Anderson Video - Uniform Circular Motion

Professor Anderson
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>> Hello, class. Professor Anderson here. Let's talk about uniform circular motion, and let's start this discussion by asking you guys a question. Are you accelerating right now? Sitting there in your chairs, rapt attention, probably glued to your chairs because of gravity, not because of the excitement of this lecture, but maybe. I don't know. Are you accelerating right now? Yeah? I see some heads nodding. Why? If you have a thought, raise your hand. Yeah. >> Because we're orbiting the sun. >> Because you're orbiting the sun, okay. The Earth is, of course, orbiting the sun, and therefore we are moving in a circle, but we're also spinning on our axis, right? The Earth is spinning, and so we're moving in a circle here as we are trapped on this Earth, and I would say that yeah, we are definitely accelerating because if you are moving in a circle, uniform or non-uniform, you are, in fact, accelerating, okay. So does it feel like you're accelerating right now? No, not really. It feels like we're stuck on this big ball of dust, right, and we're not moving up or down. It doesn't really feel like we're accelerating, and that's because the Earth is pretty big, okay. The Earth is pretty big, and for its size, its not spinning around that fast, right. We're actually moving about a thousand miles per hour right now, but for the size of the Earth, that's not that fast. When you get on a small merry-go-round and you spin around on a merry-go-round, you definitely feel that acceleration. All right. So let's talk about uniform circular motion, okay. Uniform circular motion is just this. Something is moving in a circle of radius r, and it's moving at a speed v. Uniform means constant speed. [ Writing on Board ] Okay. Later, we will learn about non-uniform circular motion, which just means that the speed changes as you go around, but for now, let's just consider uniform circular motion, which is constant speed. Okay. That's what our picture looks like. We just said that we think we are accelerating, and we can probably convince ourselves pretty quickly that that is true because acceleration is delta v over delta t, and delta v is the change in velocity. Is velocity a vector of a scalar? It's a vector. [ Writing on Board ] So if you have a delta v and it's non-zero, that could be due to change in speed, which is the magnitude of velocity, or it could be change in direction. Okay. When you're driving down the road and you turn your wheel to the left, you feel that acceleration because you're changing the direction of your car. You're not changing the speed of your car, especially if you don't hit the brakes. You're just changing the direction. Okay. So how does this lead to an acceleration? How does changing the direction lead to an acceleration? Well, the way we're going to visualize it is the following. Here is a section of our circle. Here is a radius vector, r, and we're going to identify this angle that you traverse as delta theta, and let's say that we're moving in this direction. [ Writing on Board ] Okay. Your velocity is always tangential to the circle in uniform circular motion, so the initial velocity is there. The final velocity is there. We have clearly changed directions for a velocity, and so that is going to lead to an acceleration. [ Wiping the Board ] Okay. So let's take a look at this picture in a little more detail. All right. Where did I start? I started at a position there, r initial. I ended at a position there, r final, okay. Vectors for position, they always define relative to some origin. We'll make the origin the center of the circle. The difference between those two is, of course, right there, okay. So what about the velocity vectors? Well, it looks like the velocity vector vi is up and to the right, but vf is down and to the right, and if I do the difference of those two, remember you can do your tip to tail approach by flipping the sign of one of them, but you can probably convince yourself pretty quickly that delta v is going to look like that. Okay. I was going up. I've changed to going down, and so the delta v has to be pointing down. All right. But these are just magnitudes now. The radius of the circle doesn't change. That side is just delta r. If we are in uniform motion, the speed doesn't change. That side of the triangle is just delta v. Both of these are making an angle delta theta. All right. This is the way we define it. How do you see this? Well, this is always a right angle, and that's always a right angle, so if I add 90 degrees to vi and I add 90 degrees to vf, they both are going to traverse the same delta theta. Okay. Let's go back to our acceleration. [ Wiping the Board ] We said that acceleration was delta v over delta t, but if I look at these pictures now, those triangles are similar, and if they are similar triangles, then I can write the following. Delta v over v is exactly the same as delta r over r, and now I can solve this for delta v. I get delta v is equal to v times delta r over r. I just multiply it across by v. And now let's plug that into a. A, the magnitude of a, we said was delta v over delta t, and here is my delta v, and so I can plug that in. I get v over r times delta r over delta t, which I carried in from there, but delta r over delta t -- that's just how far you've gone in some amount of time, which is, again, just your speed. Delta r over delta t is just v, and so we get what we now call centripetal acceleration, and it is a equals v squared over r, and we typically put a little subscript on it, either c or r. I like to use r for radial. Some people use c for centripetal. But what this means is anytime you're moving in a circle, you have a centripetal acceleration towards the center of the circle which has this magnitude: v squared over r. Okay. That's how fast you are accelerating towards the center of the circle. So let's go back to our original question, which was, "Are you accelerating right now?"
