 I have a terrific fear of heights, so I could never do what this man is about to do. Perched high above the ground, in a beautiful mountain environment over a river and standing only on a little platform, this man is going to bungee jump, he's going to throw himself into the air, affixed only back to the bridge from which he leaped with a cord. He's accelerating toward the ground and the cord will pull back on him in a gentle way to prevent him from hitting the ground. Some people find this exhilarating, I find this terrifying. Either way, where did the energy come from that eventually resulted in him gaining more and more speed, increasing his own kinetic energy? In this lecture we're going to begin to look at a concept known as potential energy to better understand the answers to such questions. In this part of the course we're going to begin to explore another kind of energy, and we have peripherally encountered it already in our previous explorations of kinetic energy and changes in kinetic energy, but we're going to directly confront this concept of potential energy now, and the key ideas that we're going to begin to dive into in this section of the course are as follows. We're going to come to understand that there is energy associated with the configuration of material objects, and this is what is called potential energy. We will understand the relationships between changes in potential energy, work, and changes in kinetic energy. We will learn to distinguish between two kinds of forces using these concepts, conservative forces and non-conservative forces. And finally we will begin to describe the potential energy that's associated with some very specific forces, especially gravitation and springs. Let's go back and think about kinetic energy. When kinetic energy is used up by an object in order to overcome, for instance, some external force, it's a fair thing to ask, where does that kinetic energy go? Does it go anywhere or is it lost forever? So let's consider again the situation of, say, you or me, holding a ball in your hand. So you can put the ball out in your hand at some initial height, and I'm going to choose to label that height as y equals zero. This is a choice that you can make in setting up and solving problems. I will come back to this point later, but as long as you make a perfectly reasonable choice, it doesn't affect the physics of what happens next. Now instead of maintaining the ball in contact with your hand and raising it up to some new height, let's assume instead that you throw the ball straight upward. So based on your own experience with this, what happens to the ball? Well our experience with gravity and motion, whether it was in the context of this course or in our own personal lives, and especially with the detailed concept of projectile motion, tells us a few things. It tells us that after accelerating the ball over a very brief period, it's going to have some initial velocity. It's going to rise up to some height, let's call that h above where it started, again from the place it leaves my hand, we'll call that y equals zero, and then to the maximum height that it rises we'll call that h. Now in between, from the moment it leaves my hand until it comes to a brief stop in the air, it's going to be slowing, that is gravity is accelerating it, but in a negative direction, so it's a deceleration. Now this height h is the highest it will ever go. After this we know it may pause for an instant of time, but it's going to begin to fall back down. Gravity continuing to accelerate in the downward direction will now reverse the direction of motion of the ball, and it will fall back down, say, toward my hand. Now let's look at this from our energy perspective. That was more of our perspective of forces and motion, and gravitation, acceleration, the stuff that we've explored in the first third of the course or so. From our energy perspective, we also know that something else is going on. We know that the ball begins with a kinetic energy given by one-half times its mass, times its initial velocity squared. That's the velocity it has just after it leaves contact with my hand, and is now freed from any forces exerted by my hand. We also know that it ends at some displacement above my hand, h minus zero, where zero was where it left contact with my hand, and h is the maximum height to which it rises. At that moment, its velocity comes to exactly zero, and so it has a final kinetic energy that is zero. It has no final velocity at the maximum height to which it rises, and so its kinetic energy there must be zero. Now once the ball stops making contact with my hand, there's only one force that is really acting on the ball. We're going to ignore air resistance. We're going to ignore drag. It's just gravity. We can then compute the work that's done by gravity on the ball using the work kinetic energy theorem. That is that the changes in kinetic energy of an object are equal to the work done by or on the object. So we have here the change in kinetic energy of the ball, which will be equal to its final minus its initial kinetic energy. Well its final kinetic energy was zero. Its initial kinetic energy was one-half m v naught squared. So we have zero minus one-half m v naught squared, which just yields a negative one-half m v naught squared. Now one-half is a positive number, mass is always a positive number, speed squared is a positive number, and so this overall is a negative number. That is kinetic energy declines. It decreases. It changes in the negative direction. It's sapped from the ball by gravity if you want to think about it that way. This must necessarily be equal to the work done