You Did University Level Physics

[00:00:03]I'm bored. I'm bored. I've done it all.

[00:00:06]I've done everything. I've solved physics, created every single chemical reaction known and unknown to men, cured every disease, solved every problem, and even gave you a bottle of Jack Daniels.

[00:00:19]Sentience.

[00:00:21]What else is there for me to do? All the mountains have been conquered. All the valleys discovered.

[00:00:28]Ah, it's lonely at the top.

[00:00:34]Teach others, you say. Well, that's tempting. Although I can't just give them all the answers to the universe right away. No, no, no. If I do so, they will stagnate. They will never reach for the greater heights. They are capable of achieving my greatness. But I need to guide them. We need to start small. We need to start with something simple, something graspable, something to roll them forward, to show them the way, to show them what they can achieve.

[00:01:01]That's a great idea. Jackie, initiate the carousel protocol.

[00:01:16]And so there we go. That is it. The big carousel exercise. Now you all should be familiar with it by now. And oh, what's that? Oh, statistically, you haven't actually looked at the document. So, we need to do a whole segment in which I just explain to you what the document is.

[00:01:35]You do realize that you're wasting the website is right there. You could have just gone and check it out. You do.

[00:01:40]Fine. Fine. We'll go over it. Never mind everyone. Never mind. We first need to go over the document. Just give us a second. But first, let's start with our carousel. A carousel of a specific size which will be able to move on two specific axes. The global rotation axis and the seat displacement axis. Now, intuitively from our real life observations, we do already know that once our carousel gets to spinning, the seats will be pushed away from the center. But why does that happen? In order to answer that question, let's first start with the first exercise of the paper, which reads as follows. Above is a schematic drawing of the carriage on the carousel that includes a vertical main axis of rotation, a horizontal bar of length R, and a diagonal non-elastic rope of length L at angle theta that connects the carriage of mass m. The carriage is a length of r away from the main axis. Include all the applicable forces on the carriage in the right directions. gravity pointing downwards, centrifugal force outwards, where omega is the angular velocity of the rotation around the main axis, and not forget the tension of the rope t in the direction of the rope.

[00:02:52]Now, whilst technically an exercise, it's really more of a setup, which is why I'm giving you the answer right away. But the following exercises won't be quite as easy. So, for example, exercise number two. Find a guttric expression of R in terms of R, L, and theta. Write the force balance in the vertical direction to express T in terms of G and theta. Do the same for the horizontal direction and substitute your expression for R and T into this.

[00:03:18]Now, these are exactly the kinds of exercises that throw people off.

[00:03:21]Econometric expression, force balance, this equation, but you can't be distracted by physicists being obscure pricks. I promise it's simpler than it looks. But the point is, we should be left with this equation, which concludes our Newtonian mechanics segment. Now it's time to forget about forces and get lagranjon. If you recall the lagranjun video, we don't care about forces here.

[00:03:44]We care about energies, the kinetic and potential which you need to find respectively in exercises three and four. And then exercise 5 requires us to combine it into the equation for a lagrion which we then solve using the oil lag granch for exercise 6. Then again for exercise 7. And then finally we need to interpret our result. And okay, I know it might have seemed like a lot, but I promise it's a lot more reasonable and intuitive than it first seems. And once we get to solving, you'll realize just how natural all of these answers will be. But first, before we get there, we need to get into the right mindset to do physics.

[00:04:30]Now, here's the thing. I'm going to be frankly honest with you. In this segment, I was supposed to explain Newtonian mechanics. That's because I need to give you proper context for the entire exercise to make sense. And I wanted to do it using the whiteboard, but as you can see, it's taken. And actually, it would be a real shame to erase this because this is something I've been working on for quite a while.

[00:04:52]You see, in physics and quantum mechanics, especially, the concept of spin is widely regarded as one of the most confusing and difficult to explain concepts. But I think I've actually cracked it. I think that I do have a method to explain it in a way. So that >> mate, what the >> Yeah, but like this was the >> godamn.

[00:05:20]>> Since the board has been wiped, we might as well proceed. How do forces work?

[00:05:26]Well, the entire stick of the Newtonian formalism lies in the relation of forces and acceleration. So, if we were to just ignore the mass for a second, force is just equal to the acceleration.

[00:05:37]Basically saying that if we're blind to how massive our objects are, if you just push it, that's how it moves. This does seem like quite an obvious fact, but most people tend to really quickly forget the consequences of that. For example, gravity. So as I throw things around the gravity is pushing them down.

[00:05:56]Now remember this is the force of gravity and force is equal to acceleration not velocity which means that me throwing the apple up is acting on the d force and it's me accelerating the apple upwards. Whereas gravity is accelerating it downwards. Accelerating not setting the velocity too. Okay. The velocity is moving up or moving down but not being set to by a force. And what I mean by that is basically that if we imagine that we were in an empty universe, then with this apple, it would just keep going forever, never stopping.

[00:06:30]In the empty universe, there would be no gravity. So the apple wouldn't accelerate. It would keep moving at a constant pace forever.

[00:06:43]But last, our universe is filled. And as such, the force of gravity will always be acting on the apple, always pushing it down back into my hands. Meaning that if we were to draw our apple standing on the table, how would we draw all the forces? That's simple. We just need to draw the force of gravity. So, let's draw it. There we go. The total force acting on the apple is just equal to the force of gravity, right? Well, not quite. And I'll be honest, it's a really easy mistake to make. It's a mistake that a lot of people make because we tend to conceptualize the force of gravity different from other forces. You see, the force of gravity we see as very active, right? We see as everpresent, we see it as natural part of life. Other forces, which are a tad more passive, for the lack of a better word, we don't see as forces. And I'll demonstrate what I mean. Let me take this candle right here, light it up real quick, like so.

[00:07:43]and then place it on the table. As you can see, I can drop it. The force of gravity is pushing it down. But once it meets the table, the force of gravity is still there. It's still acting on it.

[00:07:57]Which means that if the total force acting on the candle would be just the force of gravity, that would mean that the total acceleration acting on the candle would also be down. Which means that this isn't what we'd expect to see.