>> Hello, class. Professor Anderson here. Let's talk about uniform circular motion, and let's start this discussion by asking you guys a question. Are you accelerating right now? Sitting there in your chairs, rapt attention, probably glued to your chairs because of gravity, not because of the excitement of this lecture, but maybe. I don't know. Are you accelerating right now? Yeah? I see some heads nodding. Why? If you have a thought, raise your hand. Yeah. >> Because we're orbiting the sun. >> Because you're orbiting the sun, okay. The Earth is, of course, orbiting the sun, and therefore we are moving in a circle, but we're also spinning on our axis, right? The Earth is spinning, and so we're moving in a circle here as we are trapped on this Earth, and I would say that yeah, we are definitely accelerating because if you are moving in a circle, uniform or non-uniform, you are, in fact, accelerating, okay. So does it feel like you're accelerating right now? No, not really. It feels like we're stuck on this big ball of dust, right, and we're not moving up or down. It doesn't really feel like we're accelerating, and that's because the Earth is pretty big, okay. The Earth is pretty big, and for its size, its not spinning around that fast, right. We're actually moving about a thousand miles per hour right now, but for the size of the Earth, that's not that fast. When you get on a small merry-go-round and you spin around on a merry-go-round, you definitely feel that acceleration. All right. So let's talk about uniform circular motion, okay. Uniform circular motion is just this. Something is moving in a circle of radius r, and it's moving at a speed v. Uniform means constant speed. [ Writing on Board ] Okay. Later, we will learn about non-uniform circular motion, which just means that the speed changes as you go around, but for now, let's just consider uniform circular motion, which is constant speed. Okay. That's what our picture looks like. We just said that we think we are accelerating, and we can probably convince ourselves pretty quickly that that is true because acceleration is delta v over delta t, and delta v is the change in velocity. Is velocity a vector of a scalar? It's a vector. [ Writing on Board ] So if you have a delta v and it's non-zero, that could be due to change in speed, which is the magnitude of velocity, or it could be change in direction. Okay. When you're driving down the road and you turn your wheel to the left, you feel that acceleration because you're changing the direction of your car. You're not changing the speed of your car, especially if you don't hit the brakes. You're just changing the direction. Okay. So how does this lead to an acceleration? How does changing the direction lead to an acceleration? Well, the way we're going to visualize it is the following. Here is a section of our circle. Here is a radius vector, r, and we're going to identify this angle that you traverse as delta theta, and let's say that we're moving in this direction. [ Writing on Board ] Okay. Your velocity is always tangential to the circle in uniform circular motion, so the initial velocity is there. The final velocity is there. We have clearly changed directions for a velocity, and so that is going to lead to an acceleration. [ Wiping the Board ] Okay. So let's take a look at this picture in a little more detail. All right. Where did I start? I started at a position there, r initial. I ended at a position there, r final, okay. Vectors for position, they always define relative to some origin. We'll make the origin the center of the circle. The difference between those two is, of course, right there, okay. So what about the velocity vectors? Well, it looks like the velocity vector vi is up and to the right, but vf is down and to the right, and if I do the difference of those two, remember you can do your tip to tail approach by flipping the sign of one of them, but you can probably convince yourself pretty quickly that delta v is going to look like that. Okay. I was going up. I've changed to going down, and so the delta v has to be pointing down. All right. But these are just magnitudes now. The radius of the circle doesn't change. That side is just delta r. If we are in uniform motion, the speed doesn't change. That side of the triangle is just delta v. Both of these are making an angle delta theta. All right. This is the way we define it. How do you see this? Well, this is always a right angle, and that's always a right angle, so if I add 90 degrees to vi and I add 90 degrees to vf, they both are going to traverse the same delta theta. Okay. Let's go back to our acceleration. [ Wiping the Board ] We said that acceleration was delta v over delta t, but if I look at these pictures now, those triangles are similar, and if they are similar triangles, then I can write the following. Delta v over v is exactly the same as delta r over r, and now I can solve this for delta v. I get delta v is equal to v times delta r over r. I just multiply it across by v. And now let's plug that into a. A, the magnitude of a, we said was delta v over delta t, and here is my delta v, and so I can plug that in. I get v over r times delta r over delta t, which I carried in from there, but delta r over delta t -- that's just how far you've gone in some amount of time, which is, again, just your speed. Delta r over delta t is just v, and so we get what we now call centripetal acceleration, and it is a equals v squared over r, and we typically put a little subscript on it, either c or r. I like to use r for radial. Some people use c for centripetal. But what this means is anytime you're moving in a circle, you have a centripetal acceleration towards the center of the circle which has this magnitude: v squared over r. Okay. That's how fast you are accelerating towards the center of the circle. So let's go back to our original question, which was, "Are you accelerating right now?"