Types of Acceleration in Rotation

Hey, guys. So that we're gonna talk about the four different types of accelerations you'll see in rotation problems. You can get a little overwhelming because it's a lot of letters, a lot of variables, a lot of equations. But I'm gonna try to simplify this as much as possible. Let's check it out. All right, So there are four, as I said, types of acceleration, rotation problems. Most of them have two names. So in total, if you look, you have seven different names, and you have to know which one how the pairs, which pair names go together. Okay. Centripetal acceleration. A c also referred to as radio acceleration a rad. Okay, those are the same thing. Um, tangential acceleration is 80. Um, sometimes this is referred to as linear acceleration. Um, so I'm actually add that there. Sometimes you might see this called linear acceleration. Um, the other one is total acceleration, which sometimes is referred to either as total acceleration or just acceleration. Sometimes it's referred to as the total linear acceleration. Okay, so this is just a it's a total acceleration. Um, and then the last one is rotational or angular acceleration. You used to this one, This is Alfa. I want to point out that these guys were measured in meters per second squared and this is measured in radiance per second squared all these three here. These three are linear accelerations, so they're represented with sort of a straight line, as opposed to Alfa, which is an angular acceleration. So it's measured this way. Um, if it's an A, this is a C a rad a t a right. If it's in a it's meters per second eso English letter So it's meters per second. If it's an outfits Greek letter, it's in radiance rains per second. That's one way to think about it. Cool. All right, now there's these four types, but depending on what kind of situation you have, you're not gonna have all four types. And what that means is that some of them will just be zero. So first you always have VT and a C. So if you're going around a circle, got a little object that's going around this circle, let's say this way a to this point, this object has a has a, um, tangential velocity, and it has a centripetal acceleration pointing towards the middle Okay, Um, and it's spinning this way with Omega. Remember that at this point, V t equals R Omega. Remember? Also that a C equals V squared over r. This is from a while ago, and what I can do is I can actually rewrite V as our omega. So let's do that real quick. So it's gonna look like this. Are Omega squared over R um, the square goes on our and the W so there's actually to ours and to W's, but one are is going to cancel with er at the bottom. So it's gonna be like this are without the square who made a square. So this is old school old news. But this is a new version. Okay, this is ah, brand new version of writing a C. You can write in terms of r and W instead of the okay, he's always exist. You always have a C because a C. If you remember him from F equals, they may. You have a C because you have a FC a centripetal force. There's a force that pulls you towards the middle. Therefore you have a centripetal acceleration. A C is responsible for maintaining the circular path. So a C maintains circular path. Another way to think about this is that it keeps the object spinning. If a C doesn't exist, you can think of it as there is no longer a force no longer force pulling into the middle. So this object is gonna go straight up, um, in this direction. Okay, so a c um, it's there, and it maintains a circular path. So as long as you move in a circle, you have to have a seat and you have to have VT. Um, so these things always exist and you have a W because obviously you're spinning. Now, if you're accelerating, okay, if you're accelerating, you're also going to have a tea in Alfa. The the converse of that is that if you're not accelerating these guys, they're zero. Okay? Otherwise, let me put that here otherwise meaning if you're not accelerating than 18 in Alfa equal zero, So what does a T look like? The idea is that you're not just spinning in place, but at this point, the objects actually getting faster, right? So if this thing is rotating like this, you can think of it that to get faster. You have to like something has to push the object this way. Some force. Okay, I'm gonna raise that not to make a mess. So if you push the object this way, it's going to spin faster. So you have an acceleration here, which is a tangential acceleration. Okay? And if the object is going faster this way, it's also rotating faster. So there's also a new Alfa this way. So you got an Omega and you got an Alfa in that same direction. Okay. All right. So you've got these two guys here now, remember, you always have. You always have. You're a c. So this sets up an interesting situation where you have a name. 80 this way. And they see this way. They make an angle of 90 degrees with each other right there. Okay, I'm gonna delete that. So what happens is you have two arrows this way so you can combine the two of them using vector compositions. Pick a different color here. I can make a little triangle here, and this combines using vector composition. And this is what we call the total acceleration. Sometimes referred as the total linear acceleration or sometimes referred simply as acceleration. So if you don't see if you see just acceleration without the word centripetal radio tangential, linear, uh, rotational. Angular. Right. If you don't see any of these words, you can assume it's just regular acceleration. The total acceleration, which is a combination of these two. It's a by itself. Okay. And this is just factor edition. So a is the square root? Um, it's the Pythagorean of the two sides. So 80 square plus a C squared. Okay, so that's what the total acceleration. So whereas a C maintains your circular path, these guys here are responsible for, um, changing angular speed. So these guys 80 changes angular speed or velocity. Okay, technically changes angular velocity. So let's right that there changes. Angular velocity now changes means it could be going faster or slower, right? All right, uh, two or points that we're gonna do an example. Um, this equation eight er Alfa, which is one that I gave you earlier, is a good way to remember that 80 and Alfa are connected. Okay, They're linked up together, which is what we see here. Okay, um, you have this in this, or they're both zero thes guys air both Not either non zero together or they're both zero together. Okay. And in the last point I wanna make is that if you're a t is, in fact zero, which, by the way, please this, By the way, this would mean that Alfa zero just look at this equation right here. Right then your total A which is usually the square root of 80 square plus a C square simplifies. Okay, if