Hey, guys. So now we're going to talk about the 4 different types of accelerations you'll see in rotation problems. It can get a little overwhelming because it's a lot of letters, a lot of variables, a lot of equations, but I'm going to try to simplify this as much as possible. Let's check it out.

Alright. So, there are 4 types of acceleration in rotation problems as I said. Most of them have two names. So in total, if you look, you have 7 different names and you have to know how the pairs which pair of names go together. Centripetal acceleration, *a _{c}*, also referred to as radial acceleration,

*a*. Those are the same thing. Tangential acceleration is

_{rad}*a*. Sometimes, this is referred to as linear acceleration. 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. So this is just

_{t}*a*. It's a total acceleration. And then the last one is rotational or angular acceleration. You used this one. This is

*α*. I want to point out that these guys are measured in meters per second squared and this is measured in radians per second squared. All these 3 here, these 3 are linear accelerations, so they're represented with sort of a straight line as opposed to alpha, which is an angular acceleration. So it's measured this way.

Now there are these 4 types, but depending on what kind of situation you have, you're not going to have all 4 types. And what that means is that some of them will just be 0. So first, you always have *v _{t}* and

*a*. So if you're going around a circle, that little object that's going around the circle, let's say this way, at this point, this object has a tangential velocity and it has a centripetal acceleration pointing towards the middle. It's spinning this way with Ω. Remember that at this point,

_{c}*v*equals

_{t}*rΩ*. Remember also that

*a*equals

_{c}*v*squared over

*r*. This is from a while ago. And what I can do is I can actually rewrite

*v*as

*rΩ*. So it's going to look like this,

*rΩ*squared over

*r*. So it's going to be like this,

*rΩ*squared. These always exist. You always have

*a*because

_{c}*a*, if you remember from

_{c}*F*equals

*ma*, you have

*a*because you have a centripetal force. There's a force that pulls you towards the middle. Therefore, you have a centripetal acceleration.

_{c}*a*is responsible for maintaining the circular path.

_{c}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's no longer a force no longer a force pulling into the middle. So this object is going to go straight up, in this direction. So

*a*, it's there and it maintains a circular path. So as long as you're moving a circle, you have to have

_{c}*a*and you have to have

_{c}*v*. And you have an Ω because obviously you're spinning. Now, if you're accelerating, if you're accelerating, you are also going to have

_{t}*a*and α. The converse of that is that if you're not accelerating, these guys are 0.

_{t}So what does *a _{t}* look like? The idea is that you're not just spinning in place, but at this point, the object is actually getting faster. 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. And if the object is going faster this way, it's also rotating faster, so there's also an α this way. So you got an Ω and you got an α in that same direction.

Now remember, you always have your *a _{c}*. So this sets up an interesting situation where you have

*a*this way and

_{t}*a*this way. They make an angle of 90 degrees with each other right there. What happens is you have 2 arrows this way, so you can combine the two of them 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.

_{c}So if you don't see if you see just acceleration without the words centripetal, radial, tangential, linear, rotational, angular, 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 2. It's just vector addition. So *a* is the square root, it's the Pythagorean of the two sides. So *a _{t}* squared plus

*a*squared.

_{c}So that's what the total acceleration. So whereas *a _{c}* maintains your circular path, these guys here are responsible for changing angular speed. These guys,

*a*changes angular velocity. Now changes means it could be going faster or slower.

_{t}This equation, *a _{t}*

*α*, which is one that I gave you earlier, is a good way to remember that

*a*and α are connected. You have this and this or they're both 0. These guys are both either nonzero together or they're both 0 together. And then the last point I want to make is that if your

_{t}*a*is in fact 0, which would mean that α is 0, look at this equation right here, then your total

_{t}*a*, which is usually the square root of

*a*squared plus

_{t}*a*squared simplifies. If your

_{c}*a*is 0, look what happens. You get simply 0 plus

_{t}*a*squared, which is the square root of

_{c}*a*squared, which is simply

_{c}*a*.

_{c}I'm just pointing that out in case you see that, and you don't think it's weird that there's 2 *a*'s inside of the square root, but one goes away, and then your *a* becomes just your *a _{c}*. That's totally doable. In fact, that's what happens here. Here, your

*a*is only

*a*. So it's like you have these two arrows stacked on top of each other.

_{c}Four accelerations, and I gave you 2 new equations, one new way to rewrite *a _{c}* and an equation for the total

*a*. Alright? So let's do a problem here. It's got 5 parts. It's kind of annoying, but hopefully you see that it's not that bad. I have a carousel, a carousel 10 meters in radius. So a big circle radius equals 10 meters. I know I don't have a lot of space here, but you actually don't need that much room. But let's write small

Complete one cycle. One cycle every 75 seconds. So that's the period. *T* is 45 seconds. I want to know what is the tangential velocity. There's a boy that stands at the edge. So if the boy is at the edge, the boy is at a distance *r* equals 10. This thing is 10 meters long in radius. If you're sitting at the edge, you sit at the 10-meter distance from the center. And I want to know his tangential velocity. So tangential velocity is *v _{t}*. And remember, the tangential velocity of a point, a person, an object, whatever, on a circle, on any kind of spinning object is

*rΩ*, where Ω is the Ω of the boy, which is the same as the Ω of the disc.

So the boy is at the distance 10, but I don't know Ω. However, remember, I can get Ω from *T* because Ω, frequency, *T*, and rpm are all interconnected. Ω is 2π over *T*. So 2π divided by 45 seconds. And if you do that, you get 0.14. And that's what goes here, 0.14. Therefore, the answer is 1.4 meters per second is the velocity that the boy experiences.

For part b, I want to know what is the angular acceleration. Angular acceleration is α. Now remember, you only have α if you're actually speeding faster. Once it says that it completes one cycle every 45 seconds, right, first 45 seconds, one cycle, second 45 seconds, another cycle, It implies that this is a constant, it's a constant movement at a constant rate, at a constant velocity. So α is actually 0.

Radial acceleration is *a _{rad}* which is the same as

*a*which is

_{c}*v*squared over

*r*or

*rΩ*, whichever you prefer. So I'm going to use

*rΩ*, just so I don't have to square this number, but it's the same exact thing.

*R*is 10, and Ω is points I'm sorry. It's

*rΩ*squared. I was like, hey, that's the same thing as that. So I'm going to have to square something either way, 0.14 squared. But anyway, if we do this, we get that the answer is 0.196 meters per second.

For part d, keeping it tight there, the tangential acceleration is *a _{t}*. Remember,

*a*only exists if you're actually pushing this thing to spin faster. Another way to think of this is that

_{t}*a*is

_{t}*rα*and α 0, therefore,

*a*equals 0.

_{t}So part of the reason why this question doesn't require that much space is because some of the answer is 0. The total linear acceleration is *a*, which is the square root of *a _{t}* squared plus

*a*squared, And

_{c}*a*is 0, so you're left with the square root of

_{t}*a*, which is just

_{c}*a*, which is the same thing as

_{c}*a*, which is 0.196 meters per second squared. So here's the 5 answers. This, 0, this, 0. And the total

_{rad}*a*right here is that. Alright? So just a bunch of equations and knowing how to link everything together. It's good to do some practice and make sure you know how to do this. It's pretty straightforward but it's kind of annoying. Alright? So hopefully, this makes sense. Let me know if you guys have any questions.