Potential energy and conservative forces (part 1)

0:00- [Instructor] 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 gonna fall off, but let's pause the animation at this point, okay?

0:14The 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 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.dr.

0:33Since 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, it's directed downwards. But how do I write that vectorally? 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 I hat.

0:54And here I have chosen right direction to be positive, it's completely your choice, okay? And in the vertical, the unit vector is often called j hat. 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?

1:11We can say it's negative mg j hat. 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 j hat. Remember, j hat does not add anything to the magnitude because magnitude of j hat is just one.

1:27It's a unit vector, so it's only useful in representing direction. So that's the force of gravity. Okay, what about dr? That is that infinitesimal displacement. How do we write that vectorally? Well, let's draw dr over here. We can write this dr vectorally as a sum of these two vectors, right?

1:48We can write this as a sum of dy vector and the dx vector. And I can write this vector as dx times I hat and this vector as dy times j hat. So we can write dr as dx i hat plus dy j hat. Now at this point you may be like, wait a second, wait a second. Should it be negative dy j hat because it's in the downward direction like we did for the force of gravity?

2:12Well, 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.

2:29Okay? 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 Z axis, 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.

2:48Okay? And so now what will be dr in three dimensions? Be dx, i hat plus dy J hat plus dz k hat. So let's consider the three dimensional case because it'll be the most general case that we can think about. Okay, so now what will be the work done by the force of gravity?

3:04Well, it's going to be the force of gravity dot the displacement vector and that will 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 degrees, then cos 90 is zero.

3:23That means the dot product vanishes. In other words, if two vectors are perpendicular to each other, the dot product becomes zero. And notice I hat, j hat, and k hat are all mutually perpendicular to each other. Therefore j hat dot i hat would be zero. So that term would vanish.

3:38J hat dot k hat would also vanish. So the only term that survives is j hat dot j hat. So we can now write the work done would be minus mg dy times j hat dot j hat. What's j hat dot j hat? Well, it's the measure of j hat, which is just one, because remember, j hat is a unit vector, okay?

3:59By definition it's magnitude is one times the measure of the other j hat, which is again, one, times cos of the angle between the two. What's the angle between j hat and j hat? Well, it's the two same vectors, right? So the angle is zero, cos zero is one, which means this term is one times one times one, which is just one.

4:16So 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 times the integral of just one with respect to Y. And since we are integrating only with respect to the Y direction over here, our bounds are from Ya to Yb.

4:35So let me just change those bounds from Ya to Yb. 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 substitute the upper bound minus the lower bound. I get minus mg Yb minus Ya, and if I simplify, I'll get the work done to be mg Ya minus mg Yb.

4:57This is the work done by gravity. Now here's a question. What if instead of going along this particular path, this object went along a completely different path altogether, 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.

5:18All right, my initial reaction to this would be, okay, we have to repeat the whole thing because we have a different path altogether. But wait a second, 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.

5:35So 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. This is interesting. This means it doesn't matter what path is taken, as long as we are going from this point to this point, the work done by gravity will be exactly the same.

5:59And so we see a feature of the work done by gravity. 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. They only depend upon the initial and the final path.

6:19Now, I'm sure at this point you have a lot of questions, but before addressing them, let's take another example. This time we are seeing a top view of a person who's pushing a box on a rough floor. And again, the question is what is the work done this time by the force of friction?

6:35Because 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. We know the magnitude of the sliding friction equals mu K times n, n is a normal force. Now mu K is the same everywhere.

6:49N 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 fictional force. It's going to be the same everywhere. But what about the direction of the frictional force?

7:01Well, 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. Over here it's sliding this way, so the frictional force is in this direction. So look, in general, the frictional force, it's magnitude is the same, but its direction is changing.

7:18So the vector, the frictional force vector, that is not a constant, that keeps changing. So that's slightly different than what we got earlier in the case of gravity. 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.

7:34Let's just consider the work done by friction in this tiny section. That would be the mantra of the force of friction, Ff, times the magnitude of the (indistinct), that's Dorian, times cos of the angle between the two. Well then in the opposite direction, the angle is 180 degrees, so it's going to be minus one.

7:50So it's gonna be minus Ff dr. 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. Therefore we can, we don't really have to formally integrate this, we can just do this, do this in our head.

8:09If we were to integrate this well, the force of friction comes out and there's a negative sign that also comes out 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 vector up, that will give me the total path length.

8:27And so the work done by friction is going to be minus the force of friction times the path length. 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.

8:44All 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. And therefor we will call friction a non-conservative force because the work done by it depends on the path.

9:03So 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?

9:19Where 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.

9:39The change in kinetic energy is the final kinetic energy minus the initial 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.

9:57So since gravity is the only force doing work, this is the total work done. So I can just plug that in over here. 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 Ka plus mg Ya.

10:14Why 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? Because we are adding it to kinetic energy. We give it a name, we call it potential energy, and we use the symbol U.

10:30So we say kinetic energy plus potential energy at b equals kinetic energy plus potential energy at a. Okay, 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?

10:48Half mv square. 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.

11:06So 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.

11:23That'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, we call it mechanical energy.

11:47Potential 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. That's why we are calling gravitational force a conservative force.

12:06Whenever 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. This is so useful in solving problems. If you're 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.

12:28I can just say, hey, mechanical energy here is the same as mechanical energy here. If I know the values of these positions, and if I know what the initial speed is, I can plug in and find the final speed. But what if along with gravity friction was also involved?

12:44Well, 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.

13:06So 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 we're here. 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.

13:31But 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'll be some heat generated over here, so it'll get converted to some other form.

13:51So mechanical energy is lost 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.

14:06One question we could have is, hey, kinetic energy is zero when the speed of the object is zero. 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 zero.

14:21But where exactly is Y equals zero? Well, you are completely free to choose where your Y equals zero. In this particular case, we are choosing this as Y equal to zero. But you can also choose this to be your Y equals zero. But wouldn't that change the values of the potential energy?

14:39Sure, now notice Yb equals zero because this is zero and Ya would be shorter, but that's okay because what we care about is not the potential energies, but the difference in the potential energy. That's what really matters in our calculations. And therefore, the absolute value of potential energy doesn't matter.

14:58So you're completely free to choose where you want your Y to be zero. In general, you're completely free to choose which configuration you want to call as your zero potential energy. Okay, the final thing now is how do you calculate potential energy in general?

15:15I mean for gravity, when you're very close to earth, we can say potential energy is mgY, 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?

15:36Gravitational potential energies. And now let me just multiply the minus sign on both sides, so I'll get minus Wg equals, this will get flipped. You'll get Ub 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.

15:52So I can write minus the work done by gravity equals change in gravitational potential energy. But I can write this in general 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.

16:08This 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 dot dr. But we are talking about conservative forces only. So this is in general how you calculate the change in potential energy.

16:28Long story short, when the work done by a force is path independent, we call it a conservative force. Why? Because we can now assign a potential energy for that particular force. And if only conservative forces are acting on an object and the total mechanical energy stays conserved, and I can use that to solve problems much quicker.

16:48In 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.