Introduction to rotational motion
Angular motion variables
- [Instructor] I found that for many people the hardest part about solving a rotational motion problem is just keeping track of all the new names for all the rotational quantities that there are. So in this video, I want to go over all the different rotational variables like angular displacement, angular velocity, and angular exhilaration. We'll explain what they mean, how they're defined, and how you can solve for them, so let's do this. Let's consider this example. Say take a tennis ball, you tie a string to it, and you whirl the tennis ball around in a circle. If you did this and you wanted to start defining motion variables that would describe the rotational motion of this tennis ball, maybe the most basic quantity you'd come up with would just be how much angle has this tennis ball swung through during its motion. So if we imagine the tennis ball starting there and it rotates over to here, we could define a quantity that just says how much angle has this thing gone through. And that would be what's called the angular displacement. And it's given the symbol delta theta, because theta is the angle and delta theta is the change in the angle, so this is really theta final minus theta initial. For instance if we started the tennis ball over here at zero and we ended it at 180, theta final would be 180, theta initial would be zero, so our angular displacement would be 180 degrees or pi radians. And if we started at zero and went through an entire circle all the way, and then another circle all the way, our angular displacement wouldn't be zero. It would technically be two whole revolutions, which would be either 720 degrees or four pi radians. And we don't even have to start at the zero. Our theta initial could be over here at 180, and we'd go down to 270, in which case the angular displacement would be 90 degrees or pi over two radians. So this is how we define the angular displacement and we typically measure it in radians, as opposed to degrees for reasons that I'll show you in just a second. And the name for this symbol here is theta. And we should mention that this is analogous to how we defined the regular displacement, so if you imagine a tennis ball just going in a straight line, the regular displacement was a defined b, the final position minus the initial positions, which we called delta x. And that's just usually called the displacement, which is measured in meters. Okay, so now we know how to quantify the amount of angle that this ball has rotated through, but another quantity that might be useful is the rate at which it's traveling through that angle. Just like up here, knowing about the displacement is good, but you might want to know about the rate that it's being displaced. In terms of regular linear quantities that was called the velocity of the ball, and it was defined to be the displacement per time. So down here we'll define a similar quantity, but it's going to be the angular velocity, which is defined analogously to the regular velocity. If regular velocity is displacement per time, the angular velocity is going to be the angular displacement per time. And the symbol we used to represent angular velocity is the Greek letter omega, which looks like a w, but it's really the Greek letter omega. And the units of omega, angular velocity, are going to be radians per seconds. Since delta theta, the angular displacement is in radians, and the time is in seconds. Just like how regular velocity had units of meters per second, angular velocity has units of radians per second. What is angular velocity mean? What is this omega? It represents the rate at which an object is changing its angle in time. So let's say the tennis ball starts here, and it's going through a circle at this leisurely rate, that means the rate at which it's changing its angle is very small and it has a very small omega. Whereas if you had this tennis ball going through a circle very fast, the rate at which it's going in a circle would be large and that means the angular velocity and omega would also be large. So the velocity and the angular velocity are related, they're not equal because the velocity gives you how many meters per second something is going through, and the angular velocity gives you how many radians per second it's going through, but if it's got a larger angular velocity, it's going to have a larger velocity as well. And just like velocity is a vector, angular velocity is also a vector, so I'll put an arrow over this omega. Which way does it point? Technically speaking, you'd use the same right hand rule you use to determine the direction of the angular displacement. But again if it's rotating counter clockwise, we can just consider that to be positive, and if it's rotating clockwise, we can consider that to be a negative omega, or a negative angular velocity. So let me get rid of these, and let's define our last angular motion variable. You can probably guess what it is. There's regular displacement and there's angular displacement. There's regular velocity and there's angular velocity. And then the next logical step in this motion variable sequence would be the acceleration, which was defined for regular variables to be the change in velocity over the change in time. So we'll define an analogous angular quantity that would be the angular acceleration. And it's going to be defined to be, instead of change in velocity over change in time, it's going to be the change in the angular velocity over the change in time. And the letter we use to denote angular acceleration is this Greek letter alpha, so this is the Greek letter alpha. It looks like a little fishy, and that represents the angular acceleration of an object. So what does this angular acceleration mean? Well, looking at the units, helps us to figure this out, so the units of regular acceleration were meters per second per second, so regular acceleration represented the rate at which the velocity is changing, and that's the same analogous definition down here. The units down here are going to be radians per second per second, so this is going to represent this angular acceleration is going to represent the rate at which the angular velocity is changing. What would that look like? Well if we've got this ball rotating in a circle, if it's rotating at a constant rate, there's no angular acceleration since the omega, the angular velocity wouldn't be changing. So in other words, if it's rotating at a constant rate, there's no change in the angular velocity, and that means there's no angular acceleration. But conversely, if it starts off moving slowly, and it speeds up its angular velocity is increasing, then there is an angular acceleration because there's a change in the angular velocity of this ball. And just like any