Hey, guys. So when we're doing linear motion, you may remember that you had a set of 4 equations that you would use to solve a whole bunch of different types of problems. Well, in rotational motion, it's exactly the same thing, except they're going to take different letters. Let's check it out. So as it says here, just like in linear motion, there are 4 equivalent motion equations for rotation. It's the same exact thing. They just have funny looking letters. All right, so as it says here, you often use these when you're given a lot of rotational quantities. It's usually a word problem and it starts throwing out things like the velocity, the acceleration and you would use these equations. The process is the same exact one. The equations just look a little bit different. So we're going to rewrite these equations real quick. Instead of v, I'm going to have ω or omega. So it's the same thing.
Omega final equals omega initial. Instead of a, I'm gonna write αt. Same thing here, omega final squared, omega initial squared plus 2 alpha delta theta. Right? And then Δθ equals, omega initial t plus half of alpha t squared. And then this one is Δθ equals half omega initial plus omega final times t. You can think of this as translating from linear to rotational, same exact stuff. The letters just look different, different variables. So, I have a star here, an asterisk because remember, same here. In some cases, your professor may only give you these three equations and want you to stick to 3 of them. This is the extra 4th equation. You should know by now whether your professor is cool with you using it or not. Remember also that when you're solving motion problems, you need to know, you need to know 3 out of 5 variables. Remember that one variable will be your target and one variable will be your ignored variable. And this is the one that will determine the equation you use. Equation to use. Okay? This is very straightforward. Let's do some examples.
Alright. So here, a wheel initially at rest. So initially at rest used to be that the initial velocity is 0. It still means that, but now it's initial angular velocity because this wheel is going to rotate around its central axis. So you can think of it as a big disc, something like this. Right? Imagine that's a disk and it has a central axis, meaning like some sort of stick and they can spin around it like that. Okay. So it starts from rest. So the initial Omega is 0, and it's going to accelerate with a constant 4 Radians per second is acceleration, so alpha equals 4 until it reaches 80 radians per second squared. You can think of this as meters per second, but in rotation. Okay? So that is your final velocity. It's not actually meters per second. You can just think of it that way. Omega final equals 80 radians per second. All the units here are correct. So as I mentioned, you can tell that you're supposed to use this because you start getting a lot of rotational quantities. Right? In this case, I already know 3 of them. So I know that I can already solve whatever I'm about to be asked. Okay.
Cool. So it says, by the time it reaches 80, how many degrees will it have rotated through? How many degrees it's gonna have rotated through? It's asking for delta theta, but it wants the answer in degrees, which means I'm going to get it in radians because the equations always spit out delta theta in radians, and then you have to convert to degrees. Cool? So I'm going to do what I always do, which is list my 5 variables here. Delta theta is what we're looking for and the variable out of the 5 that didn't get mentioned was delta t. So I'm going to put a little sad face here and I'm going to pick the only equation out of the 4 that is missing a delta t which is this one. There's no delta t on this one. Okay? So same thing as before, omega final, omega initial, got the squares, 2 alpha delta theta. Delta theta is what I'm looking for. I'm going to move everything out of the way. So delta theta, target varies by variables by itself. Omega final squared minus omega initial squared. This stuff comes to the other side dividing. Now we're ready to plug in some numbers. And set it up like this. 2. Now we're ready to stick the numbers inside of the parenthesis. Time velocity was 80, the initial is 0, and the acceleration is 4. So, if you do all of this, you end up with 1600 radians. Remember, these equations always spit out radians. And then we're going to convert. So I'm going to do pi radians at the bottom, and then 180 degrees up top. When I cancel radians with radians, we're left with degrees. So 1600 times 180 divided by π equals approximately 91629 degrees. That's a huge amount of degrees. It spins a whole bunch.
For part B, part B is asking how long in seconds does it take? In other words, what is our delta t? Delta t was originally my ignored variable, but now we're looking for delta t. We can use since it's the same situation, I can use delta theta. So I actually have I know 4 out of 5 variables. I only needed 3, but I know 4. And when I know more than what I need, it means that I'm going to have more flexibility with the equations. Instead of having to use one specific equation, I can use any equations that have delta t, which in this case, there are 3 of them. Okay? So, the simplest equation to use would be the first one so I'm going to use that one. Alright. And we're looking for t. Let me circle it. So if I move everything out of the way so that t is by itself, it looks like this. And t equals let's plug it in. The final is 80. Initial is 0. The acceleration alpha is 4. So the answer is 20 seconds. Alright. That's it. Very straightforward. Just like it was before, You just have to basically make the adjustment for the letters, and you see different, you know, units and, you know, it's going to say things like central axis and rotation. So it's the same thing just in the rotational world. Alright. Well, that's it for this one. Let's keep going.
