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Determine the moment of inertia and the radius of gyration of the shaded area with respect to the $x$ axis.

Physics 101 Mechanics

Chapter 9

Distributed Forces: Moments of Inertia

Section 2

Parallel-Axis Theorem and Composite Areas

Moment, Impulse, and Collisions

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you can ask to find the area moment and the radius A generation of the shaded region with respect to the X axis. Um, and our our shape looks something like this where X axis goes through the center of this member here and were given all the dimensions that we need in the in the problem. And so I divided it into four regions, 12 in blue and 12 and red. Now we need to find the area moment of each of those regions about this X axis, and we can just start doing that, you know, we can see that the, um in red here, the X axis goes through the central right, So we just get the area. Moment, for that part is simply this expression here that will be used many times. And then for the second Red Member, we have it's area moment about its center of mass. Plus, it's area times the distance from this, uh, century the center it of this to the central to this and that I called to and again we can get all of these dimensions from the schematic in the in the book and then for um, the radius, Uh, the area moment. About of X one, that Red First Member or the Blue First Member. It turns out that the X axis passes through its central aid and same with this one and that since these two are the same they have they're gonna have the same area moment about that excess. And so that's just, um 1 12 b one h one where we used blue for a different be one on H one that appear. And we have those values. And we can, um, plug everything in and we get this thing here is to point, um, 67 inches to the fourth. This here is 25 point three inches to the fourth and weaken. This is much bigger than this, because this is farther away from the X axis. Um, and then these two guys both have area moment of nine inches to the four. So we gotta multiplying by two. So we have two of them, and then if we add everything up, we get 46 inches to the fourth, then the radius of gyration is easy, because we can calculate the area of all these rectangles and had them up when we get 18 inches squared, and then we take the square root of this divided by this and we get 1.6 inches.

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