Concept: Rolling Motion (Free Wheels)10m
Hey guys! In this video, we're going to start talking about rolling motion which is the motion of a rigid body, some sort of disc-like or wheel-like object that not only spins around itself but also moves sideways. So that's called rolling motion. Let's check it out. Alright. So rolling motion, I also like to think of this as freewheels is what we're going to talk about. 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. Imagine sort of a cylinder that is free to rotate around its central axis, something like this. Think of this as a fixed wheel. These are fixed wheels. They are fixed in place. In some problems, we're going to have the 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. When I say moving, I mean they're actually moving sort of sideways. They're not fixed in the same place. We're going to think of these as free wheels that's why I called this freewheel. The best example, I think the most memorable example I can give you is actually rolls of toilet paper. If you have a roll of toilet paper that's fixed in place as it normally is, this is a fixed axis. So, here's the roll and it can spin. It has a w but it doesn't move sideways. It doesn't move sideways. I'm going to put no, which means V equals 0. V is when you actually move sideways. If you spin in place, you're not actually moving. In this case, w is not 0 but V of the center of mass is 0. The middle, center of mass in the middle, the middle of the cylinder doesn't move. It stays in place. In the case of a free axis or a freewheel would be if you had a roll of toilet paper that is rolling sort of on the floor. It's doing two things here. It's rolling let's say this way and it's not only rolling this way, but it's also moving. If you combine this with this, you get this. 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. _ is not zero and the velocity of the 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. Luckily this relationship looks like something we've seen a lot of. The velocity, let's say you're spinning with omega here. Imagine that if your wheel is spinning this way, then you're going in that direction. There's a relationship between these two. The velocity of the center mass for a wheel of radius R is simply R_ (omega). Notice how we didn't use little r. I'm going to 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. This looks very similar to what we've seen. If you have a fixed axis like this, the velocity tangential at an edge here or here, these are tangential velocities. These tangential velocities are r_ (little r 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. So, this is the most important thing you need to know. Two other things you need to know is that there's a velocity at the top here and there's a velocity at the bottom. The velocity of the center of mass is R_. You should know that the velocity at the top is going to be twice the velocity at the center of mass. It's 2R_ and the velocity at the bottom is zero relative to the floor. Probably at some point, your book may derive these equations, how to arrive at them. Your professor may derive them. 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 going to draw this again here. The top velocity at the point at the top is 2R_. Velocity in the middle is 1R_ velocity at the bottom is 0R_. So, 0, 1, 2 obviously, this simplifies into R_ and this simplifies into 0. 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 disk that spins around itself is r_ and little r is the distance. Here if you are a little edge at the top here, you are 2r_ because you're moving. The idea is that this r_ here combines with this R_ to give you two of them. I'll just mention that briefly but those are the equations you need to know. Most of the time you need to know, the green one. You don't always need to know the yellow one. The green one is the most important one but I gave you the yellow ones just in case. Alright, let's do a quick example. This is very simple. You just have to remember these three equations. I have a wheel of radius 0.30 cm. I actually meant to make this 30 cm or 0.3, sorry about that. I'm going to say that it has a radius of 0.3 m. It rolls without slipping along a flat surface with 10 m/s. So, it rolls without slipping. If it rolls, it has a _ and it rolls with 10 m/s. The wheel is actually moving. When I give you a velocity here, when I say that Vwheel is 10 and give you the velocity of the center of mass of the wheel. This velocity here, V center of mass equals 10. 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, so you can basically just ignore this equation. Conceptually you may need to know for sort of a multiple choice conceptual test that this is the condition for rolling motion. The condition for rolling motion is that this is without slipping. That's a conceptual point there. Let's go back to this question real quick. I want to know A) what is the angular speed of the wheel. Angular speed of the wheel is simply _. Notice that I know Vcm and I know r and I'm looking for _. This is very straightforward. There's an equation that connects all three of them and it is that Vcm = R_. Therefore, _ is Vcm / R. The velocity is 10 divided by 0.3, _ therefore is 33.3 radians per second. Very straightforward. Part B). The speed of a point at the bottom of the wheel relative to the floor. This is just this V bottom right here and if you know this conceptually, if you remember the equation, V bottom is just always 0 for a rolling wheel no matter what. That's simple. _ is 33 and V bottom is 0. That's it for this one. LetÕs do the next example.
Example: Speeds at points on a wheel4m
Here we have a car that accelerates from rest for 10 seconds. The initial velocity of the car is zero and it takes 10 seconds accelerating. Its tires will experience 8 radians per second. 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. But w has to do with the wheel. If the car is moving that way, the wheel is spinning this way. I'm giving you the acceleration of the wheel, so I'm going to put it separately here because this is linear and I'm going to make a column here for angular. _ = 8. If the car is initially at rest, the w initial is also 0. The tires have a radius of 0.4 meters, so IÕm gonna write it down here that the radius of the tire is 0.4 meters. We want to know what is the angular speed of the tires after 10 seconds. After 10 seconds, what is w final for the tires? That's part A. This looks like a motion problem and it is. I got three motion variables here that are given and I'm asking for one. One of them is ignored. What's ignored here is the number of rotations. __ is my ignored variable sad face. This tells me that I should be using the first equation. w final = w initial + _t. w initial = 0, _ = 8*10, the answer is 80 rad/s. By the way, nothing new in this question. We've done stuff like this before. The part thatÕs new is Part B. Part B weÕre being asked for the speeds at the top, center and bottom of the tire. The tire is in rolling motion or you can think of it as the tire is a free axis or freewheel. 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. Vtop will simply be 2Rw. 2, the radius is 0.4, w is 80. V center of mass in the middle is 1Rw, and V bottom is zero always. If you multiply V top, you get 64. The top is double what's in the middle so the middle must be 32 and the bottom is 0. That's it for this question. We have to find w final which is old stuff, then we have to find V top, Vcm and V bottom and we got them. Let me get over here so you can see the numbers. That's it for this one. Hopefully this makes sense. Let me know if you guys have any questions.
Problem: 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.5m