13. Rotational Inertia & Energy
Moment of Inertia of Systems
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Hey, guys. So let's check out another example of a moment of inertia problem. So here we have a solid disk. Remember, a solid disk has the same moment of inertia as a solid cylinder. And when I tell you the shape, I'm telling you which equation to use for I. So I of the disc is half mr squared. I'm also given that it has a radius of 4, so r equals 4 and a mass of 10. Okay. And then we're going to add 2 small objects on top of it. The fact that it's saying 2 small objects and it's not giving me a shape for the object, it's an indication that these are going to be treated as point masses. Okay? And these are the 2 objects here. So I'm going to call this, 1. So this is m1 and this is m2. Okay. So the object on the left has a mass of 2 kilograms and it's placed halfway between the disc's center, so the center disc is here, and the edge. Now the distance between the center and the edge is the radius, right, which is 4. If you are halfway between the center and the edge, you your distance is half the radius. Okay. So this distance here is half the radius. So I'm going to call this r one because it's the distance for mass 1 and it is half of the radius, which is 2. And the other guy, the other object is 3 kilograms in mass, 3 kilograms, and it's placed at the edge of the disk. So if you're all the way at the edge, your distance, let's call this r 2, is the same as the radius, which is 4. Okay? So I'm giving you all the information you need to calculate the system's moments of inertia. Now system is a combination of the disc with the masses. So I system is I 1 plus I 2. Right? Object 1, object 2, plus I disk. And remember, for every one of these, you have to determine is this a point mass or is this a shape? Well, 1 and 2 are point masses. We talked about this here. So I'm going to write m 1rone squared plus m 2r2 squared. And the disc is a shape, and it has moment of inertia given by half m r squared. Half big m big r squared. Now all we gotta do is plug in the numbers, very straightforward. So what I'm going to do is plug the masses. The masses are 1, 2, and 10. So, I'm sorry, 2, 3, and 10. 2, 3, and 10. So I'm going to do 2, 3, plus half of 10 r squared. Okay? And then all we have to do is plug in the r's. This r here is the radius. It's very straightforward. We know that. The radius is a 4. So I'm going to put a 4 here. Now these r's we have slow down for a little bit. These are the distances between the center, the axis of rotation, which is in the center, and where the object is. Little r's are the distance between the object and the center. And we already had these figured out here. It's 24. So the 2 kilogram has a 2 meter distance and the 3 kilogram has a 4 meter distance. Okay. So let's just do this real quick. This is going to be 8. This is going to be 3 times 16. So that's 48. And this is going to be 80. Okay. So we have 8 +48, plus 80, And this is going to be 136136 kilograms meter squared. Cool. That's it for this one. Hope you got it. Let me know if you have any questions.
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