by the forces on the ball. Well the only force acting on the ball once it leaves my hand with an initial speed v zero is gravity. So this must needs to be equal to the work done by gravity. Well in general the work done by a force is the sum of all of the products of the force at any given place times the displacement over which it has acted. And because these are tiny displacements we can sum them up using the integral calculus technique. So we do an integral of sum from zero to the height h of the product of the gravitational force which is weight which I've denoted with w with a vector over it, dot product the displacements which are dy vector. Well this is a vector dot product and so there are many ways we can work this out but since the weight if you take a look at the picture here always points in the negative y direction while the displacement of the ball up into the air is in the positive y direction we know that weight points straight down all the displacements along the line of flight point straight up. The sum of all the little dy's is equal to the total displacement of delta y or h minus zero in this case and so we know that there is an angle between the force due to gravity which is straight down and the displacements which are straight up and that angles a hundred and eighty degrees. You can figure that out from the picture weight always points down displacements are always going to point in the upward direction until the ball comes to stop at its maximum height h. The angle between this vector the displacement vector and the weight vector is always no matter what the degree of the weight is or the degree of the displacement is is always a hundred and eighty degrees. So we can write the change of kinetic energy again and say that that is equal to the integral from the height zero to the height h of negative mg. Weight is mg. We have a dot product of mg and dy as vectors the angle between them is a hundred and eighty degrees so the cosine of the angle is negative one. So that dot product and you should try this out on your own on paper works out to be negative mg times dy the little displacements the little chunks of y that make up the total distance y that the thing travels. Well m is a constant g is a constant gravitational acceleration is constant near the surface of the earth and so these are just constants of integration. They play no role in what happens next and we can simply pull them out of the integral and then we just have the integral from zero to h of dy and if you work through that you'll find out that the integral of dy the indefinite integral of dy is just y. It's a function that gives you exactly the answer you're looking for. So we then have to just evaluate y between zero and h. Well this is h minus zero when you evaluate this. So we have negative mg h minus zero which is just negative mg h. So grouping all this together we have a minus sign on the left side in the change in kinetic energy. We have a minus sign that's in the work done by gravity over on the right hand side. Those minus signs cancel out from both sides of the equation and at the end of the day we find out that one half mv naught squared is equal to mg h. That is the change in kinetic energy is equal to the work done by gravity and this is the final equation we get. Now I could have also done one step further here. I could have eliminated m from both sides. Notice that mass appears equally on both sides of this equation. So really I just have one half v naught squared equals gh a much simpler equation. But nonetheless this is correct either way. And so we have that the change in kinetic energy delta k is equal to negative one half mv naught squared is equal to negative mg h and we see that energy has gone from its kinetic form into some form negative mg h that is associated with the configuration of the ball and the earth. As the ball moves further from the surface of the earth rising higher up above the ground it is depleted of its kinetic energy and this energy goes someplace. It goes into a form associated with that displacement. So as we reconfigure the ball earth system with the ball now further from the earth and with gravity wanting to pull it back toward the center of the earth we have some energy stored in this new configuration. Where did that energy come from? It came from the depletion of the kinetic energy of the ball and to get to this new farther distance from the earth given it is an initial velocity v zero the ball had to give up that kinetic energy and in doing so it handed it off to the configuration of the ball with respect to the earth in the earth's gravitational sort of pole. So we can assign a new kind of energy to the ball. It is energy stored in the configuration of the ball and earth system and we can say for instance in this specific case where I have thrown a ball up into the air and let it come to its highest point where it has no more velocity at that moment we can say that the ball started with no potential energy at its original height y equals zero. It thus began only with kinetic energy one half m v not squared but after rising to its maximum height h all of its kinetic energy is now gone and it possesses only this potential energy which is given by mgh. So let's explore the consequences of this concept further. Now let's compare kinetic energy and potential energy again in this special case of throwing a ball up into the air. So we can write the initial and final kinetic energy of the ball in this previous example I've hinted at this verbally before but let's be kind of emphatic and systematic about this now. The initial kinetic energy was indeed one half m v not squared. Once