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Determine the moments of inertia of the shaded area shown with respect to the $x$ 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

Simon Fraser University

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.

05:44

Determine the moments of i…

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Determine the moment of in…

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Determine the moment of i…

08:23

on this problem we're after determine the area Moment, Uh, the area moment of the shaded area shown with his back to the x and Y axes. So I've drawn the the area kind of into pieces. And so what we're gonna dio is the area in the book is kind of had basically this rectangle with the semicircle added on a semi circle at all cut out. So we're gonna find the, um, area moment of this shape and then of this shape about X and Y and subtract this from this and that will give us the area moment of the shape that's in the book where this region here is removed. So that takes basically So I'm grieving all the dimensions we need, Um, in terms of a the radius, the radius of these semi circles. And so, for this section here, you just have 1/3 to a but for about X to a times a cube and about why we have 1/3 a times to a cube and then for, um to All right, we have, um about the central right. Okay. The central right is say of this section here is up about here. And we have that, um that is the were We haven't from the book that we have the radius of the area moment about on access here, but we want an access through the center of mass. So we got the area moment about this axis. Is this the area of this thing? Is pi over to a squared and then the distance from this This point here at the center of this semicircle Quite the center of the circle that from this semicircle to this is for a over three pie. And so we bought the place by that squared, and we subtract it because this is the area moment about this point and we want about the center of the century. And so it multiply all that out and we get pi over eight minus eight, not 8/9 pi a to the fourth. That one's pretty w this one, um, again, we can figure out about it centrally, which is here, and we're given that in the book because this access passes through the central right, and that is just pi over eight A to the fourth. So we want to find the area moment about this point. So we take the area moment of this cross section about its central and add to it the the area pi over to a squared and add to it the distance to from here from this X axis to the central it'll X axis. And that is a times for a over three pie and then square. And so we do all this out when we get 3.30 many to the fourth. And now we want about the why access. So we have about the central it'll why access? And then we come to the area times the distance from this. Why access to the central? Why access is just a So we get five pi over eight times eight in the fourth, and now we need to do a bunch of other calculations. And this one isn't too bad because this is just a rectangle when you do. And then we're going to subtract off that. So that rectangle, um, has an area moment about this x axis again. It's just at the corner here, so that's 1/3 to a times a cube and then about the y axis is just 1/3 a times to a cube so that ones, that one was pretty easy. And now we need to take it. This red area here, Although I've used blue here, but, um, in this red area again. So we need to we know, um, the area moment about this point here, but for the x and Y axis from the book. And we have we can also figure out that, you know, what about the central using this formula here? Okay, so about the central and we get this value, um, and this value again, because it's the same respect, its central rate. It's the same as this one respected its century. So we have this. We have this. And so now we need the distances from here to here to the centrally. And this distance is a in this distance, back up from here is for a over three pie. So we get one minus for, um, over three pi square times a a squared. And so then the area, and then that all comes out to be 0.630 84th and from the central. Why access to this? Why access we get? We have a distance. A So that one's easy. So we get, um, five pi all over 88 to the fourth. And so we can see that, in fact, that one about the y axis, Um, this one and this one are going to cancel out because basically, we've removed, um, stuff with respect to the y axis in the same in the same kind of we've added and we've removed and they're all just shifted along the y axis. Now, the X axis is gonna be different because this one's added farther away and this one is added subtracted closer. So we can just We have all these values now and we can just simply at at these three up and then subtract off that cut out. And what we get is, oddly, a very, um, very even numbers. Um, the area moment about X is 4.0, eight of the four, and about why is 16 3rd aided the fourth? And I would assume that because those air so clean of numbers that there may have been a simple a way of calculating this, but I didn't see it, so we got her answers. You

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