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Determine the polar moment of inertia of the area shown with respect to $(a)$ point $O,(b)$ the centroid of the area.

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

Simon Fraser University

McMaster University

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.

03:05

Determine the polar moment…

03:43

04:03

08:04

02:32

in this problem were asked to determine the polar area moment of the area shown with respect to 0.0 or zero and the central eight of the area. And so we have a in area that's like an egg shaped, um, one side. This this one side is a semi circle. The other side semi ellipses. Um, the raid ei for the semi circles, Um, the outer one is 42 um millimeters and the inner one which just cut out is 27 millimeters. These are also the minor axes of the semi ellipse semi ellipses and then the major axes of the semi ellipses are 84 millimeters and 54 millimeters. We can figure out what the area moment is about for the outer, um, the outer semi ellipse, Um, and 0.0 is actually the centrally of the semi ellipse. So we can just take a look up that value when it is pi over eight, a one b one plus times a one squared plus B one squared and we can look up the, um for the cut out part. That's, um, basically the same thing, but with a to B to replaced so this is That would be a negative area moment because it's a cutout region. And now for the semicircle region again, point. Oh, is at the origin Is that the central rate of that region? So we can just look it up and we get pi over four are one the fourth. And for the cutout region, we get pile before our to to the fourth. So the total, um, pull area moment about the point. Oh, is, uh, this minus this? Plus this minus this. If we plug in our numbers, we get that. That is 12.16 times 10 to the sixth millimeters to the fourth. Now, we need to figure out where he century, ISS and the central it is going to be on the X axis because of symmetry. But we need to figure out how far it shifted, um, in the, uh, in the X direction. So why bar is gonna be zero, but expert is not so we need to figure that out, and we have all of these things. Okay, we can look up, um, where the central leads of each of these shapes are, um, individually. And then we can look up the areas and we have in the end that that gives us minus 22.31 millimeters. And finally, if we want the area moment about the center of, um, the central of the total region, we can just take it about the central. About 0.0 minus a, the area of the coastal region, Times X bar squared. And then we get you 9.73 times 10 to the sixth millimeters to the fourth.

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