13. Rotational Inertia & Energy
Intro to Moment of Inertia
Intro to Moment of Inertia
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Hey, guys. So in this video, we're gonna talk about moment of inertia, which is really just the rotational equivalent of mass. So in linear emotion, you have mass in rotational motion. You have moments of inertia. You can think of moment of inertia as rotational mass or rotational inertia. The name is tricky moments of inertia. It's sort of a name from engineering, But don't get confused. Just rotational mass. Let's check it out. So you might remember that when we solved motion problems, which were those problems we solved with the 3 to 4 motion equations or kidney Matic equations? Um, those equations didn't have a mass in it, right? So they did not depend on Mass. But once you moved into energy problems or force problems, dynamics problems, um, mass was important. So here's kinetic energy is half MV squared gravitational potential Energy has a mass force F equals in May that has a mass as well. So before we can talk about energy in rotation and forces in rotation, we have to talk about mass in rotation because it's a little different again. Linear motion, mass and rotational motion. We have this new thing that we're gonna talk about. I want to quickly remind you that mass is the amount of resistance to linear acceleration. Linear acceleration is a mass. M is resistance to a and we call that inertia. When you resist acceleration, you're resisting change in velocity. Resisting change is inertia. Inertia means you want things to keep going the way they're going. You want to resist change? I can show you real quick how this looks like some of all forces equals in May. Um, I can rewrite this as a equals sum of all forces over em. So notice that the greater my mass, the smaller my a right. So the more mass I have the mawr resistance I have. So the more mass I have, the mawr resistance I have. And remember, resistance is inertia. The more mass, the more inertia. So we say that mass is the quantity of inertia. This is old stuff, masses, the quantity of inertia. Well, it's gonna be the same thing in rotation in rotation. The only difference is that in rotation, the amount of resistance depends on mass and depends on something else. In linear motion Depends all your mass, but in rotation depends on Mass, and it also depends on distance to axis. So if on object spins, um, at a distance off 10 it's gonna have a different resistance than if it spins at a distance off 20 from its axis of rotation. Okay, so if you're going like this, you have less inertia, then if you're going like this, Okay, so this combination of mass and distance to access is what we call moments of inertia, and it's the amounts of rotational inertia. You have a moment of inertia takes the letter I you can think of. This is just inertia, right? And it's the rotation, no equivalent of mass. Cool and again, you can think of it as rotational inertia or rotational mass cool. So depending on the kind of problem you have, if you have motion linear motion or any kind of linear problem, you use massive any kind of rotational problem. You're going to use rotational mass, which is called moment of inertia. Cool. There's two types of objects and two types of problems. You can see you can have point masses again. A point mass is a tiny little object that goes around a circle of radius um, are. And we're going to say that the mass itself has no radius. Okay, so remember this distance here is little our little ours. Distance. Um, big R is the radius, so it's a tiny object that doesn't really have a radius that doesn't have a volume. Usually you hear something like a small object andan. The other type is when you have a shape or a rigid body, these air like a cylinder with the radius or something. So the radius here is not zero. And the reason I put shape is because it's usually going to the problems. Usually they tell you what kind of shape this is. It's a solid cylinder. If it says it's a solid cylinder, you know it's one of these guys and not a point mass quote. Now, if you have a point mass theme Moment of inertia is given by an equation, which is M R. Squared M is the mass of the object and are again, is the distance to the axis of rotation. Okay, distance to access. All right. And if it's a rigid body, I will be given by a table. Look up a table. Look up. What do I mean by that? Well, your textbook has a table off moments of inertia, and it's gonna say for a solid cylinder theme Moment of inertia, for example, is let me write it here for a solid cylinder. The moment of inertia is half m r square, so this will be given to you. Most professors don't require you to memorize this. They'll give this to you in some way. All right, So look to your book. Find the table. It's got some pretty shapes. Um, something like this. I pulled this from Wikipedia. It shows you the shape on git shows where it's rotating. Okay, I want to point out that if you spin here, if you spend this object at the end of the objects, so imagine that you're spending sort of like here, right at this edge right here. So it's doing this. It's different than if you're spinning in the middle like this. Notice how these evacuation equations air different. This is 1/3 m. L squared, and this is 1/12 ml square. It's because for a moment of inertia where you spin matters Okay, um, one last point here is that the most moments of inertia will follow this general