 okay guys do this you have an inclined plane this is angle theta okay this is the height edge fine and now I am removing this incline and making it a wavy path like this fine without touching the initial and final point initial and final point remains the same okay this is height edge now you're keeping a mass here fine so what is the work done by the gravity in both the cases in both cases what is the work done by the gravity yeah it is mgh gravity force is acting down and displacement is also down so the work done by the gravity is mgh in both the cases fine so you see that the path is changed but then still you're getting the same work done by the gravity all right so since the work done by the conservative force do not depend on the path it only depends on where it is right now and where it is finally okay so it depends on only two points so it is like a state function okay and since it is it depends on only where it is and where it will go all right hence we can define something which is a function of only what is a location of the object fine so we need not calculate the work done by gravity by looking at the path fine so we can cater to the work done by the gravity in a different way all right or let us say work done by any conservative force we can find out quickly if you know its initial state and the final state fine do you guys recall any other physical variable that you have encountered till now in physics that depends only on the state initial and final can you message it quickly which variable which physical parameter you have learned that is only the function of initial and final states exactly internal energy right so we have learned correct so we have learned this physical parameter delta u which is just equal to n cv delta t in case of an ideal gas fine all right it only matters what is a change in the temperature okay so change in internal energy is a state function now similarly you can define the you know a parameter which only depends on initial and final points okay and I think you guys know what I am talking about here I am talking about potential energy here right now the way potential energy is defined okay it is potential energy is the energy right down it is the energy the mass has its position say shape and size okay so how it is located where it is located what is its shape and what is its size okay so it it is the energy that is possessed by an object because of these factors fine so if if it is constrained as in if you take a let us say if you take a mass if you take it at a height h okay you're keeping it at that height all right you're keeping it that height so what I'm saying is just by the mere fact that it is at some height okay it is at some level it has some energy okay just the fact itself that it is there it has some energy all right and if you remove all the constraints or all the other you know surrounding things that is you know making it possess that much potential energy then ideally every object wants to just give up all the energy it has all the potential energy it has and comes to the state where it has the minimum possible energy fine this is how every object in the universe behaves right so natural tendency of every object is right down natural tendency is to lose all potential energy or to minimize minimize the potential energy so by the way potential energy is represented by a letter capital U so that's something which everybody has accepted okay how does shape affect the potential energy so I think the spring is the best example of potential energy due to the shape just compress it its shape changes right so if its shape is changed right I'm saying that it has energy and how do you know it has energy which is really said and if you have a mass connected with the spring it will transfer its potential energy and create the kinetic energy for the mass fine all right so every object has a natural tendency to minimize the potential energy okay so whatever potential energy an object has given a chance it will just give up all the potential energy it has and that will manifest in the form of kinetic energy fine now I can take up multiple examples right so if I take a mass m like this in air okay if I release it it will gain kinetic energy it'll come down and will gain kinetic energy right so if I assume that there is nothing called potential energy that exists in the universe then how come energy gets created energy got created out of nowhere that is against the universal law of energy conservation fine so there must be some sort of energy that is getting converted into the kinetic energy okay fine and that energy we are defining as potential energy all right so enough of the introduction to the potential energy I'll directly get into the definition mathematical definition of the potential I think you have you already know what is potential energy you might have used it in your lower grades all right so the way potential is defined is this you say that change in potential energy change in potential energy is negative of the work done by the conservative force so change in potential energy is defined as negative of the work done by conservative force and work done by any of the force is what integral of force with the displacement fine so this is the definition of the change in potential energy getting it now this definition is true no matter what potential energy we are talking about we may be talking about the spring potential energy yet we may be talking about the gravitational potential energy or let's say we'll talk about electrostatic potential energy what are potential energy you may talk about the definition and all the potential is this that change in potential energy will be equal to negative of the work done by that conservative force for which you are defining the potential energy getting it fine so if potential energy is there if there exists potential energy there has to be a force corresponding to it isn't it that's how potential is defined if there is a potential energy there has to be a force corresponding to it right so the way potential is defined is this now same thing I can write in a differential form I can say that du is equal to minus f dot dr fine now what is f dot dr you can write du is equal to minus of magnitude of f into magnitude of dr into cos of theta getting it dr cos theta is a displacement along the direction of force let's say displacement along the direction of force is dl getting it no but a du is not a vector it's a dot product between force and the displacement isn't it so dot product will give you a scalar only all the energies no matter what it is all the energies they are scalars all right