[00:08:11]Instead, this is what we'd expect to see. We'd expect the candle to still accelerate downwards, which I don't think I need to explain is not what happens in real life. Which is why we need the balance of the forces.

[00:08:28]If the total force is pushing down, that means our object is going down. If the total force is pushing up, that means our object is going up. If our object is standing still, however, that must mean the forces acting on it cancel out. And there we go. We found the force that the table exerts on the apple, which is basically all the matter making up the table stopping the apple from falling down. Now, this was all great and informative and all, but how does that help us? Okay, let's get back to the carousel and let's just ignore the whole rotating carousel aspect of it. You know, let's just put it to the side.

[00:09:05]Basically, our carousel is still spinning. We just spin the camera along with it. What forces would act onto our carousel on this plane? First, we have the simplest one. The force of gravity ever present pulling our seat down like so. What else is there? As we've just learned, the seat isn't constantly accelerating downwards, which means that we need another force to counterbalance the gravity. And that would actually be the tension of the rope. So, the rope holding the seat up. That should make sense. But as mentioned previously when carousel rotates faster and faster the seat goes up. So why does that happen?

[00:09:41]Well that happens thanks to the centrifugal force which takes an equation of this form and as you can see is proportional to the mass of the seat.

[00:09:49]The radius from the origin but also the angular velocity of our carousel. And so here's where we can see why the seat moves higher and higher. But I'm getting ahead of myself. The first exercise of the paper. Yes, however many minutes into this video we are. I don't know why I'm still recording, but yeah, there we go. We finally got to do the first exercise of the paper. The first exercise of the paper asks us to draw these three forces. And so the idea is simple. Gravity points down. Centrifugal points outwards. But since on this plane the seat isn't really moving, that means our forces have to cancel out, which happens thanks to this force right here.

[00:10:29]And the way you can see that all of them cancelled each other out is by placing all the arrows at the end of other arrows. Which is because in linear algebra vector addition is isomorphic to placing these arrows at the end. And this all happens because of these really neat identities where mathematics in my physics video. Get this out of here. Exercise two. Come on. Let's go.

[00:10:56]For the second one, we'd like to be a tad more precise. We have these forces.

[00:11:00]We'd like to note down exactly how much they're worth. So, where should we start? The exercise tells us to start by finding a gonometric expression of R in terms of R, L, and theta. And um quick question. What's a gonometric expression?

[00:11:14]>> Great question, dear listener. In fact, it's a question so great that when I for the first time read's draft for this document, I had to look up what gonometric meant. And if you wouldn't mind me actually right now leaving the gray color scheme, this is actually a tangent I wanted to go on for a bit because in physics and mathematics and all sciences of all sorts, you really often find words like gonometric or scary sounding sentences, words or equations doesn't matter. And this is actually something that's really important because I've seen a lot of people spooked by it which is why I feel the need to address this. Here's the thing. When you're learning a topic in this scenario, physics, but it does apply to other fields as well. Then there are basically two ways you can study it. First of all, the specializing way. So you're trying to specialize in physics. Eventually becoming a physicist kind of like Ramco or the second one where you just find it interesting and you like to study it because you find it funky which is the more generalist way and it is what I'm basically doing. Now in the first scenario, words like gonometric you need to learn them simply because you're going to waste literal hours if you write out everything in elaborate ways instead of using the singular wall for it. So yeah, instead of gometric, you could just in this scenario say equation describing angles or whatever and that would be fine. But if you have to replace these words couple hundred times per hour then you're really wasting time. But don't worry if you are specializing in the topic like physics for example then you don't have to go out of your way to memorize them. You can just keep doing physics okay you encounter the O lranch equation the first time you're confused.

[00:12:53]You encounter it the second time you're confused. You encounter it the 300th time and you already have it in your brain written on the side of the wall and you cannot get it out. So don't worry there. You don't have to memorize them. Just keep doing physics and eventually over time you'll remember all of these scary words and equations. If however you're going with the general approach then once again don't go out of your way to memorize them. You won't be able to okay you won't remember words like guttric you'll have to constantly keep rechecking what the oil lanch says.

[00:13:25]But that's fine because you're not trying to be a physicist eventually and all of these things are things that you can easily look up. It's a lot more important that you understand how to read them and you understand what they mean rather than remember what they are.

[00:13:40]So yeah, don't be spooked. Even I have to look them up. And even though I'm not perfect in physics by any means, I'm still good enough to teach a bottle of Jack Daniels how to do quantum mechanics. So when you approach it from this side, it's really Hey, pay attention. Goodness sake. I'm explaining important stuff here. We're looking for an equation which describes this length right here in terms of this angle. This length and this length. So let's try that. We can start by drawing all the lengths. Okay. Right away we can see that we can divide our lowerase R into two portions. This one and this one. The left one is just capital R. So the total equation will be equal to R plus something. But what is that something?

[00:14:23]Well, how would we get this distance based on the angle and the length? If you've seen any of my recent videos on the topic, then thank you very much. I really hope you enjoy them. They took a lot of effort. They're part of the largest thing, and I really hope you uh learn something, but the answer should be obvious. So, toa in this scenario, meaning that this other side of R will be sin * L, giving us R + sin theta L.

[00:14:49]Great. So, we've gotten this equation.

[00:14:51]What next? Next, the excise tells us to write the force balance in the vertical direction, expressing T in terms of G and theta, as well as doing the same for the horizontal. What does that mean?

[00:15:03]Well, a neat part of vector addition is associivity, which basically states that a vector like this is just equivalent to two vectors added together. This once again follows from the extremely elegant isomeorphism that is the again. Oh, come off it. Anyway, the point here is we take our tension and divide it into the upwards tension and the sideways tension. And now you can see that the upwards tension has to be equal in magnitude to gravity whereas the sideways tension has to be equal in magnitude to the centrifugal. So we can note down all these properties as follows. But since we know what the centrifugal force is equal to and since we know what the force of gravity is equal to, then we can get both the tensions.

