 Hi, I'm Zor. Welcome to Unizor Education. I would like to continue talking about energy and today's topic is potential energy. Kinetic energy we have already discussed in the previous couple of lectures. That's energy of the movement. Now, potential energy is energy of the position of the object relative to other objects. Now, this lecture is part of the course called Physics 14's presented on Unizor.com. If you found this lecture on YouTube or anywhere else, I do suggest you to go to this site actually and take it from there because every lecture on the site contains also has textual description which basically is like a textbook. All lectures are presented in logical sequence as a course. There is a prerequisite course on Unizor.com. It's called Mass 14's, which I kind of assume that you are familiar with all the topics of that course, especially calculus and vector algebra. And by the way, the site is completely free Unizor.com. It has no advertisement, no financial strings attached. You don't even have to log in or anything like this. Although it might actually help if you would like to have some additional functionality of the site. So, potential energy. Okay, so as I was talking about this in the beginning, potential energy is related to a position of the object relative to other objects. Kinetic energy, if you remember, is energy of the movement which will cause certain work done if you somehow change that movement. For instance, slow down the object. Now, if you slow down the object, there is some kind of a force which is being developed from this movement and it results in some work. In case of potential energy, we also are talking about work. So, this is the work which can be potentially performed by the object if left alone wherever it is right now. So, this potential energy is basically an energy of the object which it can perform if left alone in that position. Now, obviously it implies that in that position there are some forces which are acting on this object. Now, when I was saying left alone, well, it means it's left to the forces of all these other objects around it. Now, the perfect example of this is if we will, if we lift an object above the level of the ground. Now, we are supporting this object with our hand. Let's say it's this marker, okay? So, I'm supporting this object right now. What forces are acting on it? Well, obviously, the gravity force goes down and my support goes in opposite direction with the same magnitude and that's why the object remains in the same position relative to the earth. But as soon as I take out my support and leave it only to the earth and the marker. That's it. What happens? Well, obviously, the object falls down. What it means is that the earth gravity actually is moving object and as it moves object, it exerts certain force onto this object and that's why the force of gravity is performing the work. So, the potential energy of this object in this particular position is a quantitative characteristic of the position which is equal to the work which it will perform or it will be performed if I will just left this object completely alone by itself, subject to other forces. In this case, the force of the gravity. Okay, that's my explanation. Now, and I will use this particular example to do some quantitative things. Now, let's assume that this is the level of the ground of the earth, right? Now, this is my object. Now, obviously, I will put my system of co-ordinates upwards and this is initial height of the object above the ground, h. And let's assume that mass is m. So, what can we say about forces which are acting right now on this object? If left alone, so I do not support this object anymore with my hand or anything else, the force of gravity which acts at this particular point will pull the object down. So, since there is a force which is acting and there is certain distance it will fall down, this force will perform the work. Now, let's assume in this particular case that this height is not very large relative to the radius of the earth which is like, whatever, 6,000 kilometers, whatever. Now, assuming that the force is constant. Yes, in theory, you know that the gravity depends on the mass of one object, mass of another object and the square of the inversely proportional to the square of the distance and the distance is changing as it moves, but it's changing so small relative to the radius, it changes from r plus h to r and if r is very large, which is radius of earth, we can actually neglect this change. So, we assume that the force is constant. So, the force is constant and it's equal to the weight of this object which is mass times acceleration of the free fall on the earth, on the surface of the earth. So, this is the force. Now, the force is acting all the way as it falls down and the result will be the work which is performed. What happens at the very end? Well, at the very end the force of gravity will be neutralized by the reaction of the ground opposite in direction and equal in magnitude, so no more force acting, which means that whenever this object is at this position, there is no potential energy because left alone, well in this case on the surface of the earth, it will not perform any work. It doesn't move anymore. So, we are talking about potential energy by definition equal to amount of work which is performed by forces around this object if this object just left alone to these forces. In this case, the only force which is in this case is the force of gravity. Alright, so that's fine. That's easy. That's basically a definition we don't really have to prove anything or anything like this, right? But now, let's just think about what happens as the object falls down, how its potential energy is changing and what else is changing. Well, what else, obviously, is its kinetic energy is changing. Why? Because it has a speed. Speed in the beginning is zero, right? So, the kinetic energy in the beginning is zero. But as it falls down, the kinetic energy will grow because its speed grows. So, kinetic energy, by definition, is mv2 over 2, right? So, what we are going to do right now is we will examine how potential energy and kinetic energy are changing as the object falls down. Okay, first of all, we know the acceleration of the free fall, that's G. So, in this particular case, we know that acceleration of this object is equal to minus G. Minus, obviously, because the x-axis goes up and the object goes down and it's speeding up all the time. So, this is acceleration. Now, if I have an acceleration, I have