 Hi, I'm Zor. Welcome to a new Zor education. I would like to spend some time today, and how should I say it? Theoretically, proof, but this is not really a proof in the mathematical sense. It's a proof in the physical sense. Let's put it this way. We will conduct some thought experiment and see how this experiment leads us to the law of conservation of angular momentum. This lecture is part of the course called Physics for Teens, presented on Unisor.com. On the same side, you will find Math for Teens, which is basically a prerequisite for this course. It's a relatively complete course of mathematics for high school, maybe a little bit higher than that. I do suggest you to watch this lecture through this website, because for every lecture, the website provides complete notes, details in a textual format, like a textbook, basically. The site is completely free, no advertisement, etc. Let's talk about a thought experiment, which not maybe proof, but very strongly confirms the law of conservation of angular momentum, which I actually touched during the previous lecture. Let's consider this is our thought experiment. Now, let's completely abstract out other external forces, like gravitation or air resistance or anything else. We consider this experiment in the space somewhere far from gravitational fields and anything else. Let's consider that we are holding two threads in our hand here. This is the center of rotation. Now, on a shorter thread, we have an object of, let's say, an SM, and it rotates with some angular velocity omega-1. Now, we also have in our hands a longer thread, which is basically also connected to the same object, but it's longer, so this is R and this is R. R is greater than lowercase R. The lowercase R is stretched to full extent, and the object is rotating, and this one is just hanging, basically, the longer one. Now, at this particular point, let me put this point on a 12 o'clock, at this particular point, what we do is, this is our object, what we do at this particular point, we cut or let go the shorter thread and see what happens. Well, the object was rotating around the circle, and at this moment, the connection was basically cut off, and this connection is the only force, basically. The tension on this thread was the only force which kept our object on its trajectory, centripetal force, by the way, it's called. Now, so what happens at this particular point? Well, as soon as we cut this, there is no force which keeps our object on its circular trajectory, which means that it will go on a tangential line towards infinity. However, we do have a longer thread, which is also connected to this object, and this longer thread will not let this object to fly to infinity. It will stop it whenever this distance would be equal to r. So, this is my new position of my object. Now, this is the length of the longer thread, and we don't really keep this one, this shorter thread, it's just hanging somewhere. Oh, you don't really need it. So, what happened at this point? Well, intuitively we understand that at this point, since we stopped the object short, something should really force it to go in the circular orbit of the radius r, right? Which is true. Well, let's consider how this is happening in more details. Now, the velocity of this particular object v is equal to lowercase r times its angular velocity, right? So, this is my velocity. Well, any vector I can represent as a sum of two vectors, right? So, one vector would be radial, and another vector would be tangential. So, one is along this radius, another is perpendicular to this radius. So, this v as a vector is equal to sum of two vectors. Now, let's see what happens when this tension of this thread starts actually playing its role. So, at this moment of time when this object flying along the tangential line to a smaller circle, whenever it reaches the point which is on the distance capital R from the center, its speed will be exactly the same as before. Omega 1, r times omega 1. Now, let's consider this is angle phi. Well, this is angle phi, that means that this and this also are angle phi, right? Am I right? Yes, I'm right. Because this is perpendicular to this, and this is perpendicular to this. Same thing here. So, this is phi, and this is phi. And this is, by the way, phi over 2 minus phi, because it's a right triangle. And this is also phi over 2 minus phi. Alright. So, what happens with these two vectors? Well, the tension of this thread will definitely affect this particular component of the speed vector, velocity vector B. It will stop, right? So, this radial component of the velocity will be stopped by the tension. Now, if the tension is T, then we consider that the thread is very, very little stretchable, almost non-stretchable. So, during a very short interval of delta T, this would be the impulse, which will actually be equal to mass times this particular radial component of the speed, right? Because in the beginning, this moment of inertia, this particular object in this direction is equal to m times Vr radial component. And at the end of deceleration, which is a very, very short deceleration, because the stretch, this particular thread is almost unstretchable, right? So, during a very short period of time, this will be nullified, right? So, the impulse of this force will nullify this. That's why the whole impulse is equal to basically change of the speed, right? And the change of the speed is from m Vr to 0, right? At the same time, this tension would not affect this component, the tangential component of the initial vector of velocity, right? Now, so what actually happens is that the shorter this period of time, the closer moment of nullifying the radial component of the speed would be to this particular point. And in some ideal situation, and we are talking about a thought experiment, which is ideal, and the thread is completely unstretchable, this moment of time actually be infinitely small, and deceleration along this direction would be infinitely large, right? Because from certain constant, which is equal to Vr, we are reducing speed to 0 practically in an infinitesimal amount of time, right? So, that's why acceleration, which is speed divided by time, would be almost infinite, right? So, in the ideal situation, and again, this is a thought experiment, we can consider