the ball leaves my hand and my hand is no longer accelerating the ball it has a at that moment of fixed velocity and that velocity is v not. When it reaches its maximum height I call this the final location of the ball for this part of the problem. It has no more kinetic energy it comes to rest in the gravitational pole of the earthy. The earth is effectively one the velocity has been reduced to zero by the acceleration of gravity it now has no more kinetic energy. Final kinetic energy is zero. Now we can similarly write the initial and final potential energies of the ball and for now I'm going to choose to associate that starting height of y equals zero meters I defined that to be zero in my coordinate system equally with a location of zero potential energy. So I will state the initial potential energy of the ball was zero. The final potential energy at the maximum height h is mgh. Now we can compute the changes in each kind of energy. So we know that the change in kinetic energy which is k final minus k initial is zero minus one half m v not squared. And that's what gave us the negative one half m v not squared that we saw earlier. Similarly we would define the change in potential energy u as u final minus u initial. Although u final was mgh, u initial was zero so this winds up being mgh. Well this is interesting the change in kinetic energy in this specific example was a negative number. Remember mass is positive, speed squared is positive and one half is obviously a positive number so the overall minus sign that's picked up in the k final minus k initial remains and kinetic energy the change in kinetic energy is less than zero it's a negative number. On the other hand the change in potential energy uf minus ui results in a positive number. Mass is a positive number, g is a positive number, the height to which it goes is a positive number so mgh all together is a positive number. So delta u is greater than zero. So we see that while delta k is less than zero, delta u is greater than zero. So how do we relate them? We saw in the previous equation that they can be related to one another, right? One half mv not squared equals mgh so what's going on? It must be true that the changes in kinetic energy are equal to the negative of the changes in potential energy given our previous discovery that one half mv not squared is equal to mgh. Now because the changes in kinetic energy are equal to the work done by forces on an object this also implies that the changes in potential energy are equal to the negative of the work done by the forces. So we have some relationships here now between changes in kinetic energy, changes in potential energy and the work kinetic energy theorem. Now note that I made a definition in the previous example, choosing to set the initial potential energy to be exactly zero joules for the ball at its initial height which I had also defined to be the origin of the y-axis in the coordinate system y equals zero meters. So it turns out that choosing to set the potential energy to zero at a specific point should never alter the conclusions of the problem because all that really matters in questions of physical reality are changes in energy. It's only these changes that matter and we're going to come back to this in a little bit when we look more closely at the general form of gravitational potential energy which is mg delta y, so that just literally contains changes in height causing changes in potential energy, mg delta y, but it's sufficient for now to simply commit to memory to remember that the absolute degree of potential energy is not per se well defined. I can't tell you what the right absolute number of potential energy is for this ball but I can tell you that because only the changes matter in terms of what happens to the kinematics of the ball, what happens to its state of motion for instance, that it doesn't matter and we can make a convenient choice of where to put zero to help us solve the problem and answer the question. Now there are forces for which it is in fact convenient to make a very specific choice, kind of a convention in space where we would then say that is a convenient zero point for potential energy. We'll come back to this when we revisit gravity and dig more deeply into exactly what gravity is and where it comes from, but in general those choices are merely conventions. They're arbitrary, you agree on them, you use them to solve problems, but it doesn't affect the conclusions about what happens in a physical situation. Nature doesn't change because you choose to change where you elect to place the zero point of potential energy. That's a good thing, but nonetheless it's important to keep in mind that you do have that freedom to choose a zero point that's more convenient to helping you solve a difficult problem. This is one of the benefits of using the energy concept because it is a scalar, it's a number that can be added and subtracted from other energy numbers. You have this sort of freedom to elect to state that I choose this point which is a convenient location in space to be the zero point of potential energy. Another way to think about this is from the perspective of relative motion. Even kinetic energy, right? Even kinetic energy doesn't per se have an absolute definition. For instance I could be standing on a platform next to a dart train here in Dallas and I might throw a ball forward along the x direction, so horizontally along the platform, and I might throw it at two meters per second. I would say, aha, well now once I release the ball it has a kinetic energy of one-half times its mass times two meters per second squared. Just before I throw the ball