initial position, so x of zero equals h. I have initial speed, v of zero is equal to zero. Object, basically, the beginning is at rest. And then I know the acceleration. From kinematics, I know my formula for the distance this object covers. It's equal to initial h minus, since the speed is zero, initial speed is zero, so it will be minus g t squared over 2. So, that's obvious kinematic equation. Hopefully, you'll know this. And if you don't go back to the beginning of this course when I'm discussing the kinematics and how the distance is changing in constant acceleration case, depending on the time t, t is time. So, this is the formula. Great. Now, if we know this formula, so we know the position at any moment of time. And since we were already talking about the work performed, if the object is at certain height over the earth, this potential energy is equal to this, which is the work actual which is being done. Now, if my, instead of h, my position is x of t, my potential energy as a function of time would be equal to mg multiplied by the distance above the earth, which is x of t. h minus g t squared over 2. Okay, that's my potential energy. Now, let's talk about my speed. Now, speed is first derivative by time of the distance, which is equal to minus g t. Again, minus because it's directed downwards. And its absolute value is equal to g t, which again, obvious from kinematics, I don't even have to take the first derivative. We know that if it's acceleration with the constant acceleration g, in the beginning we have zero. So, at the time t, we will have the absolute value of the speed g t, which means we can find out our kinetic energy. Kinetic energy, k kinetic. Also, function of t would be equal to what? m v square over 2, which is m v is this. So, it's g square t square over 2. And now, let me give you a remarkable fact. Now, the remarkable fact is that something which we call full mechanical energy, full mechanical energy, which is equal to its kinetic energy as a function of t plus potential energy as a function of t. And what is it equal to? If I will summarize this as you see, m g m g square t square over 2 will cancel out. And what remains is m gh constant, not dependent on t. So, what happens as the time increases from t is equal to 0 to t is equal to maximum whenever our object falls on the ground. Well, my potential energy as t is increasing, my potential energy is decreasing, right? Because there's a minus sign. So, if t is equal to 0, it's m gh. And then as t grows, this is the positive, it will reduce my potential energy. Now, what happens with my kinetic energy? So, potential goes down, kinetic obviously goes up as g is increasing. In the beginning, it's equal to 0 because speed is equal to 0. And at the very end, it will be something whatever the maximum speed is. However, their sum is constant. Now, later on, we will talk about the conservation of energy. There are certain laws in physics which basically are all about the conservation of energy. In this case, it's very obvious because we're talking only about mechanical energy and we have two different kinds of energy, the kinetic energy and potential energy. And as we see, potential energy in the beginning which is equal to m gh is decreasing as the object falls down, kinetic is increasing, but their sum remains the same. That's the conservation of energy. Okay, now, let's just think about what should happen as the object falls down at the very, very end. Well, we can first of all find the time it takes to fall down. Well, that's basically x of t is equal to 0, right? Okay, what is the t? Which means that h is equal to gt square over 2 and t is equal to 2h divided by g square root. So, this is my moment of time, let's call it tg, ground, whatever. So, this is the time moment when my object touches the ground. Okay, now, question is what's the potential energy and kinetic energy in this particular case? Well, potential energy is equal to 0, as we know, because the height is equal to 0, right? So, potential energy is h mg, h minus gt square over 2. And at the moment t ground is equal to, if I will substitute this, obviously I will have 0. h is equal to gt square, so this will be 0. So, this is equal to 0. Now, my kinetic energy is equal to mg square t square over 2 and my kinetic, this is potential, my kinetic energy at the moment t tg, the time it touches the ground, it's equal to, so if I will substitute here, what will I get? t square would be 2h over g, so g goes down, 2 goes down, so mgh. Well, as we expected, actually, this is the initial potential energy and as we know, since we are dealing with full mechanical energy being conserved, so if my potential energy goes down, then everything should go to kinetic energy, obviously, so that everything is mgh, the initial potential energy is completely converted into kinetic energy at the very end. So, as the object falls down, potential energy is converted into kinetic. Converted means that kinetic is increasing, potential is decreasing, but their sum remains the same. That's what the word converted, because it's probably not exactly, there's no conversion, so to speak. It's actually just form of expression which basically shows that one energy kind of disappears, but another appears. Well, you can call it conversion, but their sum remains the same, which means whatever has disappeared from the potential appears on the kinetic side. And that would be it for today. That's all I wanted to explain about the potential energy. I would like to make a very important point. In this particular example, which I'm just using as an introduction into potential energy, I assume that the force, in this case the force of gravity, is constant. And all my calculations are very simple. The next lecture will be devoted to potential energy of the spring. And in this case, the situation is a little bit more difficult, because the force is changing as the spring is returning to its neutral position. There is a Hooke's law, if you remember. So, in this case, it will be a little bit more difficult, but this is just an introduction. What kind of energy we are talking about? What is the potential energy? Again, it's a quantitative characteristic of the position. And in theory, if this particular object, which has certain potential energy, if it's just left by itself to its own way of dealing with surrounding environment, then there is some work which will be performed by this surrounding environment, like in this case by the force of gravity. That's it. Thanks very much and good luck.