so dl is what dl is a displacement in the direction of force fine so I can say that du is minus of f into dl so from here I can get f is equal to du by dl fine this is a special derivative what you're doing you're taking derivative along the direction of force only fine so we call this as partial derivative what is partial derivative you will learn later on that when you take up the partial derivative suppose you're taking derivative with respect to y okay suppose u is let's say x square plus y okay now force along y direction is minus of du by dy okay I am differentiating with respect to the direction along which I need to find a force so when you differentiate a partial differentiate with respect to y you take x to be constant the f y will come out to be minus 1 Newton's and similarly fx will come out to be minus of du by dx that will be equal to minus 2x so fx is this and f y is that so total force will come out to be minus of 2x icab minus jgab fine so we'll take a few questions on the you know on the partial derivative I mean there is nothing more to it it just a matter of couple of questions and then you will start feeling comfortable with it so what we have done till now is that we have got a relation between change in into a change in the potential energy and the force for which you are fine in the potential energy fine now let us try to define potential g due to the gravity and potential g due to the spring because these are the two forces that you will be dealing again and again when it comes to you know solving a mechanics question fine so let us try to quickly find out the potential energies for the spring and for the gravity so first we will try to define write down potential energy due to gravity all of you might be aware of the formula already that potential g due to the gravity is mgh okay we'll get into more details about it how it comes out and what is the implication of this particular formula okay so we'll take a simple you know example let's say you have a ground like this your ground like this okay now you take a mass this is mass m okay so what you're doing is you are lifting this mass m from this point to a point which is at a height h this is height h okay we are trying to find out the formula for potential g for the gravity alright so we will start with the definition of potential energy potential energy is equal to negative of the work done by the force for which we are finding the potential g and we are finding potential g due to the gravity force right and gravity force magnitude is mg okay so work done by the gravity force is how much all of you how much is work done by the gravity work done by the gravity is minus of mgh right because you go from point number one to point number two the displacement is h and that is against the direction of gravity the work done by gravity is minus mgh okay so according to the definition this will be equal to the change in potential energy okay so according to your definition negative of the work done by the gravity is change in the potential energy fine so negative of the work done by the gravity which is equal to mg h fine this is equal to change in potential g which is what u2 minus u1 fine so I can say that u2 minus u1 is equal to mg h okay now I am interested in finding the absolute potential energy I want to know what is the potential g at point two but what I am getting is difference in the potential energy alright I'm getting difference in potential g but I want to know what is the potential energy okay so one thing you guys please understand is this there is no concept of absolute potential energy fine so you will never know what is the zero potential energy okay hence in order to find the potential g you just you can randomly assume anything to be zero okay now now I'll explain you how the answer will not change because of that first please write down that we need to assume a location where potential energy becomes equal to zero fine then only we can talk about absolute potential energy fine although it is not absolute but then you know it will help you to visualize better that's it so you can assume that u1 is zero so if you assume u1 is zero then u2 will come out to be mg h are you getting it then u2 will come out to be mg h so potential g we have defined at a location which is at a height h from where the height h from a location where you have assumed the gravitation potential g to be zero okay so it need not be zero you know the height need not be zero this could be the case like this in that that it is on a table which itself has a height of small edge are you getting it it it could be on a table that is at a height of small edge from the ground and this ground could be the second floor of the building okay so you can assume any horizontal line to be zero potential getting it so every question going forward understand I'm talking about with respect to problem solving so every numerical or every question if you have to find the gravitational potential energy fine we need to first assume a horizontal line that represents gravitation potential energy to be zero getting it so once you assume a horizontal line in a particular question then after that you can write down the potential g relative to that horizontal line so if it is at a height h above then potential g will be equal to mg h and if it is at a depth h below if it is h below of that line the potential g will be minus mg h fine so potential g could be positive if that is above the horizontal line and potential g could be negative if that point is below that horizontal line okay so like that in every question you can have a new horizontal line and you can say that that is my zero gravitation potential energy all right now once you assume a horizontal line for which you say that gravitation potential is zero you cannot change that line in between the question are you getting it so for entire question that becomes the zero gravitation potential g line fine so for different different question you can have a different horizontal line up to you what is convenient to solve the question you can assume any horizontal line to be zero potential g but once you assume a zero gravitation potential g line which is horizontal you cannot change it midway of the question okay so I hope I am very clear about this in case you have any doubt please type in you have any doubts type in yes or no so understand one thing very very clearly h is the distance which is vertical and in a particular question if you're not sure what is vertical it just find out what direction the accession due to gravity is I'm talking about the distance along the line of accession due to gravity