maybe the dart train starts to accelerate and it accelerates up to a speed of two meters per second in the direction that I throw the ball and so a person watching me throw the ball will see me moving backward at two meters per second. The ball will appear to simply hang there in air relative to the dart train not moving at all. A person on the dart train might say, well no, you started moving backward at two meters per second so you have a kinetic energy of one-half times your mass times two meters per second squared. But the ball just sat there in air not moving relative to me so I would say that it had no kinetic energy. So even kinetic energy can have a relative definition to it as to where you set the zero point. Objects in motion with respect to other objects will measure different speeds and as a result of that can obtain different values for what the kinetic energy is. There is no absolute definition of these things because it depends on what's moving, who's moving and who's making the observations. So keep this kind of thing in mind, this is a bit more of an advanced concept but nonetheless a crucial one to energy concepts is that if it's convenient for you to redefine where zero is an energy as long as you're consistent with how you then apply that definition it won't have any effect on the physical outcomes that you predict for the problem. Now because of the energy concepts we can revisit forces. We've looked at the drag force, we've looked at the friction force, we've looked at gravity, weight, we've explored a little bit the force associated with the compression or stretching of a spring, the so-called spring force, Hooke's law. We can leverage this previous example of throwing the ball up into the air to now consider two kinds of forces that are known to do work and how the work done by those forces affects the way we think about those forces. So for instance in stopping the upward motion of the ball, gravity has work done on it by the ball. The ball gives up kinetic energy, it starts with positive kinetic energy, it ends with zero kinetic energy, its change in kinetic energy is negative. So that means something has to gain energy and we'll come back to the concept of the conservation of energy in a bit but something has to gain energy in return. What gains energy? Well what gains energy is the configuration of the earth ball system. We can say that gravity has had work done on it by the ball and energy is now stored in the configuration of the ball with respect to the earth which is managed by gravity. So we can call that work, work one, so work with the subscript one. So in the process of the ball losing its kinetic energy, it converts that to go up higher, the earth ball system has gained potential energy. Now after reaching its maximum height, the ball's stored potential energy will be converted back to kinetic energy. Gravity will now do work on the ball and re-accelerate it downward toward the center of the earth. The work done by gravity on the ball is now positive, so it's doing work, it's giving up energy from the earth ball configuration and it's putting it back into motion for the ball. We can call that work work two or w sub two. Gravitational potential energy is released as the configuration changes once again and the ball will be accelerated by gravity. Now how do we categorize forces? Gravity is a force, friction is a force. How do we categorize forces using this kind of information? What we'll see is that there are basically two kinds of forces, forces where one statement is true and forces where a different statement is true. So if it's true that the work done in the first situation, work one, is equal to the negative of the work done in the second situation by that same force, say gravity in this case, then it turns out you're dealing with what is known as a conservative force and we'll explore this in a bit more detail in a moment. That is to say any energy that was, say, lost from kinetic energy is put into potential energy, like putting money into a bank to save it. And then later that money can be retrieved, in this case that energy can be restored from the earth ball configuration and put back into the kinetic energy of the ball, again by altering the configuration and allowing the potential energy to become the work done by the force, re-accelerating the ball. So we'll work through this a little bit. You'll see an example with friction, you'll see an example with gravity. You will see that for gravity it is true that the work done in the upward moving situation is the negative of the work done in the downward moving situation when gravity is involved. But if it's instead true that the work done in the first half of the motion is not equal to the negative of the work done in the second half of the situation by the force, then you are dealing with what is known as a non-conservative force. Any energy that has been lost from kinetic energy is not stored in an associated potential energy. It rather will go into other forms where it's not as easy to have it retrieved back into, say, the original kinetic energy of the ball or any object. It could be heat. Heat is the vibration of atoms in a material. It's taking the kinetic energy of the ball and instead putting it into the motion of atoms. Those motions involve collisions of large numbers of atoms. It's extremely unlikely you'll ever get that back in the form of the atoms all kicking the ball back in the same way to re-accelerate it back to the speed it was at the beginning. Sound. Well, sound is also the vibration of atoms, but it's when the material vibrates and causes air to vibrate and then the vibrations in air are