[00:15:47]But here's the neat part. And by the way, just as a quick side note, uh I really want to nail this point down because it's pretty important. F is equal to ma, total force equal to acceleration. Which means that if the force of tension wasn't equal to gravity or centrifugal on either of the axis. So for example, if the force of tension was stronger upwards, so the rope was elastic in some way, then the force going up plus the force of gravity would not cancel out and instead would give us a little bit left and the seat would be accelerating upwards. And uh this is why the force balance has to be there because as you can see from this animation, the seat isn't actually moving on that plane. It's moving on the other plane, but you know, rotation is not something we're doing right now. So there we go. We got both tangents. But here's the neat part.

[00:16:37]There exists another way for us to note this down. That's because we can also try the method of polar to cartisian coordinates. And if you don't know what I'm talking about, watch this video of mine. Wow, all those videos with callbacks and connected topics almost as if some smart people were in charge of all of this to make sure that all the explanations click together. Anyway, the point is if you have some distance, let's call it L and some kind of angle, let's call it theta, then this is the coordinate expressed in polar coordinates. Now this same coordinate can also be expressed in terms of cartesian that is using s and cosine.

[00:17:14]And the way you can think about it is this s and cosine is there to tell you how much length matters based on the angle. If the line is flat, it matters completely for the x coordinate, which is why it's a one. But it doesn't matter at all for the y-coordinate, which is why it's a zero. If you move it upwards, the rules reverse, and if you move it to an angle like so, you get the in between. Anyway, the point is the reason why I'm mentioning it is that if we consider the example to go with to be the tension, then we do have a value representing the magnitude of the tension kind of like the length and we do certainly have the angle at which it operates. So, we can rewrite the x and y components of the tension as follows.

[00:17:58]Now, this may seem fancy, but once again, we're just switching polar to cartisian. Why is that important? It is important because it allows us to not only write down the cartisian coordinates in terms of polar but write the polar in terms of cartesian. Since we know what this cartisian would be equal to, we can just replace them like so. And there, congratulations, we've gotten the equation list in the paper.

[00:18:28]And so there we go. Halfway point. I hope you're enjoying it so far. I hope you're not tired yet because there is still a fair bit of the paper to go through. And if you don't mind me for a second, I would like to take this time to mention this. As you can see, this whiteboard has been changing a fair bit.

[00:18:44]And that's because it actually took me around half a year to finish this video.

[00:18:48]Yeah, around half a year. And as decent as this video is, it is not a half a year's worth of effort. You know, it's decent, don't get me wrong, but half a year. And the reason why it took me half a year is because I kind of had a little bit of a side gig going, let's call it, little bit of a side project, side hustle, you know, a little thing on the side. Uh, I kind of sort of started a charity. Last episode, I have been in the Netherlands with Rem is we're going to define an angular momentum texture and this is going to be defined as >> as one does. It is a science communication charity and the idea is simple. We're going to make educational science material. We already have started doing a couple of things. First of all, we created the library which is an online repository where you can upload your resources, have other people verify them or verify resources of other people. It is great filled to the brim with scientific material. And a couple of notable papers I would like to mention is first of all the OG silver's paper if you're into the French geometry. Uh Joelle's paper is honestly really great about amateurized analysis.

[00:19:59]It's a very lovely like pure computer science but still on a pretty decently introductory level. So it's a really nice read. And then Rogues who wrote a paper. It's a really good paper. In fact, I'm referenced it in this video because it goes way more in depth about the Lrangeen than I'm doing it. And honestly, if you want to learn about Lranian formalism, read that paper.

[00:20:23]that's going to give you a lot more information than this video. Another thing we're working on still is physics material. There's going to be a lot more scientific resources on physics coming soon. And also on the Make It Community channel, uh I'm kind of hosting different people who wrote these papers, for example, Joel, for example, Silver, etc., etc., etc. We have so many more projects, so many more projects in the works that I want to finish. But I'll be honest, we need help. And if you would like to help us, great news. You can you can do it in many ways. You can either a just sign up with us, join the charity, uh just like Caper and Edu are working on the website. Remco is working on the scientific resources just like Adrian, Comey, and Zaki are helping Remco. like DDB is joining in with the website work.

[00:21:15]You too can join us and help us with all the different things. But I understand that not everyone has the time. And if you don't have the time, you'd like to support us. The answer is simple, a bit boring, but still very needed. We do need it. That is financial support. You can just donate on the website. There is donations.

[00:21:36]Would really appreciate it.

[00:21:39]Anyway, I won't take any more time of this video. That was just a quick ad about the charity. Do check it out. Link in the description. Make it.WTF or sliceofpie.eu.

[00:21:50]We even have a professional link. Now, we we kept both because well, make it WTF historic. Back to the video. Now, lranian formalism is a lot less known than Newtonian. And funny enough, this video is probably called something along the lines of UDI did university level physics. And the reason why it's called that is because this is university level physics. But first, we need to cover our basis because despite me making an hourong video on the lranian formalism, statistically you haven't seen it. So, let's start with generalized coordinates.

[00:22:24]Funny enough, this topic is what spurred this project in the first place. And that's because a carousel is a perfect example of a generalized coordinate already. You see, when you consider a seat on your carousel, you can describe the position of the seat in three dimensions like so. But this causes the movement to be extremely complicated and convoluted, especially when acceleration and deceleration gets involved. So another thing we can do is instead of describing it in terms of cartisian coordinates we can describe it in terms of spherical like this. Now what you may notice is that r stays constant which makes sense the seat isn't moving any further or closer away which means that we can just ignore the r and keep it with two angles. This for obvious reasons simplifies things a lot. We've reduced the amount of coordinates from three to two. But even better, if we look at the graph of the carousel spinning or a graph of the carousel accelerating, all of a sudden it makes a lot more sense and it's a lot more clear.

[00:23:31]What the langent formulism revolves around is the energies, the kinetic and the potential. And in order to cover them, we first need to talk about what they are. So what is energy? I feel like everyone has a decent enough intuitive basis of what energy is, but it's still useful to put it in words. So for me, the most useful definition of energy is potential for change. Basically, our universe has a certain concept of lack of change. This doesn't necessarily mean that things have to be standing still.

