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Two steel plates are welded to a rolled W section as indicated.Knowing that the centroidal moments of inertia $\bar{I}_{x}$ and $\bar{I}_{y}$ of the combined section are equal, determine $(a)$ the distance $a,(b)$ the moments of inertia with respect to the 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

Cornell University

University of Michigan - Ann Arbor

University of Sheffield

University of Winnipeg

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.

03:45

Two $20-\mathrm{mm}$ steel…

So in this problem, we are told that we weld two steel plates here in here to an I beam here, and we want to make sure that the area moments about the central axes of this whole section are equal to each other so that it essentially has the same bending rigidity in both about about this access. So if we try to bend it this way or we try to bend it this way, I'm gonna have the same strength. We're told that the plates are 20 cents in 26 inches long and have a thickness of one inch. Well, given the type of I beam we have here, and so then we know what we need to find is we want to find how far from the center of the plates we should well, the center of the I beam so that the two area moments are equal. Here's the area of this. I beam. We can look it up. Now we want, um, the area moments about the central little X and Y axes to be the same. We know I'm gonna set up a coordinate system here at the center of the ibeam and So because of symmetry, we know that why bar is zero. So this axes and this axe isn't going to be the same because the central right in in the Y direction is going to be great at the middle of these two things. And we know that we can find the centrally in the X direction. So this distance here So that is the total area of this whole section times this distance. And since the central, I think I've set up the X coordinate on the century of the I beam. We don't have a contribution in there because expert 10 And so we have twice the area of the plates times the sent the distance from this coordinate to its central. And that is a So we find out that X bar is to a to all over a times a okay, so we can look up the area moments for the, um, W section here and about the about. It's about these axes, a bit central axes in the X direction. It's 385 inches to the fore, and again, that's going to be the same as about this axis. So we've got that one. No need to use a parallel axis young for that. And then about the y axes about this y axes. It is 26.7 inches to the fourth. Now the total or there for the for the plates for the the why we need to shift. It s oh, here. We need to to get their central the area moment about this. Why access? We need to use the parallel, access the arm, and we get that. Um, it's this value about about the central right here. And then we need this length here, which is X bar squared. So we have We can find that in terms of a at least now, for the plates, we have their central It'll, um, area, uh, central area moment, properties. And again, we need to shift things. Um, so in for each plate, we need to shift by an amount by this amount here because the central right of the plate is here, and that's and we need to shift over to here. And this distance here is a minus X bar. So we have this shift and we can we know x bar in terms of a so we can substitute that in. And so now we have i Y two as a function of a Likewise, we can figure out about the central X axis. And so in that case we have we have to shift by by this value, and that is this distance here. Okay, so this distance over to plus the thickness over to All right, So we get this here. No, the total area moment about the central X axis is just the sum of all the pieces. And we want to make sure that that is the same as the total area about the y axis, which is the some of those pieces. And what we get is that this value here you plug it in is, um, 3353 inches to the fourth and the why one depends on a and so that is 2956 plus 9.2 to a square. Interested the fourth and setting this equal to this, we can solve for a and we get that it is 6.57 inches. So we need to take this. Um, and probably so it looked 6.576 6.5 is actually at the quarter point. So just a little over the quarter point, we need to to well, this ibeam.

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