transmitted to other places in the room, good luck getting that energy back, concentrated on the ball to re-accelerate it back up to its initial speed, V0. So these are examples of the ways that energy can be lost and if there's a force associated with those things, heat and sound, it will wind up being a non-conservative force because the work done by the force in the first case is not equal to the negative of the work done in the second case where you try to move the object back to where it started. Okay? So for non-conservative forces, it's extremely difficult, if not impossible, to recover back energy only in the form of the kinetic energy that was lost in the first half to the force. Let's take as an example of this exercise in identifying conservative forces and non-conservative forces the following case, the action of the friction force. So consider the following situation in which two processes, two cases are executed. In the first case, I'm going to take a block of some mass, m, and I'm going to push it, accelerating it quickly enough so that the time required to get it up to its final constant velocity, Vx, is not very large. It's a small part of the overall journey. I'm going to quickly accelerate this block up to a speed, Vx, moving in some direction along a surface. Now, that surface in contact with the block, whatever these materials are, will have some coefficient of friction. In this case, because we're talking about a moving object, it's kinetic friction. And I am displacing the object, for instance, to the right by some distance delta x1. It starts out a distance x along the horizontal direction. We can always say that it started at 0 at the beginning. And so delta x1 is x minus 0, the final position minus the initial position, the displacement. So this is case one. I have a block of mass, m, sitting on a surface. The surfaces are in contact, so there is a kinetic coefficient of friction, because I am moving the block of mass, m, at a constant velocity, Vx, to the right, displacing at a distance, x, along the surface. That's case one. Case two is now starting from the endpoint of the journey, starting from x. I push the block back, attempting again to move it at the same constant velocity, but in the other direction, so constant speed, but opposite direction for the velocity. And I push the block back to its starting point, x equals 0. Again, on the same surface, with the same coefficient of kinetic friction, with the same constant speed, but now with a constant velocity of negative Vx. And now, of course, delta x2, the displacement for case two, will be negative x. I start at x, that's my initial position. I end at 0, 0 minus x is a displacement of negative x. In each case, we can, as I said, assume that this constant speed is reached nearly instantaneously so that the time to accelerate is negligible. The question we want to answer is the following. What is the work done by the friction force in each case? And considering what we learned a moment ago about conservative and non-conservative forces, what can we say about the nature of the friction force? So let's consider case one. In this case, the block is moving to the right. The force of friction always opposes the motion independent of speed and independent of the apparent area of contact. So we have a situation in which the friction force opposes the direction of motion of the block. The block has a velocity to the right, so the friction force acts to the left. It points in the direction opposite the unit vector that represents the direction of velocity for the block. Thus, the angle between these two vectors must be 180 degrees. This would be a good place to pause, sketch yourself a picture of what happens in this case, and convince yourself that indeed, if the block is moving to the right, whatever the magnitude of the friction force is, its direction, because it always opposes the motion of an object, must point in the negative v hat x direction. That is opposite the direction of the velocity of the block, and thus the angle between these two vectors, the velocity of the block, and thus the displacement of the block, and the kinetic friction force is 180 degrees. So we can compute the work that is done by the friction force. It's going to be the force acting over the displacement. This is a dot product, so we have the vector f sub k dotted into the displacement, delta x1. Now we can rewrite this in terms of the shorthand for the magnitude of the dot product, and that is the magnitude of the friction force times the magnitude of the displacement times the cosine of the angle between them. As I said, in that case, the one we're considering right now, the block moves to the right, the friction points left, the angle's 180 degrees. We'll come back to the cosine of that in a moment. What's the magnitude of the friction force? It's given by the coefficient of kinetic friction times the normal force, and in this case, that's the full weight of the block. What's the magnitude of the displacement? It's the distance x over which this has moved. So if we substitute for the magnitude of fk and the magnitude of delta x1 with the values using the coefficient of friction, mass, acceleration due to gravity, and the distance it moves to the right, we have mu k mg times x times cosine theta 1. And as I said, theta 1 is 180 degrees. The cosine of 180 degrees is negative 1. So finally, we arrive at the work done by the friction force in case 1. It's negative mu sub k times m times g times x, where m and g and x and mu k are all positive numbers because we used magnitudes to get to this point. The only minus sign comes from the cosine of theta 1. That's