 Hi, I'm Zor. Welcome to Unisor Education. This lecture is about gravitation. We are talking about different forces in the universe, and this is just one of the forces. And we're going to learn what it is and how it works, etc. So, this lecture is part of the course dedicated to presentation of classical physics for teenagers, like high school students mostly. It's presented on Unisor.com website, and I do suggest you to go to this website to listen to this and other lectures, and there are other courses actually, like for instance mathematics for teenagers and high school students on the same website. It's all free, so I do recommend you to go from the website because it gives you a complete picture, not just this lecture which you can find on YouTube. It's basically a complete course with hierarchy of topics, etc. Alright, and it's free by the way. Okay, gravitation. What do we know about gravitation? Well, it exists. We all feel it, right? We are standing on this earth because of the gravitation. Now, it looks like, I think everybody knows the word, gravitation, gravity, etc., which is the movie about this. However, if you ask any scientist what is the gravitation, why it exists, they will have problems actually answering that question. In some way, it's similar to what we know about mathematics, for instance. We can prove certain theorems based on more basic qualities, and then eventually we go down to the axioms, which we just accept without any proof. Well, in physics there is a similarity. We accept this world as it is created or it exists, we don't really know, and our purpose is to learn the properties of this world, to properly use them to our advantage. And we don't really know about how it was really created, I mean, in all the details. In a way, it's similar to playing a computer game. To play the computer game, you really need to know the rules of the game and the controls which you have. You don't really have to go to the underlying software which basically drives the whole game. I mean, most people don't go, except specialists who are developing this software. Same thing in physics. We can learn the properties of gravitation, and we don't really go deep enough to understand why it exists as it exists. I mean, we would like to, but we just can't. We don't really know why it exists. We don't know who created it, why created it. Well, it exists, and we use it. Now, what's the most important property, which is kind of what gravitation actually is about? Matter attracts matter. So, if you have two objects in this world, they have certain attraction to each other. Again, why we don't know, but this is the fact. So, if you take one object and another object, and there is nothing else in the universe except these two objects, they will attract each other and they will tend to come together. If you have only one object, obviously there is nothing to attract it and it will just continue its inertial movement with a constant velocity in a constant direction indefinitely. But if you have another object, then the movement of this first one will immediately change in some way. Maybe microscopically, maybe in a big way, but it will change because there is a force of attraction which basically acts on this particular object from this and on this from this one. So, this force of mutual attraction basically conforms to the third Newton's law, the action and reaction. So, if object A attracts object B, then object B attracts object A with exactly the same force. They are mutually attracting to each other. Now, the Earth and a human being on the surface of the Earth are mutually attracted to each other. Now, why we don't really fall down to the center of the Earth? Well, because there is a floor or ground or whatever where we stand and it has a reaction force which keeps us on this level. Well, without the reaction force, if there is a hole right through the center of the Earth, we would fall down, obviously. Okay, next. Next is, in this course, we usually consider point objects. So, the objects which have geometrical dimensions of zero, it's a point and they have certain mass and they can move with certain velocity and there are forces which are acting on these points exactly into the point where this object is. Now, if you have a more complicated situation, let's say the Earth is not a point, right? It's a sphere filled with some matter, right? But to be precise about gravitation between this object and any other object which has a non-zero dimension, well, their mutual attraction, the gravitation between them is a combination of gravitation of all different little pieces which can be considered as a point. So, it needs certain integration using the language of mathematics. But most likely, we will not be dealing with these particular cases. In most cases, we will deal with point objects and if we have something like an Earth, we will always assume that it's a point, the geometrical position of this point is at the center and all the mass of the Earth is concentrated in this particular point. And if we are on the surface of the Earth, it means we are on certain distance from the point, from the center of the Earth. And that's basically how we can address the gravitation between the Earth and the object on its surface. So, we will