Okay. So today we started talking about the why we have days and nights every day, and we have four seasons. And we wanted to understand the four seasons based on the revolution of the earth around the sun. But then we came to the point that we couldn't quite understand how the orbiting actually works. and then came back to this idea that we have to think about this elliptical orbits that we are following around the sun, not like complete circular orbits but elliptical. So unless we understand why we follow this elliptical orbits, we haven't quite completed the story. Right? So, how do we understand that individual planets move about the sun in an elliptical orbit instead of a circular orbit. And that answer was given by Newton. So when he talked about this universal theory of gravitation, then he came up with this formula that the force of gravity goes with so called inverse square law. The farther you go, the gravity becomes weaker. If you go twice as far, then gravitational force becomes four times weaker. So that's what this equation is telling you, and this equation turns out to be the key. And as the legend goes, Newton discovered this law of universal gravitation by watching an, an apple falling from the tree, and then realized that exactly the same force is also letting the moon revolve around the Earth. And sort of the inspiration came from this kind of thinking. So if you actually throw a planet from the tree down, it ju, just goes straights down. But if you actually start about, giving a little thrust to the apple, then you would fly for all the distance, but eventually falls on the ground. But once you know that earth is round, if you actually put enough thrust, then you might actually go pretty far along before it actually falls down on the earth. And if you actually put you know, really sufficient thrust that way, then it might actually go around the Earth once. And that's exactly what the moon does. So, depending on this the initial thrust, you may find that something just falls, or something can go around the earth based on exactly the same idea of this universal gravitation. So, that's the way we understand that things can actually about an object, based on the same idea that something is being pulled by a mass due to the univ, universal gravitation. I learned that this, this legend that Newton was observing an apple falling from the tree was just a legend, that's probably not a fact. But apparently Newton did have an apple tree, and at the University of Tokyo we have a descendant of Newton's tree in our botanical garden. So if you happen to have a chance to visit Tokyo, don't miss out on this descendant of Newton's apple tree. But this law of universal gravitation wasn't quite enough, because that only tells you how much an object is being pulled. But what you also need to know is that how the object is influenced by the pull. And that is given in terms of this another famous equation called F equals ma. Which you might have actually studied in high school physics class. So, in this equation, m stands for mass. And, and this idea is called inertial mass. Namely that if you are given a force, and if the mass is bigger, you get less acceleration. So a stands for acceleration. Namely, that the mass is a measure of how difficult it is to change the motion of an object. And that I think is plainly clear just by comparing these two objects. So, suppose you provide the same push on a tank and on a little kid on this tricycle. And with the same push, you can hardly move this tank at all. Right? Because it's so massive. But with the same little push that little poor kid may fall down. So you can really change the, the course of his motion. So, which one is easier to move is really determined by how massive the object is. Heavier it is, it is very difficult to change the course of the motion. So, m in F equal ma really stands for how difficult it is to change the motion of an object. So, as you see in this animation the sun is not at the center, because the orbit is an ellipse, not a complete circle. And as the planet moves around the Sun, it gets pulled more strongly when it's closer, but gets pulled less forcefully when it's farther away, so that changes the course of motion at the same time. But the most important thing about this is that by combining the two equations we talked about. F equals ma, that tells you how the motion gets affected by the pull. And how big the pull is, and that's the Universal Theory of Gravitation. If you put these two equations together, you can fully explain this elliptical orbit of individual planets. And to work that out with Algebra, it's a little complicated, I'm not going to actually show that, precisely my lecture. But you're welcome to solve one of the problem sets where you can verify that this elliptical motion really comes out from these two equations. So, by combining these two equations, F equals ma and F goes like the GM over r squared, that's the inverse square law of gravity, then you can fully understand these elliptical motions of individual planets. And therefore, you'll fully understand the origin of four seasons. So that's the way a physical theory is supposed to give you a deep understanding of what's going on in our universe. And it actually, happens at incredible speed. If you measure how fast the earth revolves around the sun, it's at the speed of 30 kilometers per second. You might feel dizzy about this. It's incredibly fast, but nonetheless we're not throwing away from the earth, because again we're bound tightly by the gravity of the earth towards the earth, and we're not we don't have to worry about this, its fallingoutside from, from the planet. And the individual planets are also moving you know, as fast. And the closer it is, like Mercury, it's moving at a much faster speed. The farther away it is, it's moving at the slower speed, because the, the pull of the gravity is weaker. And they fall on a very simple curve as you can see. So, as a function from distance, the orbital speed goes down. And it goes like one over square root of the distance. So, that's something that you can verify right away by knowing these equations that we talked about. So, we make an approximation. And we make a lot of approximate methods in, in physical theories. Let's assume that it's not quite ellipse, let's say it's a complete circle instead. It's a pretty good approximation for this purpose. So, if it is a completely circular orbit, then the distance from the sun doesn't change. So r is just a number for each planet. On the other hand, when things are moving about, then the pull by the gravity should be balanced by so called centrifugal force that is trying to pull us away from the sun. And there's a formula for that too. So, mv squared over r. Where v is the velocity of an object. M again, is the mass of the planet here. So, this pull by the sun due to gravity, and the centrifugal force that is trying to pull us away from the sun should balance against each other. So we set them equal. And here's the pattern, read the power of having an equation. By setting them equal to each other, you can solve for the velocity of the planet. And that goes like this. And this big M is the mass of the sun, that's common to all the planets. We experience the same force of gravity due to the same sun. G is Newton's constant, that tells you how strong the gravitational force is. So these two numbers are fixed for all the same planets in the solar system. The only difference is this r, distance from the sun. So that gives you a very precise prediction, that the velocity of an individual planet goes down like one over square root of the distance from the sun. And that's exactly what this data show, they follow on the same smooth curve. That velocity goes down like one over square root of the distance. So outer planets go much more slowly, inner planets goes much faster. And we're, we can understand all of these planets on the same equation, same curve, on a single footing. That's the power of this Newtonian gravity. So a question to you then, is that is there anything else we can do about this theory of gravity, and, and you might start asking some simple questions that's the subject of our next clip. Now, why do all these objects follow the same idea of Newtonian gravity when their masses are so different from each other. Mercury is pretty light, Jupiter is humungous, they differ a lot in their masses, but they still seem to follow the same curve. Do you have any idea what's going on here? So we'll talk about that next.