form. Here it will be some fraction, like half or two thirds or 2/5 or whatever M r squared. In this case, this is a thin rod. So what matters in the Rod is not the are not the radius of the Rod because it's very fence. It's small, it's negligible, but it's the length of the rod. But even then you see that it's instead of m R squared ml squared. So you should expect, um to see something like that. Okay, so we know this quick example here. So you see how this works and we'll practice problem. Um, it's a system is made of two small masses. The one on the left right here. It's this guy has a mass of em. Let's call it M left equals three and then mass in the right. Right here. Em right equals four, and they are attached to the ends of a 2 m long, thin broad. So it's this guy right here. I'm gonna write it like this. Uh, length equals 2 m. That is massless. So it's a thin rod that has no mass at all, and we wanna calculate the moment of inertia of the system if it spends about a perpendicular access to the center of the rot. There's a lot of words here, and you're gonna get used to this. But I'm gonna start slowly here. Okay? So I wanna know the moment of inertia. So I equals question mark of the system. Let me put a little system here. Moment of inertia, of the whole system. Um, if it spins so it's spinning about a perpendicular axes. You're gonna see this all the time. Perpendicular axis perpendicular means 90 degrees. Perp equals 90 degrees. This isn't Remember the symbol for perpendicular cool eso? What does it mean? That it's a perpendicular axis? Well, here's the rod. Right? Perpendicular axis means that it's making 90 degrees of the rod. So it looks like this cool like that's now. This is also perpendicular because it's also making 90 degrees. So sometimes it's hard to tell which ones have to be careful. So it says perpendicular access through the center of the rod. So it means it's perpendicular, makes 90 degrees, and it goes like this, you could have a perpendicular axis, or you can have a parallel axis Parallel axis would look like this. It would go with the Rod. But then the Rogers is spinning around itself, and that doesn't do anything. Okay, so the access will be like this through the middle, which means the rod is spinning around itself like this. Okay, so it's a very visual, uh, chapter, A very visual topic. Um, so I'm gonna draw it like this. And the idea is that this guy is spinning around itself like this. Okay, The moment of inertia of a system is the sum of the individual moments of inertia. Okay, so we have three objects, but the Rod has no mass. And look at the moment of inertia. Moment of inertia is either half m r squared or if you are shape. It looks like this. Both of these guys have masses. What it means is that if you both of them require masks. So if you have no mass, you have no moment of inertia. Moment of inertia is rotational mass. If you have no regular, massive, no rotational mass either. So what that means is that we're only gonna really have two guys here. I left. Plus I right. Okay, so now we're gonna expand this. What is what goes here and what goes here. So what goes here and what goes here? Now you have to decide, or you have to figure out these thes masses a tiny point mass, or they sort of a bigger shape. And here it says to small masses, small is a hint that these are point masses PM points Masses PM, Which means the equation is m R squared. OK, so there's two clues here. One, It's a small that's a dead giveaway. And the other one is it didn't indicate a shape. So if I tell you a small solid cylinder, it's still a rigid body because I said that it was a solid cylinder. I gave you the shape here. I tell it small and I don't give you a shape. It's a point mass. Okay, so what we're gonna right here is Emma of the Left, um, are left square because that's the equation. M r Square. Something here, Emma R squared. But this is for the rights. Okay, So the masses are a three and a four, three and four, so I like to set it up Oops. This is a two right there. I like to set it up this way. What I've done is I've written the mass here mass here, this mass this mass. And I've left a space for us to plug in the ours. Okay. I left the space for us to plug in the arse. This is where you have to slow down. Make sure you find the right number. R is the distance between the object and its axis of rotation. So it's not the to are is this so this is our for the right ball or write object. And this is our for the left object. The distances are one for both because it sits right down the middle, so it's gonna be one in one. Okay, One squarish one. So the answer here is simply seven. Now, let's talk about units because we haven't done that yet. Um, if you look at, I equals m r squared, which is the eye for a point mass. Um, the units are gonna be kilograms because of the mass and meter squared because of the distance squared distances in meters. Okay, so it's gonna be 7 kg meter squared, all right. Um, that's it for this one. We I want to point out that we actually didn't use this table. Right? We had a rod here. We didn't reuse this table because this rod didn't have a mass. If it had a mass, you would have used this equation right here. Okay, You would have used this equation here. I'm gonna right here, but no mass. Sad face, so we didn't get to use it. But you would have used this one because it's spinning around its middle middle point. Cool. That's it for this one. Let's keep going.