acceleration, this angular acceleration can increase the angular velocity and speed something up. Or it can slow the object down and decrease the angular velocity. But if the angular velocity is remaining constant, in other words it's rotating in a circle at a constant rate, then the angular acceleration is zero and that means alpha equals zero. And just like the rest of these motion variables, angular acceleration is a vector, just like regular acceleration is a vector. And the direction that the angular acceleration points will be in the direction of the change in the angular velocity. So in other words, if this tennis ball is speeding up, then the angular acceleration is pointed in the same direction as the angular velocity. And if the angular velocity is slowing down, the angular acceleration points in the opposite direction to the angular velocity. At this point, I wouldn't blame you if you weren't like why, why do we need to define all these new angular variables when we already had all these regular variables up here. And the answer is that it's the same reason we define most variables in physics, because it turns out to be really convenient to do so, and these angular variables are going to be way more convenient to describe an object that's rotating than these regular variables. For this reason, imagine you wanted to describe not just the ball on the end of the string, but all points on the string as well. If you limited yourself to only these regular motion variables, you'd run into a problem. You'd realize that his ball goes through a circle in a certain amount of time, but every point on this string also goes through a circumference in that same amount of time, so the velocity of the ball is going to be greater than the velocity of points on the string that are closer to the center. Because everything's taking the same amount of time to go through one circle, but the circle the ball goes through has a larger circumference than the circle that points nearer to the center do. And so all points on this string are going to have a different velocity the closer you get to the center of the string. So trying to describe its motion with just velocity might be a nightmare, whereas if you were just going to use angular velocity, note that every point on the string, including the ball moved through the same amount of angle in the same amount of time. They don't move through the same amount of meters per second, but they do move through the same amount of radians per second because when the ball has rotated through two pi radians, once full circle, every point on this string has rotated through two pi radians. If this ball and string are going to maintain the same shape. So that's the great thing about these angular motion variables, every point on a rigid object is going to have the same angular displacement, the same angular velocity, and the same angular acceleration. It won't matter what point you're talking about. The angular displacement, angular velocity, and angular acceleration will be the same for every point on that rotating object. Alright, so before this gets too abstract, let's try a sample problem. Let's say that the ball starts over here at rest, and it rotates all the way to this point in four seconds. So it started over here at rest, and it took four seconds for it to rotate over to this point. And let's say when the ball makes it over to this side, it's going 1.57 radians per second. Let's say that's the final angular velocity. So let's just go through and try to figure these out. What would the angular displacement be for this example? Well if the ball started here and it made it over to here, the angular displacement would be pi radians, or 180 degrees. What would the angular velocity be? Well it started at rest, so initially omega was zero at this point here, and then finally it tells us what the final omega would be, 1.57. So you might wonder, what would we do with this formula? What if we just used this formula, what would we get? Well if we used that formula there, we're going to get that it went pi radians, and it did it in four seconds, which gives us 0.785 radians per second. So you might be like wait a minute, this omega doesn't correspond to the initial omega or the final omega. What is this corresponding to? Well this would be the average omega. This is the average angular velocity between this initial and final point. This at rest initially shows that the omega started off as zero. The instantaneous omega was zero, and the instantaneous omega, or the final angular velocity would be 1.57, so you got to be careful. The instantaneous values are not necessarily equal to the average value. You can get the average value by taking the change in theta over the change in time, but it doesn't necessarily give you the instantaneous angular velocity at a specific point on the trip. And we can find the angular acceleration as well if we use this formula. The change in omega over the change in time. That would be the angular acceleration, our omega final minus omega initial over time would come out to be 1.57 as our final angular velocity minus zero was our initial angular velocity, and that took four seconds to accomplish, so our angular acceleration would come out to be 0.393 radians per second per second, or you can write that as radians per second squared. Now technically that is also the average angular acceleration during this trip, but if the angular acceleration was constant during this trip, which in almost all cases we're going to look at, the angular acceleration is going to be constant. If that's the case, this would be both the average value and the instantaneous value of the angular acceleration at every point on the trip since the angular acceleration would be remaining constant. So in this example, we can say that the angular displacement was pi radians. The average angular velocity was 0.785 radians per second. The initial angular velocity was zero. The final angular velocity was 1.57, and the angular acceleration was 0.393 radians per second squared. So recapping, the angular displacement represents the angle through which an object is rotated. It's typically measured in radians, and it's represented with a delta theta. The angular velocity represents the rate at which an object is rotating. It's measured in radians per second, and it's represented with a Greek letter omega. And the angular acceleration represents the rate at which an object is changing its angular velocity, so if an object rotates at a constant rate, there is zero angular acceleration, but conversely, if an object's rotation is speeding up or slowing down, there must be angular acceleration. It's measured in units of radians per second per second or radians per second squared, and it's represented with the Greek letter alpha.