 Hello everyone, let us start this chapter Gravitational, so gravitation is a chapter in which we are dealing with one of the fundamental forces in nature which is gravitational forces. You might have already encountered gravitational forces, for example when a mass is there it will get attracted towards the earth with a force of m into g. So we have this gravitational force we have already encountered, we have already encountered gravitational force in Vagpa energy chapter and even laws of motion chapter. So we will be talking about gravitational force between heavenly objects now, for example if we are talking about gravitational force between the sun and the earth will not say m into g the force. What will happen is that when the objects are near earth surface then you can assume that g is constant. You can assume g is a constant and if g is a constant then you can directly write m into g as the gravitational force and near the earth surface if it goes up to a distance of h then you can find out the work done by the mg force simply you can write it down as m into g into h because you can treat mg force as a constant force. So because mg is constant near the earth surface you can define potential energy easily and that is how you have dealt with it. But now we are dealing with scenario where the objects are probably very far away from the earth and because of that the accession due to gravity g may not be a constant. So we are going to discuss such scenarios and gravitation is one of the oldest branch of science. In fact when the first time scientists have all the great scientists were observing the nature their first observation was what is happening to the heavenly objects. So they thought that if they study what is going to happen with star, sun and planets they will have a better understanding of the nature. So they started the understanding of nature started with the understanding of the heavenly objects and its movements. So there are a lot of scientists who were involved in developing this concept gravitation. In fact if you hear great great scientist in today's generation also they will be either taking up study of gravitation or study of quantum physics as a whole. So this is considered to be a very involved topic and it is because of that it is very important for us to understand the basics of gravitation. So this is a very small topic in class 11th as in the chapter is a very small but if you try to understand it in greater detail you will see that many, many finer things are there to take care of it. So that is how it is and we will be taking care of only few of the basic concepts here. All right so I think we have good number of students now. Fine so you understand right it is a very, very basic topic and that is why it is there in our syllabus. You are able to hear me right all of you are able to hear me? All right so you know the first person who has systematically understood about gravitation or systematically done a research on gravitation was Kepler. So Kepler has spent his entire youth around 20 to 25 years of his life studying about the phenomena gravitation. So he used to wake up every day to wake up and watch the motion of the planet. So he has this graph paper in which he used to plot what is the location of a planet after every few days. So he has plotted the entire chart of the motion of the planets and then based on the observation he came up with few laws. So these laws are called Kepler's first law, Kepler's second law, okay so Kepler's first law, second law and Kepler's third law, second law and this is third law, okay. So we will be discussing these one by one. Kepler's first law is also called as law of orbits, fine. By the way when Kepler was doing all these studies the Newton was not there right. There was no universal law of gravitation, nothing was there. So all the observations were purely empirical. He did not come up with any law as such but he has observed few things and based on that he has come up with certain laws, okay. So his observation is like this, so Kepler said that all the planets revolve around in an elliptical orbit, okay. All the planets they revolve around an elliptical orbit and I hope you have already done some study about the ellipse in coordinate geometry. So how the ellipse gets formed, anyone, how to form the ellipse? To form a circle you just tie, you just have a rope like this, tie a pen over here, fix this end and just rotate this a circle will get formed, okay. How will you form the ellipse? Anyone, to draw an ellipse you need to have two points, okay. These are the two points, okay. Then, yeah, you fix point one, two, three. This one, two, three is a rope, okay. So you take this point and rotate or move this point around so that one to two and two to three, these points, I mean one to two rope is taut and two to three is also taut, okay. So when you move this point two and ellipse will get formed, okay. Is it clear to all of you how ellipse gets formed? Okay, so these two fixed points, these two fixed points are called focus, okay. They are focus or if you together call them, they can be called as foci, okay. Okay, so the ellipse has two foci, point number one and point number three, they are two foci and what Kepler has said that the sun will be one of the foci. Okay, Kepler will be at one of the focus, so this could be sun, okay, so this is the sun at one of the foci and the planet will revolve around like this. What is the Kepler's first law? Kepler's first law is that the planets will move in an elliptical orbit, all the planets will move in elliptical orbit with the sun situated at one of the foci, write down all the planets move in elliptical orbits with sun situated at one of the, okay, all right. Now for all practical purposes, when we talk about