negative 1. So the work done by the friction force in case 1 is negative mu sub k mg x. Let's look at case 2. In case 2, the block is moving now to the left. The displacement is to the left. But again, the force of friction always opposes the motion. So again, it will be true that the direction that the kinetic friction force points is opposite the direction of the velocity of the block and also, consequently, opposite the direction of the displacement of the block. And thus, the angle between these vectors is still 180 degrees. So let's start working through the calculation. The work done in the second case by friction is still the dot product of the friction force with the displacement. We can start substituting in here. We need the magnitude of the friction force, the magnitude of the displacement, and the cosine of the angle between them. The magnitude of the friction force is again mu k times m times g. The magnitude of the displacement is again x. And the cosine of the angle between them is cosine theta 2. Again, the angle is 180 degrees. So we again pick up only one minus sign from this entire calculation. We wind up with the work done in the second case by friction as negative mu k times m times g times x. We've learned something quite important about the friction force here. Compare the work done in case one, where we slid the block to the right, to the work done in case two, where we slid the block to the left. In both cases, the work done by friction is negative. That is, energy was put into overcoming friction in both cases. Friction didn't give anything back in either case. It's not like the energy we put into friction going to the right is given back to the block, moving it to the left. Heck no, I have to push on the block again to get it to move back to the left. It doesn't just go back there on its own. And we've learned by considering what is w1 and what is w2 that w1 and w2 are equal. We've learned that w1 does not equal negative w2. This implies that friction is a non-conservative force. This was teased. And here it is demonstrated using the work energy concept that, in fact, friction doesn't give back what you put into it. It costs you more than it gets you, at least in this consideration. And it has no associated potential energy. It is not a conservative force. It cannot store energy and release it back, for instance, into the same form that was put into it in the beginning or in some other form that is useful for doing mechanical work. We can compare this and contrast this with the raising and lowering of the ball, that example, that started out this whole video. Case one is you hold the ball in your hand at rest. You raise it to a height h and bring it back to rest at that height. The change in kinetic energy is zero. It starts off at rest. It ends at rest. Delta K is zero. We learned from previous considerations of this experiment that the work done by your hand must be equal to the negative of the work done by gravity in this case. The force due to gravity, which is the weight, always points down, while the displacement in this first case is upward. So it must be that the direction that gravity's force points is equal to the negative of the direction that displacement points. Displacement is up into the air. Gravity pulls down toward the ground. We have a case of opposites. And thus, the angle between these vectors is 180 degrees. So this is beginning to start off looking like the friction case one that we looked at before. Let's plug in what we know about gravity, weight, and so forth, and see if we can get a formula for the work done specifically by gravity in this first case. So the work one that we're going to consider is the work done by gravity. And that will be equal to the force exerted by gravity over the displacement of the ball. This is a dot product. And again, we can go to one of the answers for the dot product, which is the magnitude of f gravity times the magnitude of delta y1 times the cosine of the angle between them. The magnitude of f gravity is the full weight of the ball, mg, the displacement, the magnitude of the displacement, the distance over which it moves is h a height h. And then we have times the cosine of the angle between them. Theta 1 is 180 degrees. The cosine of 180 is negative 1. And so again, we wind up with only one overall minus sign for this case one. W1 is negative m times g times h. So it seems like this might have started off just like friction. But case two is where things begin to diverge comparing friction and gravity. In case two, we have a situation where the ball is lowered, starting from rest at height h, back to its starting position, again brought to rest. So it returns to a height of 0. This time, the displacement points not upward but downward. Weight still points downward. So now we have a situation where the gravitational force on the ball points down to the ground. The displacement points down to the ground. And so now the angle between these two things is 0 degrees. Thus, the work done by gravity in the second case is still the dot product of the force exerted by gravity times the displacement over which gravity acts. The magnitude of gravity is still mg. The magnitude of the displacement is still h times the cosine of the angle between them. But this time, that angle is 0. The cosine of 0 is positive 1, not negative 1. And so we find that in this case, lowering the ball back to its starting position, that the work done by gravity is not a negative number but a positive number. Gravity had to give up something to alter the configuration of the ball to restore it back to its original position. So we reconfigured the ball in case 