you're 80 0, look what happens. You get simply zero plus a C squared, which is the square root of a C squared, which is simply a c. Okay, I'm just pointing that out. In case you see that, you don't think it's weird that there's two ways inside of the square root, but one goes away, and then you're a becomes just your a c. That's totally doable. In fact, that's what happens here. Here, you're a is on Lee a seat. So it's like you have these two arrows stacked on top of each other. Okay, let's not draw that there, so it doesn't make this more confusing. All right. Four accelerations on guy gave you two new equations. One your way to rewrite a C and on equation for theme total eight. All right, so let's do a problem. He has got five parts. It's kind of annoying, but hopefully see that it's not that bad. So have a carousel carousel, 10 m and radius. So big circle radius equals 10 m. I know I don't have a lot of space here, but you actually don't need that much room. Okay? But that's right. Small. Alright, complete one cycle, one cycle every 75 45 seconds. So that's the period T is 45 seconds. I wanna know what is the tangential velocity? There's a boy that stands at the edge So if the boy is at the edge the boys at a distance r equals 10. This thing is 10 m long and radius. Um, if you're sitting at the edge, you sit at the 10 m distance from the center and I want to know his tangential velocity. So tangential velocity is VT. And remember the tangential velocity of appoints a person and object whatever on a circle on a on a any kind of spinning object is our omega where omega is the omega of the boy which is the same as the Omega of the disc. Okay, so the boys at a distance 10 But I don't know omega, However, Remember, Aiken, get omega from T because Omega frequency t and R p m are all interconnected. Omega is two pi over tea, so two pi divided by 45 seconds. And if you do that, you get 0 14. And that's what goes here. 14. Therefore, the answer is 1.4 meters per second. Is the velocity at the boy experiences for part B? I want to know what is the angular acceleration? Angular acceleration is Alfa. Now remember you on Lee have Alfa, if you're actually spinning faster once it says that it completes one cycle every 45 seconds. Right, 1st, 45 seconds. One cycle 2nd, 45 seconds. Another cycle. It implies that this is a constant. Um it's a constant movement at a constant rate at a constant velocity. So Alfa is actually zero. All right. Radio acceleration is a rad, which is the same as a C, which is V square over our or our omega, whichever you prefer. Okay, so I'm gonna use our omega just I don't have to square this number, but it's the same exact thing Are is 10 and Omega is points. Um, I'm sorry. It's our omega. Who? It's our Omega squared. I was like, Hey, that's the same thing. Is that so? I'm gonna have to square something either way. Um, 0.14 square. But anyway, if we do this, we get that the answer is 0.196 meters per second. OK, I almost used the one equation there. Um, either one of these works cool for part D. Okay, keeping it tight there. The tangential acceleration is 80. Remember, 80 Onley exists if you're actually pushing this thing to spin faster. Another way to think of this. Is that a TSR Alfa Alfa zero? Therefore, 80 equals zero. Okay, So part of the reason why this question doesn't require that much space is because some of the insurgent zero the total linear acceleration is a which is the square root of 80 square plus a C square and 80 is zero. So you're left with the spirit of a seat which is just a C, which is the same thing is a rad, which is 0.1 96 meters per second square. So here's a five answers This zero this zero and the total A right here. Is that all right? So just a bunch of equations and knowing how to link everything together. Um, it's good to do some practice, Make sure know how to do this. Pretty straightforward, but it's kind of annoying. Alright, So hopefully this makes sense. Let me know if you have any questions.

All right. So in the second example, we have carousel now 16 m and radius. So it's draw a circle here, and the radius is 16 and it accelerates from rest with this. So Omega initial zero in Alfa is 0.5 boy stands at the edge again. So the little are the distance between the center and the boy is just the same 60 m and we wanna know After seconds, Delta t equals 10 seconds. What is his tangential velocity Now I'm gonna put it over here to the side because tangential velocity v t is not one of his five motion equations. So I'm just organizing. It's a little bit. I wanna know his tangential velocity. Remember, tangential velocity or linear velocity of an object at a point in the disk is just given by our Omega, where r is the distance from the center and omega is the angular speed. Okay, I know are 16 but I don't have w So we're gonna have to do here is we're gonna have to find w final, which we're going to do using motion equations. Okay, variable that's missing here is Delta theta. So I'm gonna put a little sad face. And this tells me that the equation I should use is the first one. Because it's the only equation that doesn't have Delta theta or make a final equals and making initial plus Alfa, um, t and we're looking for Make a final. This is zero Alfa is 0.5 The time is 10. So Omega's just point five radiance per second. I can plug this in here. This is 16 times 160.5, which means the velocity will be 8 m per second for part B. I want a little tangential acceleration. Um, a T is our Alfa and I have both numbers, so we can split in 16 alphas 160.5 eso This is going to be zero point a I'm sorry pointing, um, 0.8. Same thing meters per second squared. Okay. Yep, that's it. And then see, I'm looking for the radio acceleration A rad, which is the same thing as a centripetal a C. And I can use V square over our or I can use our omega squared. It's gonna use this one already have V, so it's gonna be eight squared. Are is 16. So the answer will be eight square by by 16 is four. So 4 m per second squared for part D. I want to know the angular acceleration, angular accelerations, Alfa Um, angular acceleration is Alfa. We actually already have the angular acceleration, so it's just going to be 0.5 Okay, this one is the freebie. Just make sure you know what you're doing. Um, radiance per second square. And the last one is a total linear acceleration, which is a and A is the square root of the other two linear acceleration. That's another way to think about this. There's three linear. One of them combines the other two, so it's a T square plus a c squared. And if you plug this in its points eight squared four squared. You combine all of this and you get 4.1 m per second squared, and these are the final answers. All right, that's it for this stuff. Um, let me know if you have any questions.