A system is made of two small masses (MLEFT = 3 kg, MRIGHT = 4 kg) attached to the ends of a 5 kg, 2-m long thin rod, as shown. Calculate the moment of inertia of the system if it spins about a perpendicular axis through the mass on the left.
Moment of inertia of Earth
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Hey, guys. So, as you know, the Earth is rotating. Therefore, it has a moment of inertia. And if we make some assumptions about the shape of the Earth, we can actually calculate the moment of inertia. The earth. Let's check it out. So it says here the earth has a mass and radius given by these big numbers. And then I also tell you that the radio distance between the earth and the sun is this What I mean by radio distance is that, um if the sun is here, um, the earth spins around the sun in this distance here, little are is 15 times to the 11. Squeeze it in their cool. That's what I mean by that on. Then I gave you the mass of the Earth and the radius of the Earth as well. I want to know the moment of inertia of the earth as it spins around itself. And as it spins around the sun. As you know, the Earth has two motions on beacon. Calculate a moment of inertia about or relative to those two motions or for those two emotions, remember, Moment of inertia depends on the axis of rotation. That's why these numbers will be different. So if you want to know the moment of inertia of the earth around self, you would have to treat the earth as a zey as an object with a significant size you can't treat it is a tiny object. Eso What we do here is we're gonna treat the earth as a solid sphere. Okay, It's a solid sphere. So the earth is a big ball that spins around itself. Now, technically, it's at an angle like that, but it doesn't really matter. You could just do this. Okay, so it's spinning around itself, and your book would show you that solid spheres have a moment of inertia given by this equation right here. So when I tell you, solid sphere, I'm indirectly telling you Hey, use this equation for I, Okay, so for part a, we're going to do Party's over here. We're gonna say I equals to over five m r squared, and all we gotta do is plug in the numbers here. So M is the mass of the earth, which is 5.97 times 10 to the 24th, and R is the radius of the earth, which is this and not the radio distance. It's the earth going around itself. So it's the radius of the actual object to the sphere. Um, 6.37 times 10 to the six square. Okay, if you look at this number, I got a 24 and then I got a six squared. So you should imagine that this is gonna be, ah, gigantic number. And it is. I multiplied everything. I get 9.7 times 10 to the um, kilograms meters square. The earth has a lot of inertia. And what that means is that it would be incredibly hard to make the earth stops spinning. Okay, now, if you were to Google this number, you would see that that's actually a little bit off. The actual moment of inertia is a little bit off, and that's because the Earth is not a perfectly a perfect sphere. It's got different layers. It's not even sphere. Um, so but this number is a pretty good approximation for part B. We want to find out what is the moment of inertia of the earth as it spins around the sun. Now, in this case, relative to the sun. The earth is tiny, so we're gonna treat it as a point mass, which is crazy. The earth is huge thing and you're gonna just treat it as a little point massive, negligible radius. And that's because relative to the sun, the earth is negligible in size. Okay, so I'm gonna put the earth here as a tiny M Earth. Um, and it's going around the sun and the distance here. The radial distance, which is little, are big r is radius of an object and little artist. Distance to the center is 1.5 times 10 to the m. In this case, we're going to use instead of 2/5 m r. We're gonna use em are square because the earth is being treated as a point mass. Here M is the mass of the object itself, right? It's the object that spinning It's not son s. So it's gonna be 5.97 times 10 to the 24th same thing, But our is going to be the distance to the center, which is 1.5. So 1.5 times 10 to the 11th square. I got a 24 I got 11 square. This is gonna be again a gigantic number 1.34 times 10 to the 47. This number is, um, like, a billion times bigger than the other number. Right? So, as hard as it would be to stop the Earth from stopping to get the earth to stop spinning, Um, it would be way harder, right? It would be 10 to the 10 times harder to make the earth stops going around the sun. Um, and that's it. So that's a finished ones. Very typical classic problem. Hopefully, it makes sense. You should try this out on your own and let's keep going.
