Determine the moments of inertia of the shaded area shown with respect to the x and y axes.

Name: Problem 35, Chapter 9: Vector Mechanics for Engineers: Statics and Dynamics
Uploaded: 2020-05-05T16:43:48-08:00
Duration: 7 min 52 s
Channel: Eric Mockensturm
Description: Determine the moments of inertia of the shaded area shown with respect to the x and y axes.

Determine the moments of inertia of the shaded area shown...

Key Concepts

Reference Axes and Coordinate Systems

The choice of reference axes is crucial when calculating the area moment of inertia. Using a well-defined coordinate system, typically with x and y axes, allows for the precise formulation of the integrals. Understanding the orientation of these axes and their relationship with the geometric configuration of the area is key to correctly setting up and evaluating the integrals.

Parallel Axis Theorem

The parallel axis theorem is a useful tool when the moment of inertia about an axis through the centroid of an area is known, and one needs to find the moment of inertia about a different, parallel axis. It states that the moment of inertia about any axis parallel to the centroidal axis is equal to the centroidal moment of inertia plus the area multiplied by the square of the distance between the two axes.

Area Moment of Inertia

The area moment of inertia, also known as the second moment of area, quantifies how a given shape’s area is distributed with respect to an axis. It is a geometric property that indicates the resistance of a cross-sectional area to bending or deflection around that axis. Calculating it generally involves integrating the square of the distance from the axis over the entire area.

Integration Techniques

Integration methods are used to compute the area moment of inertia when analytic formulas are unavailable or when dealing with irregular shapes. The process involves setting up an integral that sums up infinitesimally small elements of area, each multiplied by the square of its distance from the axis of interest. This technique is essential for deriving moments of inertia for defined regions.

Transcript

00:01 And this problem, we're asked to determine the area moment of the shaded area shown with respect to the x and y axes.

00:10 So i've drawn the area kind of in two pieces.

00:15 And so what we're going to do is the area in the book is kind of basically this rectangle with a semicircle added on and a semicircle added out.

00:28 Cut out.

00:30 And so we're going to find the area moment of this shape and then of this shape about x and y and subtract this from this.

00:45 And that will give us the area moment of the shape that's in the book where this region here is removed.

00:53 So that takes basically so and given all the dimensions we need in terms of a, the radius of these semicircles.

01:06 And so for this section here, you just have one -third, two -a, for about x, 2a times a cubed.

01:19 And about y, we have one -third a times two -a -cubed.

01:27 Cubed and then for two, right, we have about the centroid, okay, the centroid is, say, of this section here, is up around here, and we have that that is the, we were, we haven't from the book that we have the, we have it from the book that we have the, the radius or the area moment about an axis here, but we want an axis through the center of mass.

02:18 So we got the area moment about this axis is this.

02:24 The area of this thing is pi over 2a squared and then the distance from this point here at the center of this semicircle, right the center of the circle that's from the semicircle.

02:38 To this is 4a over 3 pi and so we multiplied by that squared and we subtract it because this is the area moment about this point and we want about the center of the centroid and so we multiply all that out and we get pi over 8 minus 8 not 8 over 9 pi a to the 4th so that one's pretty ugly this one um again, we can figure out about its centroid, which is here, and we're given that in the book, because this axis passes through the centroid, and that is just pi over 8a to the fourth.

03:26 So we want to find the area moment about this point...

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