[00:24:01]Quite the contrary, in fact, things could be moving because the change we care about is, for example, in its velocity. An object drifting in space isn't changing. But if you want it to accelerate, change its velocity, you have to give it energy. Or in other words, there is a certain desired path that every object wants to take. This kind of path which we would consider in quotes standing still. Then any deviation from this path that would be changed that would require energy. And here's where we are. So this energy once again we can divide into two types. The potential and kinetic the two of which you can think of as respectively energy and moving and energy of standing still.

[00:24:46]So for example we can imagine a ball on a hill. If a ball is standing still at the bottom of this hill, it doesn't need energy to do so because it's not changing anything. If it is however moving, it still doesn't need energy because it's not changing its velocity.

[00:25:03]But in order for it to accelerate, it needs that energy. Which is why if we place our ball right here and let it roll down the hill, it will accelerate.

[00:25:12]So that right there is kinetic energy. A counter to kinetic energy would be potential because if we place our ball at the top of the hill, then we know that it has the potential to change simply because we know that if it were to lose its balance, it would roll down the hill. So we know that it has energy.

[00:25:29]It's just not using that energy to move.

[00:25:32]Hence potential. Even more curiously, energy cannot be created nor destroyed.

[00:25:38]Which is why it has to stay conserved.

[00:25:40]Meaning that if energy appears from somewhere, it has to disappear from somewhere else. For example, let's consider our ball rolling down the hill.

[00:25:49]In that scenario, the kinetic energy is increasing with each passing second. But that energy has to come from somewhere.

[00:25:56]So that means the potential has to be decreasing. The curious thing here being that the kinetic energy is increasing over time, but the potential is dropping in relation to the length. This could be a bit of a headache to get around on the first try. But think about it this way.

[00:26:13]If you have an accelerating car, this car's velocity isn't dependent on where it is. No matter where you place it, its velocity can be the same. What matters is how long it has been accelerating.

[00:26:25]If you could magically teleport this car over time and space, if you teleported it 3 m forward, it would have the same velocity. So teleportation over distance doesn't affect kinetic energy. But if you could teleported 3 seconds into the future without touching the distance, that would mean 3 seconds of acceleration. So kinetic energy would look more like this. Potential energy, however, is different. If you imagine a falling ball, teleporting it 3 seconds to the future would result in no change in position, thereby no change in potential. But remember, potential only depends on height. If you however teleported the ball up or down, it would be a drastic change in potential. And so that's why if we combine the two scenarios, we know that how quickly the ball is gaining kinetic in relation to time has to be lost in the potential in relation to space. the Oiler Lranch equation.

[00:27:25]Okay, so the Oiler Lrange equation which we have covered in the past a whole bunch, but might as well do it again. If you want a more rigorous approach, if you want an approach that is a little bit more exact, a little bit more careful, then uh I will in the description add in a link to for example Rox's paper uh and maybe some other resources. This is more for intuition sake because I think that the orion is something that's very intuitive. It we just need to learn how to read it. So first we start this concept of a lagrion. What is lagrion?

[00:28:00]is as we all know at this point kinetic minus the potential kinetic energy minus the potential energy. So kinetic energy would be for example movement of something like this pin. Whereas potential is just the potential for movement. There we go. As you can see it was standing still then it moved.

[00:28:17]Potential got converted into kinetic. I don't think I need to explain that. So if we define our lrangen as the kinetic minus the potential. Let's think about what this means conceptually. Let's think about what it would mean intuitively. Well intuitively it would mean that the higher the kinetic energy, the more movement there is the higher the lrangen. But the higher the potential, the lower it gets. Or in other words, we can think of the lranjen as kind of like a metric of how much of our potential energy already got converted to kinetic. I know this is a bit vague. So let's give a little bit more of a concrete example. Let's divide this into a grid and let's uh give different categories. So right here we're going to have low kinetic energy.

[00:28:56]Right here we're going to have high kinetic energy. Right here we're going to have low potential energy. Right here we're going to have high potential energy. Let's think about these scenarios. If we imagine a hill, then the low kinetic low potential would be something like a ball standing still at the bottom of a hill. That would be low kinetic low potential. If it's not moving, then it's no kinetic. If it's at the bottom of the hill, that's low relative potential. Another scenario would be high potential, which is same exact thing except the ball is standing at the top of the hill. In this scenario, it has the highest local possible potential, but it's still not moving. So, kinetic is still low. right here. However, we would have the ball roughly at the bottom of the hill, but it would be moving. You can see those lines. Those lines would be in movement.

[00:29:39]So, it is actually speeding away from this hill. It has kinetic energy. It's doing great speeding up. It's it's fantastic. And then finally, high kinetic high potential is the same exact scenario except once again at the top of the hill. So not only is it really speeding away, not only is it doing great high velocity, at the same time it also has the potential to get even more energy. That's the idea. Now the lranjen is growing this way. That is this is a scenario with the lowest possible lrion.

[00:30:09]This is the scenario with the highest possible lranion. Obviously I haven't assigned any values. So this is a bit arbitrary. But these two scenarios would have roughly the same lag rangon because in both these scenarios in this scenario there's no kinetic potential. In this scenario, the potential is countered by the kinetic. And so that's why we can think of the lranen as basically how much potential uh we have already, how poor we are in our potential in comparison to our kinetic. That's the basic idea. Now let's tackle the partial symbol. The symbol that everyone is really terrified of whenever they get into mathematics for some reason.

[00:30:43]Partial x in relation to y. What does that mean? What this effectively means is just a question of how much our x is changing explicitly in relation to y. It is kind of like a standard ordinary so might even say derivative except even simpler because this derivative has ramifications. This y could be dependent on all the things. It could be complicated in this derivative. We don't care. We only care about the explicit y.

[00:31:09]Let me give you an example of that n derivative. All right. Great. So right here I have a standard ordinary lab power supply on the side and right here we have plugged it in so that we can measure the voltage of this power supply. I can change this voltage. So for example I can reduce it to just 2 volts.

[00:31:27]And there we go. As you can see it is dropping. Now right here this is simply here to show the voltage that I have.

[00:31:33]And let's say that we would like to note down this voltage. We would not down the capital V like so. Now let's create another value value which we will denote using a capital E for electricity and let's say that this value will denote how much lightning there is in this room because you know different rooms have different amounts of lightning for example right now we have low lightning whereas right now there is actually quite a lot of lightning as you can see this is my camera >> now as you can see if I keep it at around roughly 2 volts uh there is no lighting whereas if I crank it up by 1 volt.

