 So, let's start this video with a little experiment. First of all, I have a bike wheel here that I want to hold up in the air with this string. What do you think happens if I just let go with my left hand to hold it up? Well, of course, it will just fold up, like this, right? Now, let me change a little thing. Now, before letting go, I'm going to spin my bike wheel like this. What do you think happens now? Let's have a look. Surprise. Let's repeat this. I'm spinning it, and then I let go, and it's turning that way. What happens if I spin it the other way? It turns the other way, but it does not fall down. Isn't that weird? In order to understand what's going on, we actually need to look at conservation of angular momentum. Once you've looked at this, we can try to explain what just happened with the bike wheel. So let's look what angular momentum is based on linear momentum. I hope you all remember that linear momentum p is mass times velocity, and the unit, according to that equation, is kilograms meters per second. Now if you think back at the equivalences in the rotational entities for linear entities, what is the equivalent for velocity? Well exactly, angular velocity, we're going to use three characters. What was the equivalent for mass? That might be a bit more tricky, but it's the inertia. Remember, mass is the linear inertia, so for the rotational case, we're just going to call this inertia i, and we have the equation. Now all that we need to do is pick one random letter from the alphabet, and we're going to do pick L, and we have our angular momentum. What is the unit of angular momentum? It's a bit more complicated. It's kilograms square meters per second, or Newton meters per second. The same could actually extend the unit for the linear momentum as Newton's times seconds. It's going to be important in a moment, but for the moment, let's just take this unit. Now angular momentum is a vector. It's a vector that points in the same direction as your angular velocity. So if you have an object that turns, let's say like this, then your vector, omega and L, are pointing in the same direction. And how do you figure out which way this vector goes? It's a right hand rule. You curl your fingers in the way the object is turning, and then your thumb should point in the direction of omega and L. Now once you've figured out that our angular momentum is the inertia times omega, the big question is going to be what is the inertia? Well, the inertia depends on the object and actually depends on the distribution of the mass of the object in regards to the axis of rotation. There is a difference whether I rotate this meter stick around this axis or around this axis. Around this axis, there's way more resistance to me getting it into rotation than around this axis. Because why? Because the mass here is further from the axis of rotation. If I turn this way, my axis of rotation is actually here pointing that way. If I turn this way, my axis of rotation is here. So the mass on average is way closer to my axis of rotation. Now how can we put this in a formula? In general, the i, the inertia, is defined as the sum of all masses times the distance from the axis of rotation squared. For example, if you have an object with mass 1 here, that is at the distance x1 of axis of rotation, let's say we're rotating around this axis here, and then we have a mass 2 here at the distance x2, you would do mass 1 times x1 squared plus mass 2 times x2 squared. If you have some object that has a uniformly distributed mass, you will need calculus and you're going to integrate this as x squared dm. But if you have not taken calculus yet, don't worry about that one. Just go with the summation of the individual mass pieces in your object. And most likely your teacher will help you figuring out what is the inertia for the most standard object. Now we have to find what is angle momentum. And as in the linear case where linear momentum is conserved, we also know that angular momentum is conserved. So we have a law of conservation. In the linear case, that was that my linear momentum at the end is my initial momentum plus the change, which we call the impulse. And impulse was force times time. This is actually how we get to Newton's seconds as a unit. Well, here we had kilograms times meters per second. Here we have Newton's per second. So those two must be equivalent. And if you look at it, if you go into the ninth, the Newton is a kilogram meters per second square times a second. You actually get exactly kilograms times meters per second. Now for angular case, instead of the L of the P, we have the L. So my angular momentum final is my angular momentum initial plus the change. Well, the change actually here doesn't have its own letter as far as I know. Maybe somebody came up with a letter in between. So we're just going to put the equivalent of force times time. What's the equivalent of force for rotational cases? What is needed to bring an object in rotation? Well, yes, it is torque. So torque times time. And then you will see why we have the unit of Newton meters torque times second. The kilograms per second came actually from the inertia here, if you think about it. It was inertia was defined as the sum of all mass times the x distance from the axis of rotation squared. So kilograms times square meters times rats per second. And rats is a unit that can disappear. So we have kilogram square meters times one over second, which is equal to, it must be equal to Newton meter times seconds. And again, if you look at it, Newton is a kilogram meters per second square. So if we convert this, that should give us exactly the same units. So that's it for conservation of angular momentum. And now let's go back and try to figure out if you can see what happened with that bike wheel. So let's try to explain what's going on by using conservation of angular momentum. So initially, my wheel is spinning that way. So my rotational velocity vector was pointing this way. Remember, right hand rules and curling the way I'm turning gives me my vector for my omega. So my omega vector points this way out from the axis initially. On the board, I'm going to draw it as seen from above. So from above, my omega vector and therefore my L initial vector goes that way. Remember, this is just my rotational inertia times my omega vector. Now the moment I let go with my hands, my bike wheel is hinged here. So what will gravity try to do? Gravity will pull down here. So what type of torque is gravity causing? Gravity is causing a torque this way. Like gravity tries to turn my wheel like that, down like this. So the torque vector, if I do the same thing, I'm turning like this. It's pointing that way. So my torque vector is pointing backwards. So if I do this in the top view, my torque vector is going to look like this. Now according to conservation of angular momentum, my final momentum vector will be my initial momentum plus the torque vector times the little amount of T. So let's make this delta T very, very small, going to calculus style, small, very small delta T. Then what I have is I take this vector, multiply it by a very small dT, and I have to add it to my initial vector of the angular momentum. So I can do head to tail. So here is my initial, is my added head to tail. So this one here is torque times my dT vector, which then gives me the final vector. The final vector is my L final here. And the only way to have an L final pointing in that direction is for my axis of rotation, for my omega final, to shift slightly to the left. So let's think about this with the wheel from the front view. My initial L points that way. Then I have a torque vector that adds going in that direction. That means my final L has to be in this direction. And the only way for this to happen is if my wheel starts rotating to the left. And once I'm at this position, the same thing starts again. Now torque is trying to push that way, moving my angular momentum vector again and again and again. So let's see, according to this, my wheel should be spinning in this direction. And it does. Actually, if you ever get your hands on a bike wheel like this, try to just move it around a bit. It does some really weird things. For example, if I'm moving or rotating this way, it feels very heavy. If I am rotating this way, it feels very light. Actually, look, I can lift it almost with my small finger here. No problem. But if I go this way, it feels quite heavy. But the weight of the wheel definitely did not change. What changes is the torque it exerts on my hand when I try to hold it. So it really feels weird. So if you get your hands on a wheel like this, try and move it in different directions. It does really unexpected things. But all based back on a pure old law of conservation, that's what we have. After is what we have before plus the change.