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Mass of re-designed flywheel

Patrick Ford
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Hey, guys. So here we have a rotational, kinetic energy problem off the proportional reasoning type and what that means. It's one of those questions where I ask you, how does changing one variable affect another variable? It's one of those. Okay, so let's check it out. I'm gonna show you what I think is the easiest way to solve these. So it says you're tasked with redesigning a solid disc flywheel on. Do you want to decrease the radius by half? So first things first. Solid disk means that the moment of inertia is half M R square. That's the equation for a solid desk or solid cylinder. And you want the new radius I'm gonna call. This are to to be half of our one, and I want to know by how much mass or how much mass must the new fly we'll have eso. It's the new mass relative to the original mass so that you can store the same amount of energy you want. The amount of energy that you store to be the same theme amount of energy stored is given by K R. That's energy stored, right, um which is given by half. I Omega squared. This is energy stored as rotational kinetic energy in a flywheel. You want this number not to change. You want this number to be a Constance Constance. Okay, So how do you do this? Well, if our changes, if our changes right here, then eyes going to change, and if I change is K is going to change. And that's bad news. So how do we change something else so that the K doesn't change well for the k not to change. Um, for the k not to change. You have to make sure that the I doesn't change. And for the I not to change, you have to cancel out changing our with changing em. Okay, so what I'm gonna do here is I'm gonna expand this equation half. Um, I is half m r squared Omega Square. So now I see all the variables that affect my K. And again, the K has to remain constant. So if my radius is becoming half as large, it means that it is decreasing by a factor of two. Okay, so but the the R is squared, which means that when I reduce our by a factor of two I also have to square this and are is becoming half a xlat. But then the whole thing R squared is becoming four times smaller. Okay, four times smaller. What that means is that if you wanna keep everything constant, mass has to grow by a factor of four X. Okay, so my new mass has to be four times my old mass, and that's the answer. Cool. So again, our decreases by a factor of two. But then you have to square because there's a square here you get a four. If one variable decreases by four, the other one has to increase by four notice. There's no squares in the M, so it's just a four. Not to nothing crazy like that. Cool. That's it for this one. Let me know if have any questions.