further concepts in this chapter, we will not assume that the planets are moving in elliptical orbit, okay. When you will solve problems, they in all the problems, we are going to assume that planets or the satellites, they are moving in a circular path, okay. But since we are talking about the Kepler laws which are like the initial laws and it is empirical in nature, so we are talking about elliptical nature of the orbit and that is actually the correct also. It is correct that planets move in elliptical orbit but you know since we are in, as in you are in class 11, so we are not going to talk about the elliptical orbit scenario. So we will only talk about the simpler scenario where the planets and satellites will be moving in a circular orbit, okay. This is Kepler's first law. It is also called law of orbit, okay. Then comes the Kepler's second law, write down. Kepler's second law is also called law of areas. So this law states that, write down, it says sun. This law states that equal areas will be swept in equal intervals of time or to be more precise about what we are talking about, we will say that the line that joins planet and sun sweeps equal areas in equal intervals of time, fine. So what is the implication of this particular law? So for example, a planet is over here. Let's say planet is there here, okay, where is the line joining planet and the sun? This is the line that joins the earth and the sun, okay, or a planet or planet and the sun, okay. So after time delta t, planet will move to this location, let us say, okay. So planet has gone from point number one to point number two. It has moved to that location, fine. So how much area is swept? This line, this is the sun, S1 has swept this much area, okay. So get stay what we are talking about, don't rush ahead, okay, otherwise you will not understand the basics properly, it will be a superficial thing. So this is the swept area, okay. Now if the planet is here, let's say this is point number three, planet has reached this point three, okay. Now the line joining the planet and the sun is this, okay. Now in delta t time, in delta t time, the planet has swept this much area, okay. Now in the same delta t time, according to the law, it should sweep equal area, fine. This line should move such a way that the area covered by this line should be same as that area, okay. Now tell me where is the possible location of point three? Is it this point four or that point five, where it could be at point four or five, right. It should be at five because it has to cover equal area, right. It has to cover equal area, so that is the reason why it should be moving slightly further away and this area now you can say that there is a chance that this area could be equal to that area, okay, fine, all right. The implication of this, these two areas, let's say this area is A1 and this area is A2, okay. So area A1 is equal to area A2. This should happen. That is why the planet has moved from point three to this point five, okay. And since the planet has moved larger distance, you can see that three to five is more distance, okay, and one to two is lesser distance, okay. So you can conclude that planets move faster as they come closer to the sun, fine. Because they have to sweep equal area, right. So if you are closer to the sun, your side length is smaller. You need to cover more area, so that is why the length covered by the planet in the same time should be more. So three to five length is more than one to two length, fine. This is Kepler's second law, all right. The Kepler did not derive anything, okay. All of this was done empirically. The Kepler used to plot the graph and he used to come up with empirical relation, all right. Clear, right. Now I am going to talk about Kepler's third law. Any doubt? Any doubt, guys? Next law is law of periods, okay. Just imagine how difficult it would be for Kepler to come up with these kind of laws, okay. There was no equation, nothing was there as such. He has come up with these laws just purely by analyzing the plots which he has done over the period of years, okay. So Kepler's third law says, write down the square of the time period of revolution of a planet is proportional to cube of semi major axis of the ellipse that is traced by the planet, okay. Time period could be in any units, okay. It is proportionality. Understood, all of you? Now it says that t square is proportional to basically r cube, all right. So if you need to compare the time period of the two planets, okay. You can say, okay, t1 square is proportional to r1 cube and you can say that for planet 2 t2 square is proportional to r2 cube, all right. And then you just divide these two expressions. So t1 divided by t2 whole square will be equal to r1 by r2 whole cube. So this is how you use the third law of Kepler. So you can utilize it to compare the time period of the two planets. Suppose you want to compare the time period of Earth and Jupiter, let's say, okay. Jupiter's time period you don't know, let's say that is t2. But you know the time period of the Earth which is 365 days, okay. So when you take the ratio, whatever unit you take t1, same unit you should take t2. When you take ratio, this term will become dimensionless. So it doesn't matter what unit you take as time period. This is equal to r1 divided by r2 cube. So r1 is a distance of Earth from the Sun which will be given, okay. And r2 is a distance of Jupiter from the Sun that is also given. These three things are given, you can find out the value of t2. So like that, you can find out, okay. Now all of you must understand that Kepler did not derive these, okay. Kepler did not derive any of these laws. Kepler has discovered these laws. Sukrit, can you not talk ahead of it? In fact, what you are saying is not correct