12. Rotational Kinematics

Equations of Rotational Motion

# Equations of Rotational Motion

Patrick Ford

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Hey, guys. So when we're doing linear motion, you may remember that you had a set of four 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 gonna take different letters. Let's check it out. So as it says, you're just like linear motion. Therefore, 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 thing. Equations just look a little bit different. So we're gonna rewrite these equations real quick. Um, instead of v, I'm gonna have W or Omega. So it's the same thing. Will make a final equals Omega initial instead of a I'm gonna write Alfa t. Same thing here. Omega Final squared, Omega Initial squared plus two Alfa Delta Fada. Hi. And then Delta fatum equals omega initial T plus half of Alfa T squared, and then this one is Delta Theta equals half or mayor 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. Um, different variables. So I have a start here. NASA risk, because, remember, Same here. Um, in some cases, your professor may only give you these three equations and want you to stay three of them. That says the extra fourth equation. You should know by now whether your professors cool with you, using it or not. Remember also that when you're solving motion problems, you need to know you need two no. Three out of five variables. Remember that one variable will be your target, and one variable will be your ignored variable. And this is the one that will determined the equation to use equation to use. Okay, this is very straightforward. Let's do some examples. Right. So here, a wheel, initially at rest so initially at rest used to be that the initial velocity is zero. It still means that, but now its initial angular velocity, because this wheel is going to rotate around its central axis so you can think of it as a big disk. Something like this, right. Imagine that's 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 zero, and it's going to accelerate with a constant four radiance per second is acceleration. So Alfa equals four until it reaches 80 radiance per second square. 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 could just think of it that way. Omega final equals 80 ratings per second. All the units you are correct. Eso as I mentioned, you can tell that you're supposed to use this because you start getting a lot of rotational quantity, right? In this case, I already know three of them. So I I know that I can already solve whatever I'm going to. I'm about to be asked. Okay, Cool. So it says by the time it reaches 80 how many degrees will it's 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 radiance because the equations always spit out Delta failing radiance. Then you have to convert two degrees. Cool. So I'm gonna do what I always do, which is list my five variables here. Delta Theta is what we're looking for. And the variable out of the five that didn't get mentioned was Delta T. So I'm gonna put a little sad face here, and I'm going to pick the Onley equation out of the four that is missing a delta T, which is this one. There's no delta t on this one. Okay, so same thing is before will make a final when you're sure that the squares to Alfa Delta theta Delta things what I'm looking for, I'm gonna move everything out of the way. So Delta Theta target varies by variables by itself, will make a final squared minus will make initial squared. This stuff comes to the other side dividing. Um, Now we're ready to plug in some numbers and set it up like this to now. We're ready to stick the numbers inside of the parentheses kind of lawsuit was 80 the initial zero, and the acceleration is four. So if you do all of this, you end up with 800 radiance. Remember, these equations always spit out radiance, and then we're gonna convert. So I'm gonna do pi ratings at the bottom and then 180 degrees of top when it canceled, ratings with radiance were left with degrees. So 800 times 1 80 is 1008 100 degrees. That's a crap load of degrees. Spends 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 Fada. So I actually have I know four out of five variables. I only needed three, but I know four. And when I know more than what I need, um, it means that I'm gonna have more flexibility with the equations instead of having having to use one specific equation. I can use any equations that have Delta T which in this case there's three of them Okay, so the simplest equation to use would be the first one. So I'm gonna use that one. All right? We're looking for teeth. Let me circle it. So if I move everything out of the way so that teased by itself it looks like this and t equals, let's plug it in. The final is 80 initial zero acceleration. Alfa is four. So the answer is 20 seconds. Alright, that's it. Very straightforward. Just like it was before. You just have Thio basically make the adjustment for the letters on. Do you see different units? And it's gonna say things like Central axis and rotation eso. It's the same thing just in the rotational world. Alright, well, that's it for this one. Let's keep going.

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