 Okay, so many times you will encounter a scenario in which you know chain is there, okay, so when it is a chain, how you will take care of that particular scenario, it's a special case how to deal with a problem in which the chains are there, okay. What is chain? Chain, we are considering a chain which has distributed mass along the length, okay, mass is distributed, fine, so let's assume that you have this chain, the mass can be uniformly distributed or non-uniformly distributed, it depends on the scenario, so let's say this chain has mass m and it has a length of l, okay, so suppose this is the ground for which you are saying that gravitation potential energy is 0, okay, by the way, the work energy theorem that we have derived is not only valid for the small masses, it is valid for the bigger masses also, fine, like in the case of chain, it is valid here as well, but the problem which you will be facing here is to write potential energy and kinetic energy for the chain, okay, because this entire chain is not located at a particular location at a particular point, right, so since it is not located at a particular point when you write mgh as potential energy, you will not be sure what edge to take, whether to take edge from here, there or there, we are not sure about it, right, so that is the reason why we have taken the special case of chain here, okay, so my question here is, find out the total gravitation potential energy of the chain, ug is what, you need to derive it, okay, no Bharat, we do not know what is centre of mass, yes, you can assume the chain is just touching the ground, the hint is, take a small mass, let us take this mass is dm and this is dy, this dy is at a height of y, okay, so yes, mgh at, no but what m, mass of what, total chain, no, so potential energy of dm is what, see dm is a point mass, it is a very very small mass, it has a fixed location, so you can write down the potential energy of dm as du, which is equal to dm g into y, okay, now the total potential energy will be nothing but the integral of this, now the problem with integral of y dm or dm y is that both m and y, both are the variables, okay, so you need to write one in terms of others, then only you can integrate, right, so the width of this dm is dy, so total length l has a mass of m, so dy will have what mass, m by l which is mass per unit length into dy, okay, so this is equal to dm, so just substitute it here, you will get g times m by l y dy, now you can integrate u and mgl integral of y dy will come and this you can integrate from 0 to l, that is how the y coordinate is changing, so what you will get here is that mgl by 2, this is what you will get as potential energy, okay, so in case you have uniformly charged, sorry not charged uniformly distributed mass in the chain, okay, then all you have to do is to find out where the center of the chain is, okay, where the center of the chain is and find out what is the height of the center of the chain, okay and once you find out the location of the center of the chain, you just multiply that with m into g, so m into g into height of the center will give you the potential energy of the chain, fine, so we will learn the distributed mass in greater detail when we talk about the rigid body motion later on, but right now you can use it as a thumb rule if the chain is uniformly distributed mass, then the potential energy is just mg into h center of mass, getting it, any doubts, any doubts to write the potential energy of the chain, kindly message, all you have to do is to find out where the center of the chain is, then entire chain will behave as if it is a point mass located at where the center of the chain is located, here we have also integration part, so this is how we have integrated, we have got dmgdy, okay, we cannot integrate it right now because it has two variables, fine, so we first write dm in terms of dy so that there is only one variable y and then I can integrate changing y from 0 to l, okay, so I will get mg l by 2, fine, so you can also write this as mg into l by 2 and what is l by 2? l by 2 is the height of the center of the chain, fine, so the thumb rule is, thumb rule is write down if chain is straight, then mg into height of center of the chain will give you the potential energy of the chain, fine, you are assuming to be straight, okay, any doubts, yes or no, please message, they go for length l, total mass is m, okay, so for length dy, what is the mass, mass per unit length into dy, okay, that's how you have to apply, getting it, all right, so let us take up a small numerical on the chain itself and see how we can apply, what we have done till now is that we have defined the potential energy of the chain, okay, and that too for a special case where the chain is straight and if chain is not straight then we have to follow the integration procedure to find the total potential energy of the chain, okay, for example, if chain is placed on a hemisphere, if chain is placed on a hemisphere like this okay, then you cannot just find out the center of the chain and just, you know, write potential energy accordingly, what you have to do now is, first you have to assume some zero potential energy then take a small length of the chain, find out its potential energy and then integrate it through the length, okay, so that you learn when you take up questions like these, okay, so let's not learn it as a theory, so once you try out these questions, message me, then I can help you with this, okay, meanwhile let us take a question, suppose you have a table like this on the table, you have a chain placed with l by 4 length hanging, this is l by 4, total length of the chain is l, fine, mass of the chain is m only, okay, this is a smooth table, it's a smooth, fine, now what will happen, the chain will slowly slide down, okay, you need to find the velocity of the chain when the chain completely come out of the table, find out velocity of the chain when it comes out completely, so see with respect to kinetic energy, the chain is a straight forward case because the entire chain, if it is sliding will be moving with the same velocity, so you can directly write down half m into v square to be kinetic energy of the chain, where v is the velocity of the entire chain, so with respect to kinetic energy, there is no ambiguity, I have seen it, others, let us assume this to be zero potential energy, this is zero gravitation potential energy over here, okay, so if you break the chain into two parts, fine, 3l by 4 part has no potential energy, u is zero for 3l by 4 and for l by 4 potential energy is what initially minus of m by 4 which is the mass of the hanging part g into center of the hanging part l by 8, so initial potential energy will be equal to the potential energy of this part which is lying on the table which is zero plus potential energy of the hanging part, hanging part's mass is m by 4, okay and its center is lying below a distance of l by 8 from the level where you have assumed to be zero potential energy, okay, so u1 will be equal to mgl by 32, there is u1, okay, k1 is what, initial potential energy is zero, okay, u2 is what, u2 is when the entire chain comes down the table, that is minus of mg l by 2, if entire chain comes down the table then the center of mass will move down to a distance of l by 2, fine and k2 you can assume it to be half m into p square, all right and w is anyway zero, there is no other force other than gravity which is doing work for which you have considered the potential energy, so you can say zero is equal to u2 plus k2 that is minus mg l by 2 plus half m into v square then minus u1 plus k1 mg l by 32, okay, you just solve this equation you will get the velocity, okay, so like this there will be, you know, I can understand there is a discomfort in taking up the problem related to chain, so this you can only master when you, you know, try to look for the questions which involves the chain, are you getting it, so if you have any books just look at the questions that have that involves the chain in it, okay, don't just count the number of questions every day, number of questions what you have solved, you should also be on a lookout of some of the unique type of questions, fine, so try to get hold of at least five or six questions related to chain and solve it yourself, okay and in case you have any doubts feel free to get in touch, okay, so we'll take a small break now, we'll take a couple of minutes break right now it is 11, right now it is 11, oh 1150, we will meet at 12, okay, is it fine, okay, so guys let me tell you we have done enough so that you can crack any question from this particular chapter, now whatever we are learning are different scenarios, like if the chain is there what we'll do, if this is there what we'll do, right, so but with respect to the concept, okay, we have learned everything.