 So, let us say this is earth, okay, the earth has radius of earth and mass of earth Me, okay. Now let us say this is the path along which I am moving an object, okay. So, suppose I am going from this point, point number 1 to point number 2, okay. This distance is let us say r1, okay, and that distance from the center I am finding the distance, okay. This distance is r2, right. Now can you use this definition of potential energy which is delta u is equal to minus of negative of the work done by the gravity force to find the change in potential energy. Between point 1 and 2, so this will be delta u, delta u is what? u2 minus u1, okay and try to use this formula, work done is integral of f dot dr, okay. This f is the force of gravitation, alright. Now try to do it on your own quickly, distances are very large, so you cannot say gravitation force is mg, because g itself is changing. How much is the gravitational force? The value of gravitational force at a distance of r is how much? Let us say mass is m, this is small m, how much is the gravitational force? gm into mass of earth, okay, divided by r square, okay. This is the gravitational force, all of you clear about it, okay. See change is always final minus initial, does not matter delta u, delta x, delta y, delta t, whatever it is, it will be always final minus initial, fine. This is the gravitational force, isn't it? So the work done by the gravitational force is integral of this into dr, r1 to r2. Is this correct? Something like this, is this correct? Is this the work done? No, there is a mistake, there is a mistake over here, can you tell me what is a mistake? This is an error, the direction of force is like this, and you are moving in this direction, okay, there is a dot product between force and displacement, okay. You are moving in opposite direction of the direction of force, so that is a negative sign will come in, okay. So what you have done is, you have taken a random point at a distance of r, found out the gravitational force, and you have assumed a small displacement along this path, let us say this distance is dr, okay. So you can assume that for that small distance dr, you can assume force is constant, because dr is very, very small, okay. So force is not constant, it keeps on changing the distance r, but for small dr, you can assume that force is a constant. So for dr amount of movement, you can find the work done as simply force into dr displacement, because dr, you can assume force is constant for that dr, okay. And then you need to integrate that total work done, okay. This is a usual question, they ask you in schools, derive the expression for potential energy for the gravitation in case of this kind of scenario, okay. Switching the limits, should you have, if you switch the limits, yeah, that is correct, what are you saying. You will learn about it later on, but right now let us keep it simple only. No doubts till now, right? Alright, integral of dr by r square, what it is, integral of dr by r square is minus of 1 by r, okay. So we are going from r1 to r2, fine. So limits are from r1 to r2. So this will give you the minus, minus gets neutralized, it is a multiplication. So this will be gm Me by r2 minus of gm Me by r1, okay. So this is what work done by the conservative force, fine. And how the potential energy was defined? Change in potential energy which is u2 minus u1 is negative of the work done by the conservative force, right. So minus of gm Me by r2, okay, minus of minus gm Me by r1, okay. This is u2 minus u1, okay. Now can you tell me what is u2 and what is u1? All of you tell me what is u1 and what is u2? Okay, can I say that u2 is this? Can I say that u2 is minus of gm Me by r2? Is this correct? Others quickly type in, my simple question is, is this correct? Saying like this, is it correct? This is not correct, okay. This is not correct, all right, fine, all right, this is not correct. But when I say potential energy is mgh, when I say potential energy is mgh, in work by energy chapter when we said potential energy is mgh, we always assume zero is something and compared to that zero, the potential energy is mgh, okay. So we never found out the absolute potential energy, we first assumed what is zero and then compared to that potential energy, we say it is mgh and whatever is the case, okay. And hence in this case also, we need to define what is zero potential energy, okay. But the problem is that now it is not just about earth surface, are you getting it? We are talking about stars, we are talking about planets, we are talking about interaction between sun and other planets. So I cannot say that let us assume zero potential energy as a surface of the earth, fine. It has to be something which is unique to everything, all right. So that is the reason why we need to define the zero potential energy in a very careful manner so that that is universal, okay. So let us try to do that. Let's say R2 is tending towards infinity. If R2 tends towards infinity, this term will go towards zero, okay. So I can say that u infinity minus u1 is equal to minus of or plus of gmme by R1, okay. Now till now we haven't assumed anything, all right. Now if you assume, okay, if you assume u infinity to be zero, then you can say that potential energy at 0.1 is minus of gmme by R1, okay. So when you say that this is a potential energy between the two masses m and capital Me, you already assumed that the potential energy between these two masses will be zero when they are separated by an infinite distance, okay. So write it down in your notebook that the zero potential energy, the zero potential energy is a situation in which the masses are separated by infinite distances, fine. So in work by energy chapter, you try to, I mean, depending on what kind of problem it is, you always try to find out where appropriately you can define zero potential energy, right. Every time you define zero potential energy in work by energy chapter. But in gravitation chapter, there is this unique situation in which gravitation potential energy is zero. You don't need to define every time every new problem what is zero gravitation potential energy. Here we are assuming that gravitation potential energy is a situation in which the two masses are infinitely separated from each other. Is this thing clear to all of you? Any doubts till now? Anything? Okay. All right. So what we have learned just now is the potential energy between the two masses is minus of gmme by R, okay. So this is