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Learn the toughest concepts covered in Physics with step-by-step video tutorials and practice problems by world-class tutors

13. Rotational Inertia & Energy

Energy of Rolling Motion


Energy of Rolling Motion (Surface vs Air)

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Hey, guys. So you may remember that if we have a disc like object, like a cylinder or sphere, and it is moving on a surface while rolling around itself much like a toilet paper, if you throw it on the floor, would do, UM, that motion is called rolling Motions. Got a special name when you have an object on the floor like that, and what's special about it is that it has a both a linear velocity because it's moving the center of mass moves in a rotational velocity because it spins around itself. It's got two motions, so it has linear, kinetic energy and rotational kinetic energy. So let's look into that real quick, all right, so, as I just said, Well, like object rotating and moving around itself is called thes type of motion. Called really motion. You have a V, which is often referred to as the V of the center of Mass, and you have a an Omega because you're rolling around yourself and there are two types of motion here. What's also important about this is that there's a relationship between V. C, M and W, which is V. C M R W and the R is big Our radius. Okay. What that means is that those two variables v n w are linked. So if one grows, the other one has to grow by the same amounts. Now it's important to make a distinction that if you have an object that this Onley happens if you are on a surface. Okay, If you have an object that is rolling on air, these two variables v n w are not tied to each other. They're not tied. V. C m is not tied to W. Okay, so basically means that you cannot use this equation right here. The green equation. This Onley happens if you're rolling on the surface. And if you're rolling without slipping, lucky for you, all problems in physics, at least for you guys, is gonna be rolling without slipping. So you can just assume that to be the case. Okay, So to summarize, if you are rolling on the surface, this equation applies. If you are not rolling on the surface, this equation does not apply. So if you throw a ball and he rolls on the floor, that would apply. But let's say if you throw a baseball and it's spinning through the air and moving. You cannot say that V. C M equals R omega. That equation doesn't work. That's it. So let's do an example here. I have a solid sphere. This is the type of shape I have. Eso that's gonna tell me the moment of inertia and I of a solid sphere. I have it here. I of a solid sphere is solid. Sphere is to over five m r Square. And I'm told that the given that the masses to the radius is 0.3 and it rolls without slipping on a horizontal surface roll without slipping a horizontal surface means that this is called rolling motion. It means that the green equation is going to work. Okay, I'm gonna make the screen to match up with the other green up there with 10. This 10 is my the velocity off the center of mass. Okay, if I tell an object moves a 10 years per second, that's the velocity at the middle of the objects. The question here is let's calculate the linear rotational and total kinetic energy. Let's find the linear First we'll plug in the other one's linear energy is happened B squared. I got all these numbers. Half the masses of to the velocity is 10 squared. This will be 100 Jules. Okay. For kinetic rotational. It's gonna be half I Omega Square. I have I I is gonna be 2/5 m r squared and I don't have omega, but I can get Omega because I have a V. And these guys are related connected by this equation right here. Okay, so let's do that real quick, so v c m equals R omega. So? So omega is V c m. Divided by our V C. M is 10 are is 100.3 eso This guy here will be WB 33. Cool. All right. So I can put w over here. Notice that the two cancels with the two and then I'm gonna have 1/5. The masses to the radius is 0.3 squared and W, which is 10 over 100.3. Now, if you want what you could also do is instead of writing 33 here, I'm gonna actually gonna write this right. If you got a calculator, just put the 33. It's faster, but I'm gonna do 10 over 100.3 squared. And that's because if you notice, this cancels with this, okay? And then I'm left with two times 10 square, which is 200 divided by five. Make sure I'm doing this correctly, So we're invited by five. Yep. So this is 40 40 jewels, okay? And then for the total kinetic energy for total kinetic energy, we're gonna have kinetic linear plus kinetic rotational 100 plus 40 1 jewels. Alright, So that's linear rotational. And the total later, he's got two types of energy. So you had a linear with the rotational notice that they're not necessarily the same. Andi, remember that we can use this equation here because it's rolling on a surface. Cool. That's it for this one. Hopefully make sense. Let me know if you have any questions.

:A 150-g baseball, 3.85 cm in radius, leaves the pitcher’s hand with 30 m/s horizontal and 20 rad/s clockwise. Calculate the ball’s linear, rotational, and total kinetic energy.