always try to reduce our problem into the problem related to the point objects just because it's simpler. In most of the cases, in certain approximate way, it corresponds to the true calculations. Okay, now, so we were talking about attraction. So, attraction is a force, right? So, if it's a force, we can measure it by how it acts onto moving objects. Let's assume that you have one particular object, let's call it a probe, a probe object A, which is going along the x-axis. Now, at certain moment of time, zero, it passes the origin of coordinate and goes to the positive direction of the x-axis with certain constant velocities. There are no forces which are acting upon this object. Now, at certain later moment, when our object is somewhere here, at origin of coordinate, I have another object B. Now, as soon as it appears, there is a mutual attraction between them, which means that inertial movement of point A should really start slowing down. So, deceleration. So, that's how gravity works. So, as soon as this object appears, this starts decelerating because there is a force which acts on this object A in this direction. Well, if this is fixed, let's say, somehow we fixed the position of this object. So, we have force which is acting into this direction, but there is also some kind of a reaction since we are fixing this object at point zero, so it stands still. But this one starts slowing down because its inertial movement will be superposed with the force which is acting towards B, its attraction, right? So, we can always measure the force by measuring, let's say, the deceleration. If I know mass of the object A, I can measure deceleration and that actually means that we can measure the gravitational force. Okay, great. So, the force is, the gravitation is a force and we can measure it. That's good. Now, what does it depend upon? Well, now we have to resort to experiments. The first experiment which is very easy shows that the further object A is from the fixed object B, the weaker the gravitation will be. Now, we can measure it, right? So, we can measure that the gravity at point somewhere there is smaller than the gravity force at this particular point. More than that, purely experimentally people basically came up with a very important numerical expression how the force depends on the distance. Well, apparently they came to the opinion that the force is inversely proportional to square of the distance. So, if R is a distance between these objects, so there is something, some coefficient obviously, some factor. But anyway, if the distance doubles, then the force is weakening by the factor of 4. If the distance triples, the weakening is by the factor of 9. So, this is experimentally established fact. We will talk a little later about maybe there are some logic behind it, but that's beside the point right now. So, dependency of the distance is very measurable and it can be actually very precisely measured. Now, obviously it also depends on what kind of objects we have here. If this is a more massive object, we will again experimentally determine that the force is stronger. So, the gravity of the bigger object in some way is greater than the gravity of the smaller objects. Not only in geometrical sense, but in some sense. So, there is something which is a gravitational property of the object. And let's not talk about what kind of property this is, but it exists, right? So, there is a gravitational property of the object which basically tells what kind of force it exhorts, right? So, what is this gravitational property? By the way, let me remind you that there is another property, there is inertial property of the object. Now, you remember that there is something which we call inertial mass or just mass, which is kind of a responsiveness to the force. So, the bigger the mass is, the less responsive it is to the force which acts upon this, right? So, there is a concept which is similar. So, the gravitation is some kind of property of the object and we have to be able to quantitatively establish it, right? Now, how can we do this? First, I would like to say is that if I will put some object, let's call it B, and measure the acceleration at certain distance, I will have some result. Now, if I will have two objects exactly identical, let's say B1 and B2 in the same place, then acceleration which this, well, deceleration, this object will experience will be doubled. So, again, if I double the object, my force will be also doubled. So, the force is not just exists as a result of the existence of the object. This gravitational force is additive, which means if I have two identical objects, the force which they exhort is twice as big. This is a very good observation because it allows us to measure the gravitational property of any object. How can we do it? Very simply. And by the way, it reminds in some way how we measured the mass, the inertial mass. So, let's take some object and call it a unit of gravitation and use it as a probe, as an A. And I also put it into this position and I put it on the distance, let's say, of unit. So, it's a unit of gravitation which is arbitrarily chosen, just a piece of metal, whatever it is, doesn't really matter. The same way, by the way, as we chose the unit for inertial mass. So, I choose a unit of