Moment of Inertia & Mass Distribution - Video Tutorials & Practice Problems

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Moment of Inertia & Mass Distribution

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Hey, guys. So this quick video, I'm gonna show you how the moment of nourish of a system has to do with how the mass is distributed, how the masses air 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 masses distributed how it's spread out around an axis of rotation. So here we have a solid disk that has small masses of This is the disk and the masses are the black dots. The four black dots on dare arranged in three, not 23 different ways. And I want to know in which of these will the moment of inertia be greater? In which of these were the moment of inertia? Be greater? Um, now this is a composite system with a bunch of different masses. Eso the total moment of inertia. This system would be the moment of inertia of the solid disc, which is a solid cylinder plus the moment of inertia of the four Masses. Okay, so something like I one plus I to plus I three plus I four 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 air 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 them are squares, right? M r squared M R squared four times. Now, if you have the same four masses everywhere, these EMS will also be the same. So it's gonna come down to the ours 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 moment of inertia the system will be. So this one has to be the one with the greatest Hi. Okay, so I'm gonna 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. See, is the smallest the lowest value of I because the masses air congregated in the middle. Here you can see four masses really close to the center. Here, you see, four Mass is really far from the center, and this guy is somewhere in the middle to or far and to our clothes. So I'm going to say that the moment of inertia of B is the greater and the moment of inertia of See 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 gonna rotate faster, right? Well, this guy's the heaviest, so it's also going to be the slowest. Okay, All right, so that's it for this one. Let's keep going

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Problem

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. (From left to right, the objects are A, B, C, and D.)

A

I_{B} > I_{A} > I_{C} > I_{D}

B

I_{B} > I_{C} > I_{A} > I_{D}

C

I_{D} > I_{A} > I_{C} > I_{B}

D

I_{D} > I_{C} > I_{A} > I_{B}

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