Ratio of energies of cylinder on surface

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Hey, guys. So some of these rotational questions we'll ask you to find the ratio off one type of energy over the other. So I want to show you how to do one of these. So here we have a solid cylinder. Solid, similar cylinder tells us that we're supposed to use I equals half and March Square of Mass M and are so it's not actually giving us the numbers. This is going to be a literal question. Or just with letters variables, it rolls without slipping on a horizontal surface. This is called rolling motion, and it means I can use this equation. V. C m equals R Omega. Okay, because it's rolling like this, so v c. M is tied to Omega, all right. I wanna know the ratio of its rotational kinetic to its total kinetic energy. So what you do is you follow what's saying here, and you set up a racial like this. So it's saying rotational kinetic energy. So we're gonna right que rotational. That's the top to total kinetic energy. That's the bottom. So ratio of top to bottom K total, which is K linear plus k rotational. And now what? We're gonna do is we're gonna expand these equations as much as possible. What I mean by expanding is well, what is K r? Stand for K R stands for I half I Omega squared K l is half MV squared and KR is half Iomega Square and we're gonna expand this as much as possible. Meaning we're not gonna stop there. We can replace I with this right here. Let's do another color. I can replace I with this right, and I can also replace Omega with something else. The problem here is I have V's and Omega's. There's too many variables. Whenever you have a V and in Omega usually want to replace one into the other. So the equals R omega And what we're gonna do is whenever we have V or Omega, Um, whenever we have V and Omega, we want to get the Omega to become a V. Okay, so I'm gonna write Omega is V over R. And we're gonna replace this here. The reason we do this so that we have fewer variables, so it's easier to solve this question before I start plugging stuff in. I wanna warn you, you cannot cancel this with this, right? That's not that's not a thing. So don't get tempted to do that. What you can do is you can cancel the haves over here because they exist in all three of these guys here. Okay, so you can cancel the haves and the simplifies a little bit. So we're gonna do now is expand. I I's gonna become half. It's the half from here. Not this half half has already gone. So it's the one inside of the I m. R squared. And remember, we're gonna rewrite W as V over r. And then this whole thing is squared. I don't do the same thing at the bottom, but before I do that, you might notice right away that this art cancels, right? So that's another benefit of doing this thing here. Another benefit of doing this thing here is that it's gonna cause the arse to cancel. Okay, so at the bottom, I have simply MV squared. Plus I, which is half m r squared, and then omega, which is V over R squared and again, just like it did at the top. The arse canceled. Okay, The ours canceled. Let's clean this up a little bit and see what we end up with. Um, I end up with half M v squared, divided by M v squared plus half M. V squared. And you may already see where this is going. There's an M and all three of these, and there's a V in all three of these. So everything goes away and you end up with some numbers left here. So you have half appear. And then this there's a one here, right, that stays there. One plus half. So the masses matter. The volume doesn't matter. I'm sorry. The mass or velocity? Not volume. They don't matter. So all you have to do is do this thing here. Okay? There's two ways you can do this. If you like fractions. You could do with fractions. I'm gonna do that first. So I'm gonna rewrite this as a to over to and then I have 1/2 divided by. I got a two at the bottom here. And then I can add up the tops, the top here. So it's two plus 13 So I have 1/2 divided by 3/2. I can cancel this too. And then I end up with 1/3. If you don't like fractions. One thing you can do with this particular case, you're gonna be right. This, like this or half is 0.5. This is a one. This is a 0.5, right? This is better. Maybe feel a calculator. 0.5 divided by 1.5. And if you do this in the calculator, it's 0.333 which is the same thing as this. Okay, same. So that's it. The ratio is one third, and by the way, that ratio will change if you have a different I because this half here ends up showing up here and here, or actually, that half ends up showing up here in here. Right? So if you have a different shape, this will be a different fraction, and then you're finally it will be different. So the ratios change depending on what kind of shape you have, right? That's a finished one. Let me know if you got any questions.

A hollow sphere of mass M and radius R rolls without slipping on a horizontal surface with angular speed W. Calculate the ratio of its linear kinetic energy to its total kinetic energy.