13. Rotational Inertia & Energy
Types of Motion & Energy
Types of Motion & Energy
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Hey, guys. So in this video, we're gonna talk about how different kinds of motion give you different kinds of energy. And I'm gonna walk you through a comprehensive list of all the possibilities you might see, so that for any kind of problem, you always know what kind of energy goes with the problem. What kind of energy exists in that situation? Let's check it out. So what we want to do here is make sure that we know which energies go with a particular situation. Now. A potential problem arises when you have point masses. And that's because point masses, if you remember, if they're going to circular path, they have rotational speed. So if you have a tiny little mass here, M and it's going around a path here, um, it has a distance Little arts of the middle, right? And let's say it's spinning that way with an angular speed W. But it also has a tangential speed, which is linear. It's also has an instantaneous speed that's pointing this way. We call this V tan, Okay, so but does that mean, however, does that mean that it has lingered kinetic energy and rotational kinetic energy it has a W. So does it have a kinetic rotation? Rotational, kinetic energy? It has a V. Does it have a linear? Kinetic energy doesn't have both, and the answer is no. Nope, it doesn't have to energies. We only have here one type of motion, so we can only have one type of energy. The object only has one type of motion. It only spins around a central points. Okay, V. Tan is just the linear equivalent of W. It's think of. It is like a mirror image, right? If you look in a mirror, there aren't two of you. It's just the mirror is a reflection of you. So V 10 is just the linear reflection of W. But there's only one velocity on Lee. One motion, one type of motion, I should say, um, eso if that doesn't make sense yet that's cool. We're gonna do six examples, and that's gonna cover every possibility. So let's let's start here. So you have a box in a straight line? Does it have linear, kinetic candidates that have rotational? Kinetic candidate does have both, um, so a box in a linear in a straight line. So something like this The boxes moving. It has a V. So he has a linear, kinetic energy. The box doesn't roll around itself for around anything else. So he has no kinetic, no rotational energy Onley linear A disks spinning around itself. So a disk. Here's the axes in the middle. The disk spins around itself. Does it have kinetic linear? Does it have kinetic rotational? Every time you spin around yourself, you have kinetic rotational. Now linear has to do with you moving sideways or up and down your axes of rotation. You should say your center of mass the middle of the object has to actually move. If you spin around yourself, the middle never moves. So there is no linear kinetic energy in this case. Okay, um, what about the earth spinning around itself? So this is a disc. The earth is a sphere roughly right. And if it spins around itself, it's very similar. There's no kinetic linear, and there is kinetic rotation. There's kinetic rotation because of a shape around itself. But if you're talking about the earth around itself, just that part of the earth's motion as the Earth spins around itself, it doesn't move sideways. Now you do know the Earth does move around the sun. But that's a different motion here. We're talking about just this piece of it now. What about the earth around the sun, the earth around the sun, The center of mass of the earth does move right. The center of mass of the earth does move around the sun. So let's draw that real quick. The earth is doing this now. This is where it gets complicated because you could think of this as well. The earth has w around the sun has a w around the sun, but it also has an instantaneous velocity V So you could think of this as linear or rotational. In fact, if you solve for for it using K l. And if you solve for it using KR, you're going to get the same number, the same answer. Okay, so you could look at the energy Either way, The problem is, you have to make sure you don't count that it's both. What I mean by that is if I ask you for the total energy of the earth going around the sun. So the kinetic energy of the earth around the sun, You can't do K l plus K R. Okay, you can't double counted. All right, so here's how I'm gonna simplify this. I'm gonna say, whenever you have an object spinning around itself or around something else, we're going to call that rotational kinetic energy. And we're going to say that there is no linear energy, right? I just mentioned how you could look at it both ways because there's a V and w you just can't count it as both. Well, we're going to forget about that. We're just going to make our life simpler and always think of it as rotational energy and never linear energy. Okay, even though you do have a linear velocity going around this thing, so I hope that makes sense around itself around the sun. Now, what about the total energy of the earth? I'm gonna add a little thing here. I'm gonna call it. I got C and deep. I'm gonna call this C D. What about the total kinetic energy of the earth Kinetic total of the earth, meaning the kinetic energy of the earth around itself, plus the kinetic energy of the earth around the sun. Well, both of These are rotational. The earth has a rotational energy around itself and he has a rotational energy around. Um, the sun kr, son. Okay, so rotational is if you spin around yourself or you spit if you spin around something else. What about the moon spinning around the earth? What's the total? What kind of energy does the moon has spinning around the earth? So here you have to know that the Earth the moon doesn't spin around itself. Okay? On Dhere, by the way, I want the total energy, the total energy. And what I mean by that is I wanna know. Does the Earth spin around itself? And as it does the moon spin around itself and does it spin around the earth? So moon goes around the earth. What's what happened? The moon goes around the earth. Let's just do it like this. But you might know you should know that the moon doesn't spin around itself. So the moon on Lee has k wrote K Moon as it's going around the Earth the que moon self plus K moon Earth. The moon spins around the earth, but it doesn't spin around itself, right? And that's because the moon is locked with the earth. Okay, so the moon is always spinning facing the same side to us. That's why there's an expression called the dark side of the moon. And it's not really dark. It just means that we can never see it, because the moon is always looking at us, right? It's like if you look in the mirror, you can't see your back. You can only see your front. Okay. Um, so you should know that the moon doesn't spin around itself, so it only has a rotational energy around the earth. Now, what about a roll of toilet paper rolling on the floor? Right. So you got a toilet paper? It is rolling on the floor. It's running loose, right? Um, it has a V and it has a W. But here it actually has two types of motion. Not only it spins around itself, but it's also moving sideways. So it's doing this. This is called rolling Motion. And whenever you have rolling motion, you have two types of energy. So the total kinetic energy is going to be linear plus rotational. So the object actually has both types of energy. This is the only case where you have out of the six that I've mentioned here, where you have linear and rotational all the other cases, you have either linear or rotational, but not both. And in the case of the earth, going around the sun and spinning around itself, it has to rotational kinetic energies. One around itself, one around the sun. Who so that's it. I hope this makes sense. This basically covers every possibility. So you should be rocking from here on. All right, so if you have any questions, let me know.