[00:32:11]Now there is lightning. Beautiful. Let me turn it off. And so uh what can we do with those values? Well, what we can do is we can do a partial derivative, right? And in this partial derivative we can denote the lightning in relation to voltage. And so right now we know that at 1 volt there is no lightning. So at 1 volt right if we made like a little graph and here we have voltage then at volt one there is no lightning whereas at volt two there is some lightning you can feel it better than you can see it but up to a four okay you can't really um see it all that well the camera doesn't have a fast enough shutter speed to actually view it all but you certainly could hear it with when we cranked it up to a four there was in fact even more lightning. But crucially, it only increased slightly.

[00:33:05]And even better, if we actually were to put it at 1 volt, there would still be no lightning. It can't go any lower than no lightning. So, this right here is a function E of V because how much lightning there is depends on what kind of voltage we're giving. It depends on how much voltage there is. What the derivative tells us is how high is going up. Now a derivative is a concept I've explained many times in the past but right now what I want you to do is I want you to start thinking about it conceptually intuitively not with just numbers and functions because what matters here is to think of the derivative as change because then we can get back to our original oil lranch and we can ask the question of what is the change of the lranion in relation to position and so there we go that is the oil lranch now this is not really the standard way to note down oil lranch the standard way is to do this minus this equals zero But it's equivalent.

[00:33:55]Physicists just do it for the sake of context or whatever. And anyway, I'm done with this in real life segment because I'll be honest, I know I got it from a lot of people. A lot of people tell me that these in real life segments aren't quite as interesting. So, I won't really put any more of them in the video and I'm just I'm just done with this because this is just Yeah, let's get back to the video. Now, yes, I have rushed through this a fair bit, but that's because oil lrange is something we once again discussed a lot in the past. So, for now, I think we might as well carry over to the exercises.

[00:34:26]Starting with the third exercise. Draw the possible rotations for phi and theta coordinates. Write down the corresponding equations for the kinetic energies for f and theta. Conveniently, I don't actually have to draw out the rotations for you. I've literally created this interactive carousel for that. So, here you can see all the different rotations you could wish for.

[00:34:47]The simplest kind of kinetic energy you can note down is mass velocity squared by two. But we will modify it a bit.

[00:34:54]First of all, purely notational, we can switch the V to an X dot, which does mean the same exact thing. But the reason why we'd prefer to note it down this way is because this allows us to save velocity in relation to which coordinate we're talking about. That is really useful since we have two vastly different coordinates and it makes it easy to distinguish between them. So this is the first one, the second one, velocity of the first one, velocity of the second one. But then again, these are not just velocities, they are also angular velocities, which is why we need the r to also be there. Okay, so the expansion for it is pretty simple. So like this, right? As you can see, we have two objects with different radi spinning at the same angular velocity.

[00:35:38]But since one of them has a different radius, it ends up moving at a faster velocity. Which is why we need to multiply by the radius because the larger the radius, the faster the object is actually moving. Therefore, angular velocity is different velocity as the yada yada yada. And so back to our kinetic energies, these are the two energies we have each dependent on the different coordinates.

[00:36:00]But even better, what is this small r right here? If you recall correctly, we've calculated that before. So we can just substitute giving us a better equation for the energies. Great, we've got the kinetics sorted. Next, in exercise 4, we have the potential.

[00:36:16]Fortunately for us, the only potential we're dealing with is the gravitational.

[00:36:20]So we know that it will take the form of the gravitational constant. Now, it's also dependent on the mass and the height of the object since the higher it is, the more it can fall. The excise asks us to write this potential in terms of theta. So let's think about how we could do it. First, we have the gravitational constant. Considering the constant part of it, I don't think I need to tell you it will not depend on theta. Next, we have the mass. And since we're doing sane physics, mass will also not depend on theta and will largely remain a constant. Meaning that quite logically, the only thing that can depend on theta would be the height of our object. And just as a quick tangent, if we look at the height of the seat and the theta, we can see they're clearly related. This relation can be even proven considering how if we look at it from the side, this right here is the value of our depression. So basically the inverse of height, the longer this length, the lower the height of our object. So if we set our height as the inverse of depression then it's just a matter of calculating the minus this length which is trivial because once again soa so cosine of theta is equal to depression by l and we've gotten our equation for height which we can plug into our gravitational potential using this we've already solved xis 4 and can move on to exercise 5 write down an expression for the lranjen now this one is really difficult because lran is kinetic minus potential And so we have our kinetic, we have our potential and what we need to do is take it to kinetic and minus potential.

[00:38:01]And man, physics is hard.

[00:38:06]Oh. Oh, wait. Damn it. What are you What are you doing here? Oh, the other ring sizes. Oh, they're just calculations.

[00:38:15]So, make it didn't feel like uh animating them. All right, fine. You know what? Sure, let's let's get cleaning quickly. Most of them I will not spend much time because they are just calculations. There isn't much interesting in just the process of noting them down. They're really fun. If you want to do them yourself, I would recommend you still have time before I spoil everything. Uh but watching someone else do it may not be quite as interesting. So, let's go through it quickly. First of all, we'd like to denote our Lranon.

[00:38:43]We have our Lrion.on. It's a bit of a monster, but considering how we have built it up throughout this live video, it should be fine. you should get it just fine. And in the XI6 we were specifically looking to solve the oil lounge equation. So this one but using uh here as our coordinate. So right here we have this this will be our coordinate. So we do just denote the oil range equation except with this coordinate. And this time I'm going to denote it with the equals 0 on the right. You'll see why soon. And yeah this is what we want to solve. So in order to solve it we would like to solve these two separately ideally for the sake of simplicity. And let's start with the left one. So the definition is lranjen in relation to our variable and you may notice that theta is not our variable which means that out of these three equations considering how derivatives work that is I'll note it down on the left in red as an identity derivative of any kind of function f in relation to x. If you have f as a some kind some kind of a plus b plus some kind of c its derivative will be derivative of a plus derivative of b plus derivative of c. simple as it could be summation is that simple and so if our equation doesn't have the variable we we're differentiating in relation to it is equal to zero which means we can just skip it. So this one doesn't have FE. This one does not have FE. This one looks like it has FE but it has FE dot.

