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A channel and a plate are welded together as shown to form a section that is symmetrical with respect to the $y$ axis. Determine the moments of inertia of the combined section with respect to its centroidal $x$ and $y$ axes.

Physics 101 Mechanics

Chapter 9

Distributed Forces: Moments of Inertia

Section 2

Parallel-Axis Theorem and Composite Areas

Moment, Impulse, and Collisions

University of Washington

Simon Fraser University

Hope College

Lectures

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

03:30

In physics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Given a force, F, applied for a time, t, the resulting change in momentum, p, is equal to the impulse, I. Impulse applied to a mass, m, is also equal to the change in the object's kinetic energy, T, as a result of the force acting on it.

05:51

A channel and a plate are …

06:14

Two steel plates are welde…

03:45

Two $20-\mathrm{mm}$ steel…

05:06

Two channels and two plate…

In this case, we have a channel and a plate of welded together. So we have a channel here and it's well, look on top of a plate here, Um, so this is just a rectangular has a rectangle, a cross section, and then we can look up this C eight by 11.5 channel properties, and we want to find the tournaments, that moment of inertia, the area moments of the combined section with respect to the central X and Y axes. Well, now we are central. Why? X and y axes are somewhere kind of unknown twist. At this point, figure out where that origin is. We know it will be on the Y axis and expire will be zero because of symmetry. So the I think that there with and thickness of the plate there it's one foot and 1/2 inch. The area cross sectional area of this channel is 3.37 inches. The area moment about its central X axis, which is somewhere up here, is, uh, 32.5 inches to the fourth and about its central y axis, which is actually along this. Why? Access is, um, one point 31 inches to the fourth. Now we have We can figure out where the central lead the central ideas. Hey, um, by looking over the central right of, um, this this shape here is 0.572 inches from its I think, from its outer side here. Yeah, Why? Two bars right here, from this outer edge to its central legal to its local centrally. Um, and then we know he, too, is gonna be something we get from. We need in our calculations that we confined from this geometry. So, um, I'm gonna take the origin down here initially, and so you can find the area moment for the plate. About this origin about the X axis. This X axis is 1/3 BH Cube. And about this, why? Access is 1/12 each, be cube. So then we can look for this one, and so we get that the we need to get there. The area moments about this point. So we have it about the central. So we need to use the parallel access there and for the x axis. So we have this Who? We know the area. And so we gotta figure out the distance from this point. Get centrally to this point here and that Is this value t two, which I believe. Yeah, the two and then plus h hey, so d to is this length of this side flanges, I believe. And h is the thickness of this plate. And so then we need to subtract off the distance from this. So we have this value here is the distance from the very bottom to the very cop. And then this value is the distance from the top to the centrally. So you need to subtract that off. Um, and then about the y axis, we don't need the parallel access here because the Y axis that we're looking at is the central y axis for that at that part of the section so we can start plugging in numbers. We know this from here, and we know this from here. We have all this information and we plug in their values and we get 49.3 point 13 inches to the fore. We know he this from here. We know this from here, plugging our values, and we get 73.1 73.31 interest of the fourth. And it makes sense that this is bigger than this, because again, we're looking at this x axis and this y axis so that, you know, we have a lot of area away from this y axis and again. And once we shift this x axis up here, obviously this is gonna drop. This is gonna drop. So we need to find Figure out where that central it is for the entire cross section. And so we know X bar is zero. So we need y bar. So we know the areas. What? We all know what the areas of each cross section and the total area. Um, we know where the central rate is with respect to this, um, origin for each of the sections individually. So we have 8/2, and then we have this d two plus h minus. Why, too, for the century of the channel. And we have all that information and we can figure out that the central it is that it's about one inch up from where we had our origin before. And so if this was two point, if this whole thing here was Yes. Here, 2.76 inches. Um, where we go up about one inch. So, yeah, it went down here, which seems about right, because obviously is gonna be above this plate. But that plane has a lot of area, and so it's kind of down shifted down from the century of this channel. And so somewhere in here, it seems reasonable, so that that number seems like it's probably correct. And so, given that we know that about the why access our own why access is actually the central why access so that we just get has the same value here. And we have this and we know that total area, and now we know this. And so we get that the, um, area moment about the central X axis is 40.73 Interest to the fourth, which is less than this, but not a whole lot less, because we haven't really shifted things that much

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