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Anderson Video - Moment of Inertia

Professor Anderson
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>> All right so we keep talking about this idea of moment of inertia, but we need to explain a little bit about how to, in fact, calculate that. Okay. Moment of inertia for a system of individual particles is defined as this, the summation of m sub I, r sub i squared, where m sub i is the ith mass and r sub i is the distance from m sub i to the axis of rotation. So, if we're thinking about something simple like a particle on a string, let's say here is my mass, m, and it's sitting out there at radius, r, and it's spinning around in a circle. Okay. Your yo yo on a string you're going to spin it around in a circle. What is the moment of inertia here? Well, it's sum over I, m sub i, r sub i squared. We only have 1 particle in they is example and so this just becomes sum of m r squared and those are the variables that we used. We only have 1 term that's it. Okay. But let's say we add a little bit more stuff to it. Let's say instead of a single mass let's do a double mass like in a bar bell. Okay. So a bar bell looks like this. We've got mass, m, at one end, mass, m, at the other end, the whole length of the bar bell is L and this is our axis of rotation. We'll spin it about the center. So you hang a bar bell from a string and you spin it around like that. What is the moment of inertia? All right. It's the summation of m sub i, r sub i squared. We only have 2 particles here so it's m1 r1 squared, plus m2 r2 squared, but I know what each mass is. It's just m. And, in fact, the radius, how far is that mass from the axis of rotation? That's just half the length. So we get m L over2 squared plus m L over 2 squared. I put them together what do I have? I have an m, I have an L squared, I have 1 over 4 plus 1 over 4, which is 2 over 4, which is 1/2 and so I get 1/2 mL squared. This is a moment of inertia of a 2 particle bar bell spinning about its center. All right. Moment of inertia is to rotational motion what mass was to linear motion. When things are rotating, you need to be concerned about the I, the moment of inertia, and that takes into account not only the mass but how far is that mass from the axis of rotation. So, in general, the moment of inertia is due to mass being a distance from the axis but oftentimes we don't have a nice set of particles. We have some complicated blob, right? Here's our blob and we're going to spin it about some axis like that, how would you calculate the moment of inertia of something like that? And the way you do it is you turn your summation into an integral. So, whenever you have a summation, if you just reduce the size of your m to infinitesimal it becomes an integral, okay, and that's all we have to do here. The summation becomes an integral. The r squared hangs out, the m becomes an infinitesimal piece of mass which is what we call dm. And then you integrate this over the whole object. Okay. So in general, this is how you calculate moment of inertia and you have to do an integral. And if you do that integral for different objects, you can figure out what the moment of inertia is and there are some fairly straightforward examples. For instance let's do the wheel. Okay, and we'll give the wheel a radius, r, and we'll say that the total mass is capital M. What is the moment of inertia about the center of it? Well, let's see I have to pick a little dm so here is my little chunk of wheel. We will call that dm. Then we have to integrate all the way around the wheel until we get back to where we started. Apparently, this I is going to be integrate around the wheel of r squared, which in this case is capital R, times dm, but as I integrate around this wheel, R doesn't change it comes out in front. And so now we just have to integrate dm over the rest of the wheel over this full rotation but if I just take a little dm and I add up the next one and the next one and the next one and I do that for the entire wheel, that just becomes the total mass, M, of the wheel. Okay. So what is the axis of rotation? It's the center of mass here. What is the moment of inertia about that center of mass? It's just m r squared. That was a simple example. If you do a more complicated example like a sphere, the integral becomes quite a bit more difficult to do and if you do a solid sphere, you get something like, we're not going to do the integral here, but you get something like 2/5 m r squared. And if you do a hollow sphere, you get something different; and if you do a hollow cylinder, you get something different; if you do a stick, you get something different; if you do a rectangular sheet, you get something different. And there's a table in your chapter that tells you what the moment of inertia is for all these different objects. They did these to calculate what those are. Okay?