1 up into the air and moving it further away from the earth. Work was done against gravity, so gravity gains some energy in the process. It stored it in the form of this potential energy, negative mgh. Going back down, however, we're reconfiguring the ball earth system again, moving the ball closer to the earth. Now gravity gives up that energy in order for the hand to make the displacement happen. And what we've learned, especially in this case, is that w1 is, in fact, equal to negative w2. W1 is negative mgh. W2 is mgh. So the negative of that is negative mgh, which is exactly equal to the work done in the first case. W1 equals negative w2. And from this, we realize that, indeed, gravity is a conservative force. That is, it is possible to store energy in the configuration, for instance, in this case, of earth and the ball by moving them relative to one another. That stored energy can be restored back, for instance, to the hand. Or if we let the ball go, convert it into kinetic energy by dropping the ball, either way you go, that energy is taken back out of the configuration and put into other things. Gravity is a conservative force. It has an associated potential energy. So let's, again, review conservative and non-conservative forces. Conservative forces satisfy the situation that the work done in moving an object, for instance, from point A to point B, is equal to the negative of the work done in moving it back from point B to point A. You put some energy in to go from A to B. You get it back when you go from B to A. Non-conservative forces, however, do not restore the energy required to, say, overcome them. So friction is the example we've looked at. To slide the block along the table, I had to do work. My muscles had to give up stored chemical energy to convert into a push, to get the block moving at some constant speed. If I let go of the block, it just sits there. In order to move it back to its original point, I have to push it again, again overcoming the friction force. There is no potential energy associated with the friction force. That is not a conservative force. The work done in both cases by friction is the same. It doesn't give anything back in the example we've considered so far. So we've seen that gravity is a conservative force. We just checked this. It's associated gravitational potential energy is, so the change in the potential energy stored in gravity is equal to mg times the displacement in the direction perpendicular to the surface of the earth. So we would call that the y direction that's up or down. If I go up, I'm putting energy into the gravitational force, storing it in the configuration. If I go down, I'm taking it back out of this configuration and turning it into other things. We've considered friction. We've seen that friction is a non-conservative force. W1 does not equal negative W2. We just checked this. You can repeat this exercise for the spring force, and you'll see that it, too, is a conservative force. You can show this, for instance. And I encourage you to go and try this on your own by considering a spring originally sitting at its equilibrium position. So we would say that it is displaced from its equilibrium position by an amount of 0 meters. Then stretch it out or compress the spring. So use, imagine a hand grabbing the end of the spring and compressing it. That displacement costs your hand, your body chemical energy. Now, however, energy is stored in the spring in the form of its reconfigured atoms. It doesn't want to be compressed. It wants to go back to equilibrium. And so its atoms will push back on each other. If you then relax your arm and let the spring extend again until it reaches its equilibrium point, you get back the energy that was stored in the tension in the spring in its atomic configurations. And that can be put back into your muscles, for instance. So the spring force is conservative, and you can show that. And that means it has an associated potential energy, which we could call spring potential energy or elastic potential energy. Any body that has an equilibrium configuration and that when stretched or squashed restores itself back to that configuration is an elastic body. We'll come back to that potential energy in a moment. We'll derive it directly. It's a relatively straightforward exercise, given what we've seen already about gravity and friction. And so you can store energy in the spring. You can get energy back from a spring. If I compress a spring or stretch a spring and then let it go, the energy once stored in its configuration is released into kinetic energy, and then it can be stored again in potential energy by, say, starting with a stretch and having the spring compress and then re-expand and then compress and then re-expand and then compress. So we can go from stored energy in stretching to kinetic energy to stored energy in compressing to kinetic energy, et cetera. Finally, the drag force. This turns out to also be a non-conservative force. It always acts against the direction of the velocity of the object moving through the fluid. And so if you do the same analysis on this force that we did on, say, pushing the brick around on a surface with friction, it will yield the exact same conclusion as friction. And again, I encourage you to sit down with your own made up example of the drag force and demonstrate to yourself conclusively as an exercise that it is, in fact, a non-conservative force. Let's take a look at this example I hinted at a moment ago involving the potential energy associated with a spring. So we can take this analysis of energy that we've been applying here, work, kinetic energy, stored energy in the form of