it potential energy and we use the symbol U. So we say kinetic energy plus potential energy at B equals kinetic energy plus potential energy at A. Okay.

[00:10:38]But what exactly is this potential energy? Think of it this way. Kinetic energy is a number that we assign to an object due to its speed. It depends on its speed, right? Half mv². Potential energy is a number that we assign not to a single object but to a system of particles. system of objects like in this particular case the system of box and earth because remember not only is the mass of the box involved but g the acceleration due to gravity due to earth is also involved. So it's a number that we assign to a system which depends on the position of the objects within that system or the configuration of that system. So potential energy is the energy assigned to a system which depends on the position or you can just say the configuration. That's what potential energy is. And what we see is that when gravity is acting on an object, the sum of kinetic and potential stays the same. That does not change as the box goes from here to here. That's why this is interesting. And since this total sum total, you know, sum of kinetic and potential stays the same. We give a name to the, you know, this sum.

[00:11:45]We call it mechanical energy. Potential energy plus kinetic energy is mechanical energy. And so as the box goes from here to here, its speed changes, its position is changing, everything is changing, but its mechanical energy stays the same. It is conserved.

[00:12:02]That's why we're calling gravitational force a conservative force. Whenever the work done by a force is path independent, I can define a potential energy for that force and then the mechanical energy will be conserved.

[00:12:18]This is so useful in solving problems.

[00:12:21]If you are asked to calculate the speed at this point for example, I don't need to think about Newton's second law or or even think in terms of work done. I can just say hey mechanical energy here is the same as mechanical energy here. If I know the values of those these positions and if I know what the initial speed is, I can plug in and find the final speed.

[00:12:40]But what if along with gravity friction was also involved? Well, we now know the work done by friction depends on the path taken. So I cannot write this as the difference of two numbers. I can no longer do that. So I cannot associate potential energy along you know with respect to friction. And so now if we apply the work energy theorem on the right hand side there'll be work done by gravity and there'll be work done by friction. So I have to add it up. And when I do that look the mechanical energy is no longer conserved. The mechanical energy at this point is less than the mechanical energy over here.

[00:13:17]That's why friction is a non-conservative force. That's why if the work done by a force depends on the path, it will end up becoming a non-conservative force because I cannot associate a potential energy with respect to it. But wait a second. We've heard that energy can neither be created nor destroyed. But if the mechanical energy is not conserved, where is it going? Well, over here it's being converted into thermal energy and that makes sense. You can intuitively imagine that as the object moves over here due to sliding, there will be some heat generated over here. So, it will get converted to some other form. So, mechanical energy is lost and getting and it gets converted into thermal energy. However, if there are only conservative forces acting, then the mechanical energy stays the same. It is conserved. Okay, before we wind up, let's talk a little bit more about this potential energy. One question we could have is hey kinetic energy is zero when the speed of the object is zero.

[00:14:11]Similarly when is the potential energy of a system zero? Well in this particular case when it comes to gravity for example we could say that potential energy is zero when y equals 0. But where exactly is y equals 0? Well you are completely free to choose where your y equals 0. In this particular case we're choosing this as y equal to 0. But you can also choose this to be your y equals 0. But wouldn't that change the values of the potential energy? Sure.

[00:14:40]Now notice yb equals 0 because this is zero and y a would be shorter. But that's okay because what we care about is not the potential energies but the difference in the potential energy.

[00:14:52]That's what really matters in our calculations. And therefore the absolute value of potential energy doesn't matter. So you're completely free to choose where you want your y to be zero.

[00:15:02]In general, you're completely free to choose which configuration you want to call as your zero potential energy.

[00:15:10]Okay. The final thing now is how do you calculate potential energy in general? I mean for gravity when you're very close to earth, we can say potential energy is mg y. But how about potential energy for any conservative force? How do you calculate that? Well, let's just look at this and try to generalize it. If you look at the work done by gravity, we can now say it's equal to the potential energy at point A minus the potential energy at point B, right? Gravitational potential energies. And now let me just multiply a minus sign on both sides. So I'll get minus WG equals this will get flipped. You get U minus UA. If you're wondering why I'm doing that, well that's because now I can write this as change in potential energy. Change is always final minus initial. So I can write minus the work done by gravity equals change in gravitational potential energy. But I can write this in general.

[00:15:59]Work done by any conservative force negative of the work done by any conservative force equals the change in the potential energy of that conservative force. This is how you calculate the change in potential energy. And we know how to calculate the work done. That's just the integral of the line integral of the force dr. But we're talking about conservative forces only. So this is in general how you calculate the change in potential energy. Long story short, when the work done by a force is path independent, we call it a conservative force. Why?

[00:16:34]Because we can now assign a potential energy for that particular force. And if only conservative forces are acting on an object, then the total mechanical energy stays conserved and I can use that to solve problems much quicker. In contrast, if the work done by a force depends on the path taken, I can no longer write it this way. I can no longer associate a potential energy because the path matters. And now the mechanical energy is no longer conserved. We call it a non-conservative force.