[00:40:04]It has a dot there which is slightly different. But then we have another one which is what if we did the same kind of derivative but this time in relation to FE dot. This one doesn't have F dot.

[00:40:14]This one doesn't have V dot. This one does have F dot which means that this time we do actually need to note it down. There we go. And the simplest way to achieve that is to actually divide it into two and then do the product rule on them. The product rule states that if you have a * b and you take derivative of them, this effectively translates to a der a prime b plus a b prime. So derivative of the first one * second uh plus first one times derivative of second. Simple as it could be. All right, it was a fair bit brave to still try it on the same line, but the idea is simple. I took out all of the constants and shoved it into the left. Left only our variable on the right. And using this we know that's going to be derivative of the first time second derivative of something that doesn't contain the variable you're differentiating in relation to is a zero once again which means it will just be equal to and since right here we have zero this one cancels itself out it's a zero this two cancels out this two and what we're left with in the end is and so there we go that's what we get there is actually something interesting to mention about the equation we have gotten and that is this left side the lagrion in relation to the velocity of our coordinate and that's because this right here is something that we in the business call canonical momentum which you can just think of as momentum.

[00:41:26]That's it. Which means that this equation right here isn't just a random step intermediary step to getting a final solution. This right here is actually the momentum of our system as far as the FE coordinate is concerned.

[00:41:39]So this animation right here shows you this is the FE coordinate. And that right there, this equation is the momentum in that coordinate, which is pretty neat. And the reason why it's so neat is because the next step as you can see right here is to take its time derivative. The point is this equation is important because what it states is that our momentum isn't changing in relation to time. That's literally what it says. This right here part is the momentum and this right here is changing in relation to time then is zero or in other words change in relation to time of our momentum is zero. That is literal math to English translation of that equation and this is pretty neat one to keep in mind. Second sentence of xi6 what law of conservation could you relate to it? There we go. Conservation of momentum symbol could be well angular momentum should be pretty important I would say. Solve the oil equations for theta substituting all f do as omega.

[00:42:38]Relate this equation back to the one you found in the Newtonian case. Which one is more general? Interpret your results.

[00:42:44]Okay, this one is a fair bit more of a headache because we are still using the same exact uh lrangen other than right now instead of the fot we would use an omega color palette. That is a mess. No one will be able to tell what that is.

[00:43:01]This time it is a lot worse because our partial in relation to theta on its own we can't just skip it as nice as it would be. That's because out of these three things right here, the only one that doesn't have a theta is this because this has theta dot, which is completely different, which means that we effectively need to solve this derivative for the other two and this one is a bit of a mess.

[00:43:29]So speeding through it once again, we can first of all separate this off as a constant. Treat it as a constant and considering how we just saw what happens to our constants when differentiating one step earlier, that should be fine.

[00:43:40]and then derivative of minus cosine theta. This entire thing will simplify into mg l s right here. Well, I suppose I don't need the brackets because the minus doesn't matter. The mgl and then sign is an utter mess because in this we need to introduce one more identity which is the chain rule. So if we have a of b like so then this the chain rule would apply here which means a prime of b and then this times b prime like so.

[00:44:09]Now the reason why chain rule would apply here is because we have this sign of theta but this sign of theta is inside all of the other garbage and the result of garbage. I'm just going to skip to the solution because once again I know that these segments where I'm just drawing on paper are boring enough.

[00:44:25]There we go. Don't mind the slant I was uh writing down in cursive formatic.

[00:44:32]So that was one of them. Uh annoying enough already. I'm going to highlight it in green so that we don't lose track of our solution. And that's because right here I have some empty space and that's what I'm going to use for the other one because keep in mind that's just one. Now we need to do the same thing but with a dot. And in this scenario it is trivally simple because once again we can do the same thing as we did right here. Separate off into constant and variable. uh then derivative of the variable and we already know that this two will hop down cancel out this two and what we'll be left with is very standard derivation of kinetic energy and then once again we need the time derivative of that so of m l theta dot square right here don't forget here we do the derivative in relation to time and uh well this kind of brings up the do notation because the derivative of some kind of variable x derivative in relation to time of some variable x just means that we put a dot on top of the x. So right here this will just be m l 2 like so and then theta dot dot two dots. So instead of velocity acceleration and so there we are. We've solved it all and this is the final equation. And yes it does look quite monstrous but we've spilled it up from scratch. So don't worry it's not as scary as it may seem. First let's realize something. This right here is acceleration. Acceleration on this axis to be more exact. So if we say that our carousel isn't accelerating, we can set it to zero and be left with this. And so if we move this term to the right, divide both sides by cosine and since this is just equal to a tangent, well this feels familiar, doesn't it? And yes, it is the equation we've gotten on the very beginning. But before we get to it, there is one more exercise left, isn't there? And that is exercise 8.

[00:46:28]What happens if our lranjen doesn't explicitly depend on Q? This may seem like a weird sentence, but effectively it just means what if our kinetic to potential ratio isn't affected by position.

[00:46:40]And well, what happens then? Well, what happens then is this left side of our oil lanch will equal zero, which means that we're left with this. Or in other words, since this right here is the momentum, this just means that the time derivative of momentum is zero or translated to human, our momentum isn't changing in relation to time, constant motion. And it makes sense because it's effectively unaffected by potential. So it's just normal constant motion.

[00:47:13]But we are not solving the free particle. Remco can solve the make it forsaken free particle himself if he really wants. And so there we go. That was it. That was the lranian formulas.

[00:47:24]As you can see, it's slightly different from the Newtonian, but the basic idea is the same. It's still physics. It's still classical mechanics. And by now, you should have a rough idea of how to use it, which we will use because I don't know if you remember this entire video is about an exercise paper about the carousel. Yeah, that's the whole time theme. And funny enough, our research department is actually right now hard at work reviewing the very last bits of those papers. And once they're done, we'll finally be able to review them all.

