Find Ix for the arch in Example 3.

Name: Problem 41, Chapter 16: Thomas' Calculus, Multivariable (12th Edition)
Uploaded: 2020-05-30T00:30:04-08:00
Duration: 6 min 27 s
Channel: Jack Hou
Description: Find Ix for the arch in Example 3.

Key Concepts

Area Integration for Second Moment of Area

When determining the moment of inertia for non-standard geometries, the calculation is carried out through integration. This involves integrating the square of the distance from the axis (typically in the form ?y² dA) across the area of the shape, which requires a clear understanding of the geometry and limits of the shape in question.

Analysis of Curved Structures

Curved structures, such as arches, require specialized techniques for analysis because their geometry does not lend itself to simple formulas. Understanding the curvature and employing proper coordinate transformations or parameterizations is essential for accurately setting up and evaluating the integrals needed to compute properties like the moment of inertia.

Moment of Inertia

The moment of inertia, also known as the second moment of area, is a measure of a cross-section’s resistance to bending. It quantifies how the area is distributed about an axis and plays a key role in determining the deflection and stress in beams under load.

Transcript

00:01 Okay, folks, so in this video, we're going to talk about problem number 41.

00:04 So we're given an arch, and we're going to be looking for the x component of the moment of inertia for this particular shape.

00:13 So, as you recall, we have an expression for the x component of the moment of inertia, and that expression can be written.

00:22 In this way, we have i -x, we have i -x equals integral of delta, or a ducous.

00:30 Delta is the density function, which may or may not be constant, depending on the problem, multiplied by r -square minus x squared, multiplied by ds.

00:41 So this is the expression for the x component of the moment of inertia matrix for any shape.

00:52 And in this problem, it's an arch.

00:56 So, and what we're given in this problem is we're given a density function which varies with position.

01:07 So let's write that down.

01:09 We have density being equal to to minus z multiplied by r squared minus x squared.

01:16 But r squared minus x squared gives you y squared plus z squared.

01:21 So let me write this down.

01:22 We have y squared plus z squared.

01:26 Because as you remember, r squared is just x squared plus y squared plus z squared.

01:32 Right, that's the square, that's the square of the distance from the origin to any point on the shape that we're looking at.

01:41 But anyway, y squared plus z squared happens to be a very nice number and it's given in the problem, that number is one.

01:49 So we're just going to ignore that number.

01:52 We're going to ignore one.

01:53 We're not going to write it out every time.

01:54 Okay, so that leaves us with ds.

01:58 However, there's another way to write ds.

02:02 As you remember, ds is just an infinitesimal line segment, so we can write that as square root of d, y, squared plus dz squared.

02:14 Okay, and the picture, the picture that i want you to have in your mind is this...

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