also, all right. So you must understand that Kepler didn't derive, okay. So Kepler did not derive any of these things. He just found out using graphs, okay. Later on Newton has proposed the universal law of gravitation. Why it is called universal law of gravitation? That law of gravitation is valid in entire universe, okay. So using Newton's law of gravitation or universal law of gravitation, you can prove all the Kepler laws, okay. So we will come back to proving of Kepler's laws later on when we introduce the universal law of gravitation later on, okay. In fact, we can quickly talk about derivation of the Kepler's first law using conservation of angular momentum. So that we'll discuss and then we'll talk about universal law of gravitation to derive other Kepler laws, okay. So all of you draw an ellipse like this, draw an ellipse, okay. Then this is sun. Let's say this is sun, all right, and this is the planet. Planet and the line joining planet and the sun is this. This is the line, okay. Now can you tell me whether I can conserve angular momentum of the planet only, okay. Angular momentum of planet. Can I conserve about any point or any axis? Can I conserve angular momentum? If not, then about which axis I can conserve angular momentum of the planet? No, to conserve angular momentum what should have? What we should have? Talk about the axis. See, one axis that passes through the sun, okay. If you consider an axis that passes through the sun, the only force this planet is feeling is the gravitation attraction forced due to the sun, right, towards the sun. Is there any other force? Any other force the planet feels? There's no other force, right. Only gravitation force is there, the planet is feeling, and that is along the line joining sun and the earth, okay. So the torque because of this gravitational force is zero about the sun, fine. Is it clear to all of you? Torque about the axis passing through the sun will be zero for this planet, okay. Let's say this vector is r, okay, and velocity vector is v, all right. The angular momentum of the planet is r cross linear momentum, right. So angular momentum of the planet is r cross m into v, okay. So this will be equal to m r cross v. Anyone of you have any doubt, please quickly tell me. I'll discuss that again, okay. Don't hesitate, okay. Now, the small amount of area that is swept, let's say in delta t time, this planet has moved to this location, okay. So the area swept delta a will be the magnitude of r, okay, multiplied by v delta t, all right, into sine of theta between r and v, and half of that, okay. How that sine theta coming in, you can see that if you have this triangle like this, to find the, this thing, area, you need to have this height multiplied by this, okay. That distance, you can say is v delta t. If speed is v, the small distance the planet has covered in delta t time is v delta t. So the area should be equal to, area should be equal to half of base into altitude, base is this into altitude, which is h. Which part, which part you want me to repeat? Lakshya, which part you want me to repeat? Quickly tell me, all right. So the height is basically magnitude of r into sine theta, okay. So this is velocity direction, this is r direction, okay. This angle, sine theta of that angle, if you just look at the geometry, you'll find that, okay. This angle will be 90 minus of, let's say this is theta, this will be 180 minus theta, and this will be 90 plus theta. This is 180 minus theta, this one is 90 degree. So 90 plus 180 minus theta plus x, which is this, should be equal to 180 degrees. So x is theta minus 90 degrees. This angle is theta minus 90 degrees. So h is magnitude of r into cos of theta minus 90 degrees, which is sine theta, all right. So the magnitude of this altitude is r sine theta, all right. So that is why it is, this is correct. So h is r sine theta, so delta a can be written as half into magnitude of r into magnitude of v delta t into sine of theta, fine. v delta t is the distance travelled by the planet. This distance is that. Is it clear to all of you? Quickly tell me, this is basics, so I'll go slow in the basics, okay, great. So magnitude of area is equal to half into this, right, which is r into v into delta t into sine of theta, okay. Now you can see that r into v into sine theta, this you can write as magnitude of cross product between r and v, this into delta t is delta a, fine. And if you multiply mass both sides and divide by delta t, let me write it again in the next page, I'm running out of space. So this is delta a is half of magnitude of r into delta, r into r, what was that? r cross v into delta t, r cross v into delta t, all right. Now, let us multiply mass both sides, so you'll get m into delta a will be equal to half, now you can take m inside, so r cross m into v times delta t, all right. So from here, from here you'll get delta a divided by delta t to be equal to what? What is this? Tell me, what is this? r cross m into v, magnitude of that is what? It is magnitude of the angular momentum, right, that is angular momentum magnitude, so magnitude of angular momentum into delta t, delta t is open, left hand side man, okay, so that divided by m, all right. So you can see that rate of sweeping of the area as in d a by d t, if delta t tends to 0, this you can write it as d a by d t is equal to magnitude of angular momentum divided by 2 m, fine, this thing, all right. So clearly you can see that the right hand side is a constant, right, because angular momentum will be a constant and mass is anyway constant, so because of that d a by d t is also constant. So equal area will be swept in equal intervals of time, because rate of sweeping of area is a constant, fine, so that this you can treat as if it is a derivation of Kepler's first law.