[00:47:58]Very good. Perfect.

[00:48:06]>> Take what do we do now?

[00:48:12]>> Yep. So, we Okay. So, we review >> we have the first submission by Julio. I don't know how we should do it with the names.

[00:48:22]>> With the names, uh with the names, we can just um put them on the screen if they wish to be seen. Uh by the I would just like to mention we do know all of the names because we have spent litual months going over these. It's just that um we're still not completely certain who wants to be in the video. So just in case, we're going to just put the name on screen if >> we get complete concrete confirmation that they want to be in the video.

[00:48:48]>> So this part is also going in the video.

[00:48:50]>> Yeah, this part is also in the video.

[00:48:51]>> And we're not cutting this away.

[00:48:53]>> Yeah. No, why not?

[00:48:54]>> Okay.

[00:48:54]>> I mean, I'm I'm going to cut it if things get too weird. No, no, no. I'm going to cut the bit where you say we should have maybe prepared this.

[00:49:00]>> You're not going to cut that.

[00:49:01]>> But I'm not going to cut the bit where I say I'm going to cut the bit. This submission, yeah, this submission was really um it was really clear that the person who wrote this exercise really understood the exercise. Um it's very mathematically precise and um on point.

[00:49:17]It's got it's got the conclusions. It's got um all the the tiny steps it got the point out. So this is an excellent example of a submission, I would say.

[00:49:26]>> So beautiful uh beautiful setup. I mean, I can already see on the review the yes of exclamation marks and the amazing.

[00:49:34]>> Yes. And um it was a bit of a pain for my printer, but otherwise sublime. Like there is there is literally no remarks I have on this.

[00:49:46]Yeah, there there's nothing that went wrong.

[00:49:49]>> Absolutely beautiful job though.

[00:49:50]>> So, next we have submission. Uh [ __ ] cranky.

[00:49:58]stupid crank. You can leave that one in.

[00:50:01]>> Now they know where we are.

[00:50:02]>> Sh. The next submission is um really well done. It starts with a very hard foundation. It it works out the exercises um in good detail. The images are perfect. They they are exactly what was asked. Um there in the later derivations there are some steps that um are not mathematically clear always >> clear to say that this person does show a lot of promise just a matter of adding a little bit more of the rigor and preciseness with the um introduction to classical mechanic formalisms. It's not that bad. It's still fine. Uh but considering how this is already lagranjen and lagranjen you all did lagranjun which is impressive the whole title of this video will be something along the lines of you did the university level physics precisely for that reason. So in classical mechanics still fine but especially as you then expand and expand and expand into more and more different kind of formalisms those precise details very start to matter. So in this scenario great job and you do show great promise. So >> I'm really proud of this one. Yeah.

[00:51:10]>> I have forgotten anything you said.

[00:51:13]>> The next one number three. Uh >> they did that on the on the paper.

[00:51:19]>> Yes. So they have they copied the paper and they wrote their notes on the paper and additional calculations were on the microphone.

[00:51:30]>> So next they used they used the paper themselves wrote some notes on the paper. For example, the drawing it was readable. Some tiny calculations.

[00:51:39]uh the the the easy to calculations and then uh they called a spelling mistake. Good on them and they got the other mistake in the paper as well.

[00:51:50]They ended with a nice conclusion and all their uh calculations and all their other drawings were uh >> oh yeah >> very neatly presented.

[00:52:00]>> That is beautiful. That is absolutely absolutely beautiful. I'll need to recheck the emails because I'm not certain that everyone in the end agreed to be in the video. But I would definitely love to show the notes like that because that is very very cleaner.

[00:52:17]>> Yes. And every um every exercise has been worked out properly except for exercise number six and number seven which are the most difficult one that is second year uh bachelors. Um so it is okay that if you don't uh make these ones um we have we will show how we have shown how to make these ones in the video something however and >> the main point about the submission is I am really really glad that you submitted it despite not knowing every single thing because that also gives us an indicator on how we should approach this on how we should um >> explain and we'll continue. So the next submission is a beautiful submission. It is printed in uh black but black and white um but it has very nice colors as you can see.

[00:53:09]>> Yes. So it is printed you can see it in full color. We unfortunately are on a budget and if you would like to help us get better printers then you can go to the link in the description and donate to the charity >> THE FOUNDATION.

[00:53:20]>> THE FOUNDATION. So on on on the paper itself, it's um it's a really nice and aesthetically looking paper. It's um the visualizations are really really clear.

[00:53:32]They're really um correct. Energy reference skill uh example was done perfect. They even got the mistake. They carried out the calculations all the way to the end where they there was no mistakes in the entire derivation. they um got the final result and not only that they interpret their final result and related it back to the Newtonian case which uh I would say was the most difficult part about the entire exercise but they did it flawlessly >> there I I have no remarks no comments it's not only that there um the derivation the first uh exercise it was a really creative approach where they used geometry uh in a very >> diff in I would say an unusual way but it it got to the answer really quickly.

[00:54:25]I would say the final remark for this is the there is also the therefore symbol and peak >> that that's flexing to be fair. That is flexing that point.

[00:54:35]>> It's going to be a cut here which is fun because what >> now it's not going to be now just going to now I'm going to leave from the moment you said there's going to be a cut here. I'm going to leave that in.

[00:54:46]The fifth handin is um written in a very old school uh physics way where you take your notebook, you sit down and you start. I did that again, did I?

[00:54:56]>> You did do that again. Regardless, as I was saying, the the fifth hand in it was a very standard physics way of doing it.

[00:55:04]You take your notebook, you sit down and you just write. That is what I perceived from this. And it pays off because not only is it very elaborate in its writings, it draws on the right conclusions. It takes all the right steps and derivations. It catches the mistakes of the exercise. It's it's complete. It's it's almost like there was no assignment and they just started to write.

[00:55:31]>> Maybe they were building carol themselves.

[00:55:34]>> They they can now. They got all the physics to do it.

[00:55:39]uh with the like very clear explanation like why because we can't differentiate a function with respect uh to Q if the function doesn't depend on Q so the derivative with respect to Q is zero I like the elaboration on why the fact that the elaboration was there whether they actually went into it and it is just working out >> this looks like the page the >> yeah no so sacrifice were made, sacrifices were made for this.