Kinetic Energy of a Point Mass
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Hey, guys, in this video, I'm gonna show you how there are two ways to calculate the kinetic energy of a point mass going around a circle. Let's check it out. All right, So remember, if you have a point mass around a circle under in a circular path looks kind of like this. Around a distance of little are from the axis of rotation. You have rotational speed, omega, and you also have a linear equivalent, which is your tangential velocity. Okay. All right. Um, but you only have one type of motion. All you're doing is this Okay? You're on Lee Motion is really rotational motion. Um, your only emotion is rotational motion. So you only have one type of kinetic energy. Okay, But you can calculate using K, l or KR. You can use thean question for linear or for KR, and that's because these two equations as I'm gonna show you now or equivalent, Okay, the most important thing to do here is to make sure you don't double counting. Okay? When I ask you for the total kinetic energy of an object, you can't, uh, point mass like this. You can't look at it and say, Well, it's got a V. So it has a linear kinetic energy, and it has a w. So it has a rotational kinetic energy. It's got to kind of energies. Let's add the two of them together. You can't do that because these guys are equivalents, right? The tangential velocity is basically a mirror of W. It doesn't mean there is to. It just means that one basically reflects the other. Alright, so you can't do is double count. So let me show you how this works. Um, a small 2 kg objects, so mass tickles. 2 kg is going around with a rate of going around the vertical axis. So what is the vertical axis? Remember? Access. You can think of it as a, uh, imaginary line that you spin around. The vertical axis would look like this. So it means the object is going around like this. Okay, like this. Cool. So it would actually I could draw it like this. And the object is doing this, okay? And it does this in a rate of three radiance per second, maintaining a constant distance of 4 m to the access this distance to the axis is what we call little. Our little are 4 m and I wanna know the objects kinetic energy. And I want I want to do this using the Kael equation the KR equation. And the purpose of this question is to show you how the answer ends up being the same. And I'm gonna simplify. I'm gonna summarize it at the end so we can do que el We could do Que el, which is going to be half M V square. Okay, remember that thes two V and are are related right V and are related by V equals R Omega. So what I'm gonna do is also right. K r equals half I omega, and I'm gonna rewrite one of these questions of one of these equations and you're gonna notice how it's gonna look exactly like the other. So let's rewrite this one here half, Remember, I for a point Mass is m r Square someone. I replaced this with m r Square and I can rewrite omega as well. V equals R omega so omega equals V over R. So instead of omega here, I'm gonna put V over R. Now look what happens. This r squared cancels with this are square and we're left with half M V squared, which is exactly this equation. Okay, so you can go from one to the other four. A point mass. You could do this. Which means I could have calculated them either way. All right, so if I go here, K l equals half M V squared. Let's get these numbers. W equals three v equals R w So w equals V over R the I'm sorry, I'm trying to get V so v equals R four W three v is 12. So this is half mass is too. They canceled 12 squared. So this is 1 40 for Jules. Cool. And if I wanted to do it using K R. I already showed you how the equations turned out to be the same. Now, I'm just gonna plug in numbers differently. So if I wanted to do it this way, I could have done half m r squared omega squared, half right, which is this Half the mass is too. And the distances four square and the W is three squared. So these two cancer, I have 16 times nine, which is 1 40 for Jules. Okay, so If you calculated using linear, it's 1 44. If you calculate using rotation is 1 44. And if I ask you what is K Total? Thea answer is 1 44. Okay. And I want you to please right here, not to 88. You do not add the tude. You could get the same answer using the two different equations. Now, to make this simpler for you, I have a convention. I always think of a knob checked going around the circle like this. It has one motion. I always think of this as linear motion. I always I'm sorry. Rotational motion, not linear. So I would always do it like this K l plus K r. And I would say there's no que el There's Onley kr, and this will guarantee that you don't double counting. Cool. So this is just a potentially tricky thing. But once you understand, you get it out of the way. It's never gonna bother you again. Cool. Let me know if you any questions
The Earth has mass 5.97 × 1024 kg, radius 6.37 × 106 m. The Earth-Sun distance is 1.5 × 1011 m. Calculate the Earth’s kinetic energy as it spins around itself. BONUS:Find the Earth’s kinetic energy as it goes around the Sun.