the potential energy between two point masses or two uniform spheres separated by a distance R, okay. So once you define the potential energy, you can use the work energy theorem anywhere you may want to, okay. Now understand one thing that this is negative. There is a negative sign over here. So if R decreases, what will happen to potential energy? It will increase or decrease. Potential energy will increase or decrease if R decreases, correct. Potential energy will go down, okay. So earlier if it is let us say minus 100 joule, if you decrease the R, it may become minus 200. So minus 200 is less than minus 100, fine. There is a negative sign over there, all right. And also you need to remember few things like potential energy. You got this formula. This formula is valid for only two masses, fine. But then there can be multiple masses, okay. For example, this situation, all of you draw this. Suppose you have a square, okay. This is a square. Let us say you have four masses, masses M and the distance between the adjacent masses is A. It is a square, all right. Can you find out the potential energy of this system, okay. The formula you have is only for two masses. But right now you have four masses, okay. Now can you try to find out the potential energy of the given arrangement? Quickly do that. The hint is you can take two masses at a time and add all the potential energy, okay. Potential energy is a scalar quantity, okay. You do not need to worry about directions and all. You have four masses. Yes, you need to take care of diagonals as well. Potential energy will be potential energy between one and two. Potential energy between two and three. Between three and four. Four and one. One and three and two and four, okay. These kind of combinations can be there. In fact, you can easily apply the combinatorics. Four C2 is six, no. Six combinations will be there, okay. Now U12 is equal to U23, U34, U41. So just find out one and multiply with four. So this will be equal to minus of four times GM square by A. This is equal to sum of these four, okay. Then you need to take care of the diagonal ones as well, okay. You need to take care of the diagonal ones, these two and those two. You may think that I have already counted mass two. How can it have an energy with four? I have already counted two with one, two with three. How come it has with four also? The point is, it doesn't matter how many masses are around. The forces between the fourth and the second one is unaffected by what are the masses around? Between these two masses, the force will be GM1M2 by R square. And because of that force, only the potential energy is coming in, right? So it doesn't matter how many masses are around. You can always count one mass multiple times when you are talking about the potential energy, fine? Now potential energy between one and three is what? Between one and three, the distance is A root two, okay. The diagonal length is A root two. This is A, this is A. The distance between one and three is A root two. Same is the case with two and four as well. So these two are equal. So you can say that minus of two times GM square divided by A root two, fine? And then you can simplify it further. You can say that root two cancels a little bit like this. So minus of four plus root two GM square by A. Any doubts till now? Anything? Anything you want to ask? No doubts? No doubts. Okay. You need to now find out, find out the work done to create the system. This is a system I'm talking about. You are creating this system by bringing these masses from infinity. So initially they are separated infinitely apart, okay? So you're bringing these masses from infinity and making this arrangement, okay? Now in order to make this arrangement, how much work is done by you is what you need to find out. Properly solve it, okay? Properly solve it. Use work energy theorem. So whenever someone is asking you to find the work done, immediately this should come in your mind. Work energy theorem, okay? We are talking about the work done to create the system, okay? The first point is when the masses are infinitely separated from each other, right? And our gravitation potential energy will be zero. If masses are infinitely separated from each other, gravitation potential energy is zero. U1 is zero initially when they are at infinity. Even K1 is infinity. And then you are bringing them slowly, okay? So finally when they are static like this, then also K2 is zero, okay? So whatever work you need to do is equal to the final potential energy itself, okay? So you need to actually do the negative work, okay? You need to do this much work, whatever is a potential energy to build the system, to create the system. So work done is equal to potential energy itself. Is this thing clear to all of you? Any doubts? No doubts? Now find out work done to destroy the system. Destroying the system means you're taking these masses from this system and placing it in infinity. So totally dismantling the system from wherever you have brought it. Work done will be negative. Yes, because the forces are attractive, right? So they are trying to come close to each other. You have to move slowly. So you need to apply force in opposite direction of the attraction. So that is why your direction of force is in opposite direction of the displacement. So that is why work done will be negative. Now work done to destroy is how much? Now your initial point is this and final point is infinity, okay? So the work done which is equal to U2 plus K2 minus U1 plus K1, okay? So first point is this and second point is infinity. So U2 is zero, K2 is zero, K1 is zero. So the work done will be negative of the potential energy, right? So these kind of questions are very common in school exams, all right? So they ask you, I mean, you can understand that it is like to create a system, to destroy the system and what is the potential energy? These things are in a way they're trying to ask the same thing as in what is the potential energy of the system. So depending on that, you will need to do some amount of work to destroy it or to create it, all right? So whenever like there is a question in school where they're trying to ask you or they're asking you how much work is done to create a system, first step is to find the potential energy of the system, okay? And the next step is to use the work energy theorem, okay? So these are straightforward questions which are routinely asked in school. No doubt still now, right? Nothing.