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Determine the polar moment of inertia and the polar radius of gyration of the shaded area shown with respect to point $P$.

Physics 101 Mechanics

Chapter 9

Distributed Forces: Moments of Inertia

Section 1

Moments of Inertia of Areas

Moment, Impulse, and Collisions

Rutgers, The State University of New Jersey

Hope College

University of Sheffield

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.

02:22

Determine the polar moment…

01:26

05:06

02:32

01:20

where can ask to calculate the polar second moment of area and the polar raining situation of the sheet. It figures shown on that figure is bounded below by this function. Um, que one x squared and it's bounded above by C plus k two x weird. And so we can figure out what k one k two and CR by noting that why one at two a. Has to be to a why two at two. A is also too way. And why to at zero is a so that gives us that c must be a k one is one over to a and K two is one over for a So we need to set up the in ago and we got integrate x squared plus y squared over the area And the why goes from why one toe y two for these functions Next goes from minus to a to A and cranking through the inner grow. We find that Isa Bar is 64/15 times eight of the fourth we can substitute Put one in here and do the air calculate the Negro and we get the area. It is 8 30 a squared, and that gives us a polar radius of gyration of to times square to 50 times a or 1.265 a.

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