potential energy and so forth. And we can bring that analysis to bear on the spring and its associated spring force, Hooke's law, f equals negative kx. That is, a spring will always oppose changes in its displacement from equilibrium. And it does so linearly with the displacement. The spring force can be applied really to any material that resists changes in shape and for which the force exerted is linear with displacement from equilibrium. So really, this analysis will apply to any so-called elastic body, which I kind of defined a moment ago as one that, once deformed by a force, restores its original shape when the external force is removed. The relationship between work done by a force and changes in potential energy is given by the equation we have seen earlier. It is that changes in potential energy are the negative of the work that is done by the forces involved. So for instance, we know that the work done by a spring whose length is stretched to a displacement x from its equilibrium length of 0 is given by the following string of analysis efforts. The work done by the spring will be equal to the integral from the initial compression, which is none, to the final compression or stretching, which is x. It's the integral of the product of the force of the spring, which varies with displacement, times the tiny little displacements. So in this case, we have a situation where the displacements, for instance, could be in the positive direction, stretching the spring out. But the force from the spring will oppose those displacements pointing in the opposite direction. So if we do the dot product, we'll find that this is equal to negative kx dx. We have to integrate this from 0 to x. Now, k is a constant. It's the spring constant. It depends on the properties of the material that make up the elastic body. But it's just a constant. It doesn't change with displacement up to a limit, which we discussed earlier in the context of Hooke's Law. It is possible, of course, to deform a material so much that there's no way it can restore itself. All materials have limits. So if we do this integral, and I will leave this as an exercise to you to show that this is true, you can go back to the previous discussion of the spring force and try this out yourself. We've looked at this before. But we find out that this integral is to be very pedantic about it, negative 1 1 1⁄2 kx squared plus a constant of integration. And then we evaluate that at the end points of the integral. We evaluate it at the value of x displacement. And we subtract from that the same function evaluated at 0 displacement. And we find at the end of this that the work done by the spring in stretching it out, for instance, is negative 1 1⁄2 kx squared. So let's now figure out what is the potential energy associated with the spring. The work done in, say, stretching out the spring, the work done by the spring, is negative 1⁄2 kx squared. And if we relate changes in potential energy to the work done by the spring, we find that the change in the potential energy of the spring, which should be equal to the negative of the work done by the spring, is 1⁄2 kx squared. That is, the final potential energy of the spring when I stretch it out is bigger than the initial potential energy. 1⁄2 kx squared is a positive number. k is a positive number. x squared is always a positive number, even if x is negative, and 1⁄2 is a positive number. So when I stretch a spring, I add to its stored potential energy. If I compress a spring from equilibrium, I also add to its stored potential energy. Again, x squared doesn't matter if I compress or stretch. I add to the potential energy of the spring if I move it away from equilibrium. So, again, if we assign that an unstretched spring has an initial potential energy of zero, then we learn that the final potential energy, either stretching or compressing the spring, is 1⁄2 kx squared. Stretching a spring costs us energy in the form of positive work, but it stores energy in the stretched spring. The spring does negative work, and thus stores energy. So I hope you've enjoyed this introduction to this concept of potential energy, the energy of configuration of bodies. And you've seen that there are cases where you can reconfigure two objects relative to one another, and in doing so, if the force associated with that reconfiguration is conservative, you can store energy in the configuration, and that energy is potential energy. So we have come to understand that there is energy associated with the configuration of material objects, potential energy. We've seen the relationships between changes in potential energy and work and changes in kinetic energy by considering a few examples, and of course, we're going to have to work through more examples on our own in order to build familiarity and comfort with these ideas, but that's what practice is for. We have learned to distinguish between conservative and non-conservative forces using a relatively straightforward comparison of work done going from A to B to the work done going from B back to A. And by considering this situation where we go from start to finish and finish to start, we have learned to identify conservative and non-conservative forces. And finally, we've seen how to describe the potential energy associated with things like gravitation and with springs, gravitational potential energy and elastic potential energy, reconfiguring objects in the presence of a gravitational force, reconfiguring objects where there's an elastic restoring force, has with it the added benefit of the ability to store energy and that energy can then be released from the object into other forms of energy.