that basically it stops exactly, this movement along this direction will be stopped exactly at the moment when our body has reached this point. So, there is no stretching. Because if there is a stretching, then we will probably lose some energy, etc., etc. The situation would not be ideal, but in the ideal situation there is no stretching. Now, all we have to do right now is to see that since this angle is 5, and that means that this component will be completely 0 down after our thread stops this movement. So, this component, this component, radial component, will be brought to 0 immediately, instantaneously, and this component will not be changed, because this tension force doesn't really affect the movement along this direction. So, vt, the tangential component, actually is equal to v times cosine, which is equal to... Now, what is the cosine? Well, look at this triangle. Obviously, this is lowercase r. Obviously, cosine phi is equal to r divided by r, right? So, we can see this is r divided by r. Well, basically, that's almost it, because what is our tangential speed if our object is circulating on the radius r? Obviously, this tangential speed is equal to r times omega 2, where omega 2 is the angular velocity of rotation on a bigger circle, right? And v, in turn, is equal to r omega 1. And the cosine phi is equal to r divided by r, right? We have already established that. Well, let's just substitute everything in this formula. What do we have? We have that vt, which is r omega 2 is equal to v, and v is this, r omega 1 times r over r. Almost finished. r square omega 1 is equal to r square omega 2, right? If we will multiply by r. So, our angular speeds are obviously changing, and if radius has increased, then the angular velocity must decrease. To preserve this particular equality, right? And not just decreased, decreased inversely to radius square. Now, this is almost a law of conservation of angular momentum, because if I will multiply it by m, by mass, I will have this. Now, what is this? This is the moment of angular moment of inertia. A1, I1, omega 1 equals I2, omega 1, omega 2. So, this is angular moment of inertia. This is angular velocity, and their product is called angular momentum of rotation. And this is a proof that angular momentum of rotation is preserved. Okay, let me add a couple of more words. In the description to this lecture on Unisor.com, I will try to find this link to a small video, which discusses some kind of experimental proof of this particular law. That our angular velocity is inversely proportional to square of the radius. So, there was an experiment, which if I will find this video, you will see it yourself. But basically, it's something like this. You consider a tube or something, then a thread. And on a thread, you have some kind of an object. And let's consider its rotating at certain radius and at certain angular velocity. Now, and you measure basically it. What's the angular velocity? Just physically measure this velocity. Then if you will pull down this thread and let's say, shortening by the radius by half. In theory, you should have increase of the rotation by the factor of 4, right? The radius was reduced by 2, so our angular velocity should be by 2 square, which is 4. Now, what's interesting in this experiment is that if the person who conducts this experiment moves it slowly, he does not have this ratio of 4 times increase of the angular velocity. But if he moves it fast, he does have this result. So, this is a very interesting consideration and it's definitely a subject to think about. Because if you are moving it slowly, it doesn't really jump from one circular orbit to another. It actually moves along a spiral. And the spiral is, you know, it's a pretty long spiral if you are moving it slowly, right? And somewhere during this spiral movement, you cannot really make this picture which I did before. Because the forces will not be directed exactly perpendicularly to the trajectory. So, it's not easy to understand and to tell you the truth, myself, I don't have all the details. But since it's really obvious that experiment results depend on the speed, this spiral trajectory, moving along the spiral trajectory, is supposed to waste a certain amount of energy and that's why the final number of rotations per unit of time, the angular speed, would be less. But if you do it fast, then you will have probably more or less as close to ideal experiment as I had explained before. Well, that's it. I do suggest you to read the documentation, the description of this lecture on the website. And as I said, I will try to find actually this website and point to you. And I hope that would convince you that angular momentum is conserved in ideal situations. But if situation is not really ideal, then you might expect actually that certain deviations obviously would happen. Primarily deviations to a slower number of, slower value of the angular speed if you reduce the radius or maybe greater if you will increase it by certain factor. So the spiral rather than jump from one circular orbit to another is very important. That's where we are losing our idealness of the situation. And that's why you should not really expect this law to be really physically observable. Another situation is if you are just rotating something on the thread and then you will put a finger and the thread will go around the finger. Same thing. It will go along the spiral and the width of the finger actually changes the direction of the forces in this case. So again, I do not go through all the details and there are some details, of course, it can be done. But it's probably kind of too involved. Let's just concentrate on the ideal situation and in ideal situation experiment with two threads, which by the way was recommended to me by my cousin who I discussed certain things with. That experiment actually, this ideal experiment, thought experiment, proves. And you shouldn't have any reservations against the thought experiment and its ideals because Albert Einstein basically has come up with the special theory of relativity using only thought experiment. So everything was in his mind basically. He did not conduct any real experiments. And the real experimental confirmation came much later. All right. So that's it for today. Thank you very much and good luck.