Hey, guys. So some of these rotational questions will ask you to find the ratio of 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. A solid cylinder tells us that we're supposed to use I=12mr2 with mass m and radius r. It rolls without slipping on a horizontal surface. This is called rolling motion. And it means I can use this equation, vcm=rΩ. Because it's rolling like this. So vcm is tied to Ω. Alright. I want to know the ratio of its rotational kinetic to its total kinetic energy. So what you do is you follow what it's saying here and you set up a ratio like this. So it's saying rotational kinetic energy. So we're going to write Kr (rotational) at the top to total kinetic energy at the bottom. So the ratio of top to bottom, Ktotal, which is Kl (linear) plus Kr (rotational). And now what we're going to do is we're going to expand these equations as much as possible.
What I mean by expanding is, well, what does Kr stand for? Kr=12IΩ2. Kl is 12mv2 and Kr is 12IΩ2. And we're going to expand this as much as possible, meaning we're not going to stop there. We can replace I with this right here. I can also replace Ω with something else. The problem here is I have vs and Ωs. There are too many variables. Whenever you have a v and an Ω, you usually want to replace one into the other. So v=rΩ. And what we're going to do is whenever we have v or Ω, we want to get the Ω to become a v. Okay? So I'm going to write Ω=vr, and we're going to replace this here. The reason we do this so that we have fewer variables, so it's easier to solve this question.
Now before I start plugging stuff in, I want to warn you, you cannot cancel this with this. Right? That's not a thing. So don't get tempted to do that. What you can do is you can cancel the halves over here because they exist in all 3 of these guys here. Okay. So you can cancel the halves, and this simplifies a little bit. So we're going to do now is expand I. I=12mr2 and remember, we're going to rewrite Ω as vr, and then this whole thing is squared. Now we're going to do the same thing at the bottom. But before I do that, you might notice right away that this r cancels. Right? So that's another benefit of doing this thing here. Another benefit of doing this thing here is that it's going to cause the rs to cancel. Okay. So at the bottom, I have simply mv2+I, which is 12mr2, and then Ω, which is vr squared. And again, just like it did at the top, the rs cancel. Okay? The rs cancel. Let's clean this up a little bit and see what we end up with. I end up with 12mv2 divided by mv2+12mv2. And you may already see where this is going. There's an m in all three of these, and there's a v in all three of these. So everything goes away, and you end up with just some numbers left here. So you have 1 up here. And then this, there's a one here, right, that stays there, 1 + 2. So the mass The velocity doesn't matter. So all you have to do is do this thing here. Okay? There are two ways you can do this. If you like fractions, you can do with fractions. I'm going to do that first. So I'm going to rewrite this as a 2 over 2. And then I have 1 over 2 divided by I got a 2 at the bottom here. And then I can add up the tops the top here. So it's 2 + 1. So I have 1 over 2 divided by 3 over 2. I can cancel this 2, and then I end up with 1 over 3. If you don't like fractions, one thing you can do with this particular case is you can rewrite this like this, or half is point 5. This is a 1. This is a point 5. Right? This is better if you have a calculator. Point 5 divided by 1.5. And if you do this in the calculator, it's point 333, which is the same thing as this. Okay? Same. So that's it. The ratio is 1 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 and here. Right? So if you have a different shape, this will be a different fraction and then your final list will be different. So the ratios change depending on what kind of shape you have. Alright? That's it for this one. Let me know if you have any questions.