gravitational mass. I use it in both cases. I put it on a distance of one meter and I measure the force. Now, since the force is proportional to this particular gravitational mass, I can say that any other object for which gravitation at this point is exactly the same will have this gravitational mass of one unit mass. Now, if I have another object which is exhort certain gravitational force and I can measure it, and this gravitational force relatively to the gravitational force which I measured as a unit, let's say it's 3.5 times greater, that means that the gravitational mass of this new object is 3.5 in the units of my measurement. So, that's how I can measure my gravitational mass based on how my gravitational force actually shows me. So, the bigger the gravitational force, let's say it's bigger than the unit force in a certain number of times. Exactly in the same number of times I'm saying that the gravitational mass of that object is bigger than the unit. So, this is how I measure it. Now, here is another very interesting observation. Experimentally, it was established that two objects of the same inertial mass or something which we call just mass. At least before we address the gravity, we use the term mass which meant inertial mass. So, if two objects have exactly the same inertial mass, they have exactly the same gravitational mass. That's an interesting thing because it's completely two different properties of the object. One property is inertial mass. This is a responsiveness to certain forces which are acting upon this particular object. So, the greater the mass, the more inertia it has, the less responsive they are to the forces. And apparently, exactly the same object was exactly the same inertial mass exhort exactly the same gravitational force on whatever the distance is. However strange it is, it's the fact. As many other things in physics which we just accept without basically much thinking about it. I mean, I'm sure physicists will probably elaborate this significantly further. Contemporary physics is a very complicated thing. And there is a theory for everything. And probably there is a theory for this as well which in some way explains why inertial mass is so much coincidental with gravitational mass. But it's outside of the scope of this course. And quite frankly, I don't think I qualify to explain this in any case. So, I just accept it as is. So, two objects with the same inertial mass have the same gravitational mass. This is the fact. Now, inertial mass is also additive, right? If you have two different objects, the mass of the combined object is some of these masses. So is gravitational mass. So it's completely proportional. So if we are having this particular situation that identity of the inertial mass is related with identity of the gravitational mass, there is no reason to use different units of measurement for gravitational mass. It makes sense to use exactly the same grams, kilograms or whatever you're using for gravitational mass as we are using for inertial mass. So, basically, we can say that inertial and gravitational mass are equal. Now, they are equal not because they are equal somewhere in, I don't know, they are equal because we have chosen to measure it in exactly the same units, just because we are based on the fact that the identical inertial mass causes identical gravitational mass. So that's why we can do it, basically. Okay, next. I've covered this. Okay, now we are about to explain what exactly is the formula for gravitational force. Now, we have chosen as a unit of gravitational mass exactly the same unit we are using for inertial mass, which is one kilogram in the system internationale, right? So, what we have chosen, we have chosen one kilogram here, one kilogram there, one meter in between, and we can basically measure the force between them, right? Okay, and the force is measured in new tones. So, we have one gravitational mass, another gravitational mass, we have a distance, and that's why we have to have the force. Now, let's think about the formula. Now, we know that the force is proportional to the gravitational mass of B, right? When we are talking about, let's say, a doubling gravitational mass of this, we will have double the force in this particular case. Exactly the same thing, since gravity is directed in both directions, this to this and this to that. If I will double this guy and leave this alone, it will be exactly the same result. The gravity will be twice as big. So, the gravitational force is proportional to mass of this and mass of that. Good. So, we can say that gravitational force is proportional to M1 times M2. That's what it means. And I also know that it's inversely proportional to the square of the distance between them, which we will discuss separately, which basically implies that the formula must be something like this. And this is proportionality. Well, that's basically it, so I can write this formula. When G is some kind of a constant. And that's exactly how all the measurements to a certain precision, obviously, show how it works. So, it's proportional to one mass, gravitational mass, which we have chosen to use exactly the same units as inertial mass. So, this is basically the same thing as inertial mass. And this is gravitational mass of another