[00:56:11]>> One particular thing I would like to highlight is the approach for exercise two which is done in a very mathematical way by arguing how it should act like.

[00:56:23]So it it it argues proportionality and based on that it derives um the expressions instead of going from the usual approach that we chose. That was not the microphone.

[00:56:34]>> Yeah, don't worry. That's the edit route.

[00:56:36]>> It's very the main paper. The next exercise, um, once again, a a little bit of a a cost on my printer.

[00:56:46]Um, but it's this is a very expensive video to make.

[00:56:51]>> All I can say about this is is I would print it again. It's it's worth it. Uh, >> I mean, I would print it again. Is really good. I really like that. Like, they did say I don't have GitHub. I don't have Discord. And I said they link their GitHub. So, uh, once again, I'm gonna double check the email, but, um, if if they're fine with it, there there's gonna be a GitHub nickname on the stream. Start the repos.

[00:57:16]>> Too warm. It's getting warm. Uh, >> I know.

[00:57:19]>> Okay, we need to finish this. Let's do this.

[00:57:23]>> You You think it's getting warm, huh?

[00:57:25]>> One thing I particularly like about this hand is in the way how meticulous it derived um the kinetic and the potential energy. there is a very clear division of how to um construct the entire lranchchan and um it once again it also catches the mistake. It's uh it it even gives the range for the generalized coordinates. It solves it all the way through. It interpret interprets the results and it even has the final example of the free particle is flawless. And um as a final remark mark, your handwriting is absolutely amazing.

[00:58:04]I I'm jealous. And the next ones which is nicely typewrited in latte which bless that this makes the reading very easy. So first of all the one important or one one noticeable thing about this one is that the first exercise and the second exercise instead of using the usual approach for forces it uses uh moments which is a more generalized version of force. It is more applicable to this situation because we are moving with rotation. We are moving with extended objects rotating around some center of mass.

[00:58:40]moments are the natural way of approaching this which this paper nicely does and uh in doing so it achieves the right answer of both. They proceed with the lranchchan formalism they calculate the derivatives of the lrunchen as one choose. they interpret there is a very very um nice and indepth uh interpretation of the results which for as a as a physicist as a master of physics cut that as a almost >> I'm going to cut depending on >> no as someone with almost a master in physics um no this is an argument by authority I can't do this as a physicist this um shut off as a physicist this kind of indepth uh argumentation explanation is exactly what we are looking for is it it tells you everything that is necessary to know about the solution and there is even an extra discussion on the end that relates and concludes everything of the entire paper which very nice I very well done and then the last one which uh once again uses the exercise of the the exercise sheet itself and takes notes on this excess sheets. Um, it found all the mistakes.

[01:00:06]It found an extra mistake that I didn't even find a missing comma. The picture is very straightforward. The errors at home very straight. I could not do that.

[01:00:18]>> Um, the equation of motions are found.

[01:00:21]The Newtonian case is completed. There is a very lovely and kind word and it tears my heart to hear these sweet words. I am very grateful for your submission but it's so nice and uh for the exercise itself it's it's so complete. Every single step is a derivation but every step is shown.

[01:00:42]There is no gaps in the entire derivation. There is no um new things introduc new things popping up without them being introduced there.

[01:00:54]There are conclusions, there are interpretations of the results. It's >> yes, this is as a student and >> you would easily grade like maximum or perfect score on any assignment. I am I'm very proud of this one.

[01:01:13]>> If I remember correctly, you did grade them in Britain.

[01:01:17]>> Yes.

[01:01:18]>> All right. Because then I would have something to say.

[01:01:20]>> Okay. So basically as a bit of an anecdote which I think is really fun.

[01:01:25]This did take uh around half a year for us to make simply because so many things happened with you know Reno's masters the charity and all that stuff. It did take a lot longer than expected and it just so happened that it took such a long time that by the time Remco was grading this we were actually working on another checkpoint. We were working we were in Britain the one with the with chemistry with spin and I as such I was there in person when Remco was grading this paper and he got so excited that this is this that this paper was like the sole motivation for why the second exercise sheet is so damn long that like after seeing this you just logged in and you just wrote down the entire quant quantum mechanics exercise sheets, which by the way, it is available on the website if you'd like to see what I mean. And at the same time, go and do it because we're probably going to do a second one of this kind of video. And cuz we have this one to cover a year, I imagine uh I don't know, 2030 might be a reasonable estimate for the other one.

[01:02:34]>> In general, I am really proud and glad with all the submissions, whether they are perfect or not. They are all an opportunity of course to learn and um they give us the feedback.

[01:02:51]>> So I would say that yeah the big takeaway here is that all of the submissions at the end of the day did what they were meant to do. You did physics like some sure certain exercises may have been left not completely finished but you all gave it a shot. And I don't mean it as you know like a prize for showing up. No, I do genuinely mean it. This took efforts. This took effort.

[01:03:14]This took commitment. This took genuine work. But at the end of the day, you've done it. You've all submitted a paper on the lranjen of the carousel. And I think by now you all may be happy to know that you can with confidence say that you did university level physics >> and exercise.

[01:03:38]>> Yeah. And but since No, no, no. Because that's the title of the video. So, you know, it's going to be epic cuz it's going to end with a >> me saying the title of the, you know, it's like Yeah. It's like when when in Openheimer they said Openheimer.

[01:03:49]>> Uhhuh.

[01:03:50]>> Yeah.

[01:03:50]>> And then you're going to cut it right here, right?

[01:03:52]>> No. Yeah. No, this is >> Oh, [ __ ] Of course.

[01:03:57]>> I'm going to split the >> Anyway, uh yeah, there is more stuff that happens. I honestly don't know where exactly will be finished. So just to be safe on the website there is a library which you should check out. Um and check out all of the announcements that happen right now because um once again the charity is finally registered.

[01:04:18]So we're finally doing all the things we wanted to do for such a long time. And if you don't mind, we're recording this during a heat wave with 42° C and we had to turn off the fan for this recording for the sound. So thank you very much for watching and have a great day. Bye.