A solid disc 4 m in diameter has a moment of inertia equal to 30 kg m2 about an axis through the disc, perpendicular to its face. The disc spins at a constant 120 RPM. Calculate the mass of the disc.
Inertia of planet of known density
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Hey, guys. So here's another pretty straightforward moment of inertia. Question three Only difference here is that we're gonna have density to deal with. So let's do this real quick because we have a planet that is nearly spherical. Nearly spherical means that it's a sphere. Okay, so you can basically ignore the word nearly. It means we're going to approximate it as a sphere with nearly continuous mass distribution again, you can ignore the word nearly and assume that it has continuous mass distribution, continuous mass distribution. We'll talk more about this later, but it basically means that it's fear has the mass evenly distributed throughout the sphere. So you sometimes you see drawn like something like this. This'd is a solid sphere, and this is opposed to a hollow sphere, which is a sphere that has nothing inside. It's not continuously or evenly distributed. All the mass is concentrated on the edges. This is not what we have here. This is what we have here. The reason why that's important is because you're gonna get a different I e. Equation a different moment of inertia equation, depending on what kind of sphere you have and the moment of inertia equation for this guy here is to over five m r Square. The question didn't give you this, but you would look this up in your book or on a test. He would have to give you this somehow. Unless you're Professor requires to memorize this thes, and then you have to do that. But most of them don't. All right, so that's the equation you're supposed to use. I give you the ratings right here. Radius is eight times 10 to the 7 m, and I get to the density density you can use D. But the official variable, if you will, is row right. It's 10,000, um, kilograms per cubic meter. I want to remind you that if you have a volume, the density of the volume is mass over volume. And you could have seen this from the units. Kilograms, um, cubic meter cube. Right. So it's a volume. It's a three dimensional objects. And and that's it. That's all we're given. I also give you hear the equation for volume of a sphere volume right here of a sphere. Now, if you were looking for I and if you start plugging stuff in here you would realize you don't have em, but you have are okay, so we don't have em. We gotta figure this out. And if you look around, you realize, Well, I have another piece of information that has some connection to em. So maybe I can use this to solve for M, and that's exactly what we're supposed to do. So I wanna find em. I have 10,000, but I don't have the. But once again, I have another piece of information here that allows me to find V V equals the volume of the spheres. Four thirds pi r cube. And I know our I know are So I can get the I'm gonna novi. So I'm gonna be able to get em. I'm gonna know em. So I'm gonna be able to get I Okay, That's how it's gonna That's how it's gonna flow. All right, So what I'm gonna do is right here. I'm gonna solve for em. In other words, I'm gonna move you over here, so M equals 10,000. The and V is according to this equation right here. Um, four thirds pi r cubed. So I'm gonna get this whole thing, which is M. And I'm gonna stick it in here. Okay, so to over five times 10,000, four thirds pi r cute. This is just m And then I also have the R Square here. Tons of ours. Okay. And if you multiply this whole crap you're gonna end up with Let's see, this is 18. Um, I'm sorry. This is eight. So this is gonna be 80, divided by 15. So it's 80,000 pie. I got that right. Yep. Divided by 15 times are to the fifth. These two guys combined are to the fifth. So it's going to be eight times 10 to the seven. This thing to the fifth. Okay, So you should expect Ah, pretty big number. And I got 3.15 times. 10 to the 45. Now, how exactly you arrive at this number? Um, doesn't really matter its eyes. So it's kilograms meters square. You could have, you know, gotten a number here plugged in here. It really doesn't matter as long as you arrived here. It's a bunch of multiplication. Cool. So that's it for this one. Let me know if you have any questions.
Additional resources for Intro to Moment of Inertia
PRACTICE PROBLEMS AND ACTIVITIES (3)
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