object. This is the distance between them. So, if I will put one and one, one kilogram and one kilogram and one meter, I will have certain force in newtons, which gives me the value for this constant. And that's exactly how it was done and it was measured. And the constant is 6.674 times 10 to the minus one. Now, what's the dimension of this constant? We have to have newtons, right? So, it's newtons. We have to divide it by kilograms square, right, and multiply by meters square. And that's how we will get newtons, right? Multiply by a kilogram and kilogram will cancel this out, divided by r square will cancel this out, and we will have newtons remaining. So, this is the dimension of this particular constant. Okay. So, this is the foregone. And now, we basically can calculate knowing this constant and knowing inertial masses, which we have defined as being identical to gravitational masses. And knowing the distance between the objects, we can always find the force of gravity between them. So, this one acts on this and the gravity force is this direction, and this one acts on this and the gravity force which pulls this goes into this direction. Now, let's talk about r square. I mean, you see, I think it's intuitively obvious that if I will put double object here, then the force will be doubled. It's kind of natural. You would expect it. It's the same thing as with inertial mass or many other things. If you have, let's say, an area, if area of some kind of a triangle is whatever it is, then the area of two triangles is double, basically, right? So, again, if you have some object with some gravitational mass, it has some force and the two objects of this will have double the force. So, the fact that this is proportional to gravitational masses is no surprise. This is maybe a little bit strange. Maybe it's inversely proportional to a distance, not the square of a distance. I mean, y square of a distance. And here is how I can probably explain it. It's not a proof, obviously, of this. And it's not really a theory. It's a reasonable explanation. Think about it this way. You can model the forces of gravity as some kind of an octopus with tentacles or whatever else you can choose as an animal or whatever it is. Anyway, it's something inside, which is the object. And it has tentacles to all the different directions in three-dimensional world, right? Now, obviously, if you have a certain fixed number of tentacles, then the force of attraction, well, let's say every tentacle has some kind of a suction at the end. So, if you have certain other object, the more tentacles will suck into this, the stronger will be attraction, right? Now, if you have this object close to this one, you will have more tentacles than, let's say, the same object. You see, only one tentacles hits here. Here I have, in this object, I have maybe a certain number of tentacles. And the further the less tentacles actually goes. So, what is the law here? Well, very simply, the number of tentacles which are going into three-dimensional world are basically covering a sphere, right? So, the area of the sphere is very, very important. The number of tentacles is, let's say, n. The area of the sphere is whatever it is. And if you divide n by this area of the sphere, you will get a density of tentacles, so to speak, right? So, the greater the density of the tentacles per, let's say, square meter, the stronger will be attraction, right? Now, what is the area of the sphere? Well, area of the sphere is 4 pi r square, where r is radius. So, the area of the sphere is increasing proportionally to square of the radius. Radius is the distance, basically, between source of the gravity and the probe object. So, that's why the further you are, the greater area these n-tentacles and this fixed, that's the gravitational mass of this particular object, the less density you will get per square meter or something of this sphere. So, that's why it's reverse proportional. So, again, as I was saying, it's just some kind of a reasonable explanation. These tentacles, on the further distance, they are actually spreading into greater area and that's why you have this weakening of the force and the greater the area and the area is growing as this, obviously, the weaker the density of these tentacles and, therefore, the gravitational force. Well, that's basically all I wanted to cover, including this little octopus theory, which tries to explain this particular thing. All right, so that's it. That's a classical theory of gravitation, which was presented by more than one. It's attributed to Newton in 17th century, but there are some other people who claim the fame for this, including the formula. So, I don't want to say this is invented by Newton or anybody else, but it does have the name of the Newton on it. It's called the law of universal gravitation and it's attributed to Newton because, well, he published it and there are some other publishing works, etc. Anyway, it's just called this way, Newton's law of gravitation, the universal gravitation. All right, that's it. Thank you very much. And I do recommend you to read all these notes to this lecture on theunisor.com. It's like a textbook and it probably will clarify certain issues which you might miss maybe during this lecture. Thanks and good luck.