Moment of Inertia & Mass Distribution

Concept: Moment of Inertia & Mass Distribution

Video Transcript

Hey guys. So, in this quick video I'm going to show you how the moment of inertia of a system has to do with how the mass is distributed, how the masses are spread out around the axis of rotation, let's check it out, so it says, here, the moment of inertia I has to do with how mass is distributed, how its spread out around an axis of rotation. So, here we have a solid disk that has small masses, so this is the disk and the masses are the black dots, the four black dots, and they're arranged in three not two, three different ways and I want to know in which of these will the moment of inertia be greater, in which of these will the moment of inertia should be greater. Now this is a composite system with a bunch of different masses, so the total moment of inertia of this system would be the moment of inertia of the solid disk, which is a solid cylinder plus the moment of inertia of the four masses, okay? So, something like i1 plus i2 plus i3 plus i4. Now, these three situations have the same disk with the same mass with the same radius. So, for all of them this is going to be the same, the only thing that will change is this, so the difference will be in how the tiny masses are arranged around the disk. Now, if these are point masses, which they should be treated as point masses because it says your small mass, the equation for them is m, r squared. So, you have a bunch of m, r squares, right? m, r squared, m, r squared for times. Now, if you have the same four masses everywhere, these m's will also be the same. So, it's going to come down to the r's for each mass, in other words, how far from the axis of rotation they are, okay? So, basically the farther the masses are the greater their individual moments of inertia will be and the greater the total of inertia of the system will be. So, this one has to be the one with the greatest I, okay? So, I'm going to call this A, B and C and B is the greater one. Now, and that's because the masses are farther out from the center, C is the smallest, the lowest value of I because the masses are congregated in the middle, here you can see four masses really close to the center. Here you'll see for masses really far from the center and this guy's somewhere in the middle, two are far and two are close. So, I'm going to say that the moment of inertia B is the greater and the moment of inertia of C is the smallest, okay? So, greatest smallest and A is in the middle, this means that you can think of B as being the heaviest of the three, okay? Even, if the masses are the same it's got the most inertia, another way that this question could be asked is, you know, if you apply the same force to it, who's going to rotate faster, right? Well, this guy's the heaviest so, it's also going to be the slowest, okay? Alright, so that's it for this one, let's keep going.

Problem: The objects below all have the same mass and radius. Mass is distributed evenly in all objects. Rank the objects according to the Moment of Inertia they each have about a central axis perpendicular to them, highest to lowest.