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Ratio of energies of cylinder on surface

Patrick Ford
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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.