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12. Rotational Kinematics

Rolling Motion (Free Wheels)


Rolling Motion (Free Wheels)

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All right so rolling motion. I also like to think of this as free wheels is what we're gonna talk about and so far we haven't talked about that yet. What we've seen is we've seen either a point mass moving around a circular path or we've seen rigid bodies moving around themselves. So imagine sort of a cylinder that is free to rotate around its central axis. Something like this. Right? So think of this as a fixed wheel right? These are fixed wheels that are fixed in place now in some problems we're gonna have these rigid bodies or these shapes that are going to not only be rotating around themselves but they're also going to be moving sideways. So they're both rotating. So they have an omega because they're spinning and they're moving and when I say moving I mean they're actually moving sort of sideways they're not fixed in the same place. Um We're gonna think of these as free wheels. Okay that's why I call this free wheel. Um And the best example I think the most memorable example I can give you is actually rolls of toilet paper. So if you have a roll of toilet paper that's fixed in place as it normally is um this is a fixed axis. Okay so here's the role and it can spin it has a W. But it doesn't move sideways so it doesn't move sideways with, no which means V. Equals zero. Okay V. Is when you actually move sideways if you spin in place you're not actually moving now. So in this case w. Is not zero but V. Of the center of mass is zero. Okay, the middle center of mass in the middle the middle of the cylinder doesn't move, it stays in place. Okay. Now in the case of a free axis or a free wheel would be if you had a roll of toilet paper that is rolling sort of on the floor, right? And it's doing two things here, so it's rolling, let's say this way and if it's it's not only rolling this way but it's also moving. So if you combine this with this you get this right, so it's sort of moving this way. So I can say that it has omega and it has a velocity of its center of mass is moving to the right, so omega is not zero and the velocity center of mass is not zero either. What's special about these situations, The most important thing you need to know about these situations is that there's a relationship between these two numbers. Okay, and luckily this relationship looks like something we've seen a lot of, So the velocity, let's say you're spinning with omega here. Um the imagine that if your wheels spinning this way then you're going in that direction, right? Um there's a relationship between these two, The velocity center mass for a wheel of radius R. Is simply our omega. Now notice how we didn't use little R. So I'm gonna write big r not little R. Because in this case what we actually want is the actual radius of the wheel or the disk and not a distance from the center. It's the actual radius of this thing. Okay, and this looks very similar to what we've seen. If you have a fixed axis, if you have a fixed axis like this, the velocity tangent co at an edge here or here, these are tangential velocities, right? These are gonna make them blue, these are tangential velocities. These two velocities are little are omega but we're not talking about a velocity of a point at the edge or any distance from the center. We're talking about the velocity of the middle of this thing because this thing moves sideways. Okay too, so this is the most important thing you need to know to other things you need to know um is that the velocity, there's a velocity at the top here and there's a velocity at the bottom. So the velocity center mass is R. W. You should know that the velocity at the top is going to be twice the velocity of the center of mass. So it's two R. W and the velocity at the bottom is zero relative to the floor. Okay, now your book probably at some point, your book may derive these equations, how to arrive at them, your professor may derive um um here, just for the sake of simplicity and time. I'm just going to give you these equations without deriving here's a really easy way to remember this, I'm gonna draw this again here. The top velocity to point at the top is to our omega, velocity in the middle is one our omega and velocity at the bottom is zero R omega. Okay, so 012. Obviously this simplifies into our omega and this simplifies into zero. Okay, those are the three velocities. Notice how this is different from this situation here here, the velocity of a point at the top of a circle, of a cylinder of a disc that spins around itself is our omega and little arc. It's the distance here, if you are a little edge at the top here you are to our omega because you're moving the idea is that this are omega here, right, combines with this are omega to give you two of them. So I just mentioned that briefly, but those are the equations you need to know. Okay, most of the time, you need to know um the green one, um you don't always need to know the yellow one. The green one is the most important one, but I'll give you the yellow ones just in case. All right, let's do a quick example. This is very simple, you just have to remember these three equations. Alright, so have a wheel stride of radius 30.30 centimeters. actually may actually meant to make this 30 cm or .3 Sorry about that, So I'm gonna say that it has a radius of .3 m. Okay? Um and it rolls without slipping along a flat surface with 10 m per second. So it rolls without slipping. So if it rolls it has a w and it rolls with 10 m per second. So the wheel is actually moving when I give you a velocity here, when I say that V wheel is 10 and give you the velocity of the center of mass of the wheel. Okay, so this velocity here, V center of mass equals 10. Okay, now there's something interesting here that we need to talk about. It says rolls without slipping. Rolls without slipping is the condition for these three equations to work. These three equations are only true if you are rolling without slipping, but guess what? In all these problems, you will be rolling without slipping. Um so you can basically just ignore this equation Now, conceptually, you may need to know for sort of a multiple choice conceptual test that this is the condition for rolling motion. Okay, Rolling motion, the condition for rolling motion is that this is without slipping. That's a conceptual point there. Alright, so let's get back to this question real quick. So, I want to know a what is the angular speed of the wheel? So, angular speed of the wheel is simply omega. Okay, now, notice that I no V C M and I know are and I'm looking for omega. Well, this is very straightforward, there's an equation that connects all three of them and it is that V C. M equals big R omega. Therefore omega is V C M over R velocities 10 divided by 100.3. Omega therefore is 33.3 Um .3 radiance per second. Cool, Very straightforward. Um Part B. The speed of a point at the bottom of the wheel relative to the floor. This is just this bottom right here. And if you know this conceptually, if you remember the equation, the bottom is just always zero for a rolling wheel, no matter what. Okay, so that's that's simple. Um Omega is 33 and the bottom is zero. That's it for this one. Let's do the next example.

Speeds at points on a wheel

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All right. So here we have a car that accelerates from rest for 10 seconds. So the initial loss of the car zero and it takes 10 seconds accelerating. Okay, It's tires will experience eight radiance per second. So V is the speed of the car. Let's draw a little car here. V is the speed of the car. This is a really crappy car. Um, but W has to do with the wheels. If the car is moving that way, the wheel is spinning this way. Okay, um, I'm giving you the acceleration of the wheel, so I'm gonna put it separately here because this is linear. And I'm gonna make a column here for angular. Uh, Alfa equals eight. Uh, if the car is initially at rest, Omega initial is also zero. Um, the tires have a radius off 0.4 m, so I'm gonna write it down here that the radius of the tire is 0.4 m. You see that? Yes, you can. And we want to know what is the angular speed of the tires after 10 seconds. So, after 10 seconds, what is Omega Final Four? The tires. Okay, that's part a. So this looks like a motion problem. And it is. I got three motion variables here that are given, and I'm asking for one and one of them is ignored. What's ignored here is the number of rotations. Delta Theta is my ignore variable sad face. This tells me that I should be using the first equation or make a final equals Omega initial plus Alfa Tea or make initial zero Alfa is ate. Time is 10. The answer is 80 80 radiance per second. By the way, nothing new in this question. We've done stuff like this before. The part that's news Part B, Part B were being asked for the speeds at the top center and bottom of the tire. Well, the tire is a tire is a rolling is enrolling motion. Or you can think of it as the tire is a free access or free wheel. It's call it free wheel. So this means that on top of the other equations, we know we're also going to be able to use the three equations that we just learned. So the top will simply be, um, to our Omega. To the Radius is 0.4, Omega's 80. Um, the bottom of the center of mass in the middle is one our omega, and the bottom is zero always. Okay, so if you multiply the top, you get 64. The top is double what's in the middle. So the bottom the middle must be 32. And the bottom is zero. Okay, so that's it for this question. We have to find Omega find, which is old stuff. Then we have to find the top V C. M. And the bottom. And we got them. Okay, let me out of the way here. So you can see numbers. And that's it for this one. Hopefully, this makes sense. Let me know if you guys have any questions.

A long, light rope is wrapped around a cylinder of radius 40 cm, which is at rest on a flat surface, free to move. You pull horizontally on the rope, so it unwinds at the top of the cylinder, causing it to begin to roll without slipping. You keep pulling until the cylinder reaches 10 RPM. Calculate